Mutualistic Networks [Core Textbook ed.] 9781400848720

Table of contents :
Contents
Preface
CHAPTER ONE. Biodiversity and Plant-Animal Coevolution
CHAPTER TWO. An Introduction to Complex Networks
CHAPTER THREE. The Structure of Plant-Animal Mutualistic Networks
CHAPTER FOUR. Ecological and Evolutionary Mechanisms
CHAPTER FIVE. Mutualistic Networks in Time and Space
CHAPTER SIX. Consequences of Network Structure
CHAPTER SEVEN. Epilogue
APPENDIX A. Indices Used in Mutualistic Network Analyses
APPENDIX B. Fitting Degree Distributions
APPENDIX C. Measures of Nestedness
APPENDIX D. Measures of Modularity
APPENDIX E. Phylogenetic Methods and Network Analysis
APPENDIX F. Null Models for Assessing Network Structure
APPENDIX G. An Analytical Theory of Mutualistic Networks
APPENDIX H. Software for the Analysis of Complex Networks
Bibliography
Index

Citation preview

Mutualistic Networks

MONOGRAPHS IN POPULATION BIOLOGY EDITED BY SIMON A. LEVIN AND HENRY S. HORN

A complete series list follows the index.

Mutualistic Networks jordi bascompte and pedro jordano

PRINCETON UNIVERSITY PRESS Princeton and Oxford

c 2014 by Princeton University Press Copyright
Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data Bascompte, Jordi, 1967– Mutualistic networks / Jordi Bascompte and Pedro Jordano. pages cm. — (Monographs in population biology) Includes bibliographical references and index. ISBN 978-0-691-13126-9 (hardcover : alk. paper) 1. Mutualism (Biology) I. Jordano, Pedro, 1957– II. Title. QH548.3.B37 2014 577.8
52—dc23 2013015338 British Library Cataloging-in-Publication Data is available This book has been composed in Times Printed on acid-free paper. ∞ Typeset by S R Nova Pvt Ltd, Bangalore, India Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

To Eva and Myriam

To me the most important thing in composition is disparity. Anything suggestive of symmetry is decidedly undesirable, except possibly where an approximate symmetry is used in a detail to enhance the inequality with the general scheme. —Alexander Calder, A Propos of Measuring a Mobile (unpublished manuscript, 1943, The Calder Foundation, New York)

Contents

Preface

xi

Chapter 1. Biodiversity and Plant-Animal Coevolution Historical Overview A Bit of Natural History Coevolution in Multispecific Mutualisms Summary

Chapter 2. An Introduction to Complex Networks A Network Approach to Complex Systems Measures of Network Structure Models of Network Buildup Ecological Networks Summary

Chapter 3. The Structure of Plant-Animal Mutualistic Networks Degree Distribution Nestedness Small World Modularity Weighted Networks Comparisons with other Ecological Networks Summary

Chapter 4. Ecological and Evolutionary Mechanisms Single Ecological Traits Phylogenetic Effects Summary

Chapter 5. Mutualistic Networks in Time and Space Network Dynamics Spatial Mosaics

1 1 6 12 14 15 17 21 31 32 41

42 42 44 48 50 54 60 62 64 67 77 85 87 88 98

x

CONTENTS

Sampling and Robustness Summary

Chapter 6. Consequences of Network Structure Coextinction Cascades Dynamic Stability Global Change and Mutualistic Networks Coevolutionary Implications Implications for Nonbiological Systems Summary

102 106 107 108 113 120 126 132 134

Chapter 7. Epilogue

136

Appendix A. Indices Used in Mutualistic Network Analyses

139

Appendix B. Fitting Degree Distributions

143

Appendix C. Measures of Nestedness

147

Appendix D. Measures of Modularity

150

Appendix E. Phylogenetic Methods and Network Analysis

154

Appendix F. Null Models for Assessing Network Structure

160

Appendix G. An Analytical Theory of Mutualistic Networks Fixed Points and Local Stability Effects of Network Structure on Competition and Biodiversity

167 167 169

Appendix H. Software for the Analysis of Complex Networks

175

Bibliography Index

179 201

Preface

Observing plant-animal interactions in nature has remained one of the most fascinating aspects of our scientific activity. We recall the unforgettable experience of witnessing frugivorous birds feeding on fruits or hummingbirds pollinating flowers. These are mutually beneficial interactions: animals move the genes of the plants across the landscape, and obtain a food reward for this service. Mutualisms in nature are widespread and have played a major role in the diversification of life on Earth. A persistent challenge is to understand how these mutualistic interactions evolve and coevolve in speciesrich communities. Mutualisms form complex networks of interdependence between dozens or even hundreds of species. Understanding the architecture of these networks is very important for understanding coevolution and network robustness to global change. This monograph reviews research on plant-animal mutualistic networks that brings a community-wide approach to coevolution. It puts the emphasis on the component of biodiversity explained by species interactions, and how these interactions are dismantled through human-induced perturbations. The extinction of interactions leads to the empty forest: species are still there but the whole system loses its functionality. In Daniel Janzen’s words, “... the most insidious type of extinction is the extinction of interactions” (Janzen, 1974). Deforestation and defaunation are the main drivers of interaction loss and their effects pervade multiple ecosystem services. In our tour across the structure and dynamics of mutualistic networks we will be combining a deep enthusiasm for natural history with modern analytic techniques such as network theory and phylogenetically-informed analyses. We begin this book with two introductions, one focussing on the natural history background, the other on the network tools we will be using. Paralleling the development of this young research field, we will follow by describing the architecture of these networks, a task that depends on the analysis of large data sets with appropriate statistical tools. One such a tool is a null model, and we describe the technical aspects in appropriate appendices. This organization of the book aims at allowing interested readers

xii

P R E FAC E

to explore these details without precluding the reading of the book by a wider audience. Next, we will review the suite of ecological and evolutionary mechanisms generating mutualistic networks. To achieve this, we will be combining simple build-up models of network formation with phylogenetic analysis and ecological correlates of network structure. As in food webs and other interaction networks, mutualistic networks represent aggregates through time and space. Space and time are the next frontiers in network research, but fortunately there are a few papers that we will review, showing us the path to follow. We will proceed by exploring the consequences that the above network patterns have for coevolution and community stability by using mathematical models and computer simulations. Up to a few years ago, there was no theory for mutualistic networks as there was theory for competition or predation. We will summarize the first steps towards such a general theory for mutualistic networks. Finally, we will end up by looking at work that has taken networks a step further, moving this basic research field into a more applied science in the context of some of the most pressing questions such as biological invasions or habitat transformation. This book is the consequence of more than ten years of close collaboration between the two authors. The move of one of us (JB) to the Estación Biológica de Doñana (Sevilla) in 2000 created a wonderful opportunity to tackle the coevolution of mutually beneficial interactions in species-rich communities. The timing was very good. On one hand, there were large data sets available. On the other hand, network theory was making a splash in the late nineties, providing new tools to shed light to these complex networks of mutual dependence among species. Our early work immediately benefited from the continuous source of support, inspiration, and keen naturalistic insight of our good friend and coauthor Jens M. Olesen. This initial triad is only the core of a larger scientific network where many have played influential roles. Among them, we have been very fortunate to join forces with Thomas Lewinsohn, who independently had reached similar views to the ones we were reaching. Thomas is a renascent man and we have enjoyed his wise comments and ideas. Similarly, a most influential contribution has been provided by the generous insight from John N. Thompson. Few like John have seen the potential of the concept of networks as a way of “dispelling the naive view that mutualisms between freeliving species are diffuse assemblages that are intractable for coevolutionary analysis” (quoted from an enthusiastic e-mail written after reading our 2003 PNAS paper).

P R E FAC E

xiii

In turn, our current and former graduate students and postdocs Carlos Melián, Miguel A. Fortuna, Paulo R. Guimarães Jr., Enrico Rezende, Jofre Carnicer, Cristina García, Alfredo Valido, Daniel Stouffer, Luisjo Gilarranz, Franck Jabot, Jelle Lever, Rudolf Rohr, Serguei Saavedra, and Vasilis Dakos have been pivotal in the generation of some of the work on mutualistic networks here discussed. Other members of the Integrative Ecology Group have also contributed very useful suggestions, ideas, and discussions: Pete Buston, Arndt Hampe, Jessica Lavabre, Kimberly Holbrook, Rocío Rodríguez, Cande Rodríguez, and Abhay Krishna. Over the years, the technical support of Manolo Carrión, JuanMi Arroyo, Cristina Rigueiro, María Cabot, and, especially, Javier Escudero and Raúl Ortega have made things much easier for us in the lab. Outside our lab, we should highlight Bartolo Luque, Ugo Bastolla, Antonio Ferrera, and Alberto Pascual-García for their development of analytical techniques that have advanced the theory for mutualistic networks. Jason Tylianakis, in turn, has championed the use of networks in the conservation arena. Other co-authors in our work on mutualistic networks deserve a special mention since this is their work as well. The list of co-authors expands to include truly outstanding scientists such as Stuart Pimm, Diego Vázquez, Marcus A. de Aguiar, Sergio F. dos Reis, Mauro Galetti, José M. Gómez, Miguel Verdú, Paulo I. Prado, Anders Nielsen, and Victor Rico-Gray. We are also in debt to Roger Guimerà, Marten Scheffer, George Sugihara, Robert May, Robert Paine, Joel Cohen, Eugene Schupp, Louis-Félix Bersier, Scott Armbruster, and Wesley Silva. It has been very rewarding learning from them all. Over the years, we have enjoyed discussions with students in our courses at the Universities Pablo de Olavide (Sevilla), Alcalá (Madrid), Internacional Menéndez Pelayo, and UNESP, UNICAMP, and USP (all in Brazil). The suggestions, insight, and support obtained in these courses have been instrumental during the writing of this book. We extend our appreciation to the excellent administrative support provided by the staff at the Estación Biológica de Doñana. Facilities for field work have been generously provided over the years by the Consejería de Medio Ambiente, Junta de Andalucía. The Centro de Supercomputación de Galicia (CESGA) generously allowed us to use their computing resources for some critical simulation work. We have benefited from the generous help and advice of the technical staff at PUP, especially Sam Elworthy, Alison Kalett, and Dimitri Karetnikov. We are also very thankful to Simon Levin and Henry Horn, for inviting us to write this monograph and their constant encouragement.

xiv

P R E FAC E

Paulo R. Guimarães Jr, Jens Olesen, Rudolf Rohr, Jason Tylianakis, John Thompson, Alfredo Valido, and an anonymous reviewer have read a previous version of this book and provided wonderful suggestions to improve clarity and content. Financial support has been generously provided by the European Science Foundation through a EURYI (European Young Investigator) Award and a European Research Council’s Advanced Grant to JB, and several grants from the Spanish Ministry of Science, the Regional Government of Andalucía, and the Programa Iberoamericano de Ciencia y Tecnología para el Desarrollo (CYTED) to both JB and PJ.

Mutualistic Networks

CHAPTER ONE

Biodiversity and Plant-Animal Coevolution

HISTORICAL OVERVIEW The almost-perfect matching between the morphology of some orchids and that of their insect pollinators fascinated Charles Darwin, who foresaw that the reproduction of these plants was intimately linked to their interaction with the insects (Darwin, 1862). Darwin even predicted that the extinction of one of the species would lead to the extinction of its partner: If such great moths were to become extinct in Madagascar, assuredly the Angraecum would become extinct (Darwin, 1862, p. 202). Later on, Alfred Russell Wallace would take the examples of plant-animal interactions to illustrate the force and potential of natural selection to shape phenotypic traits. He already noted that the selective pressures derive directly from the interaction itself (Wallace, 1889). The fascinating experimental work by Darwin on plant sexuality was very influenced by the earlier work of Sprengel (1793) demonstrating the role of insects in plant fertilization (Fig. 1.1a). Similarly, his work on hybridization shows the strong influence by Köllreuter (1761; see Waser 2006, for a historical overview). Köllreuter already documented the diversified pollination service that multiple insect species provide to plants. However, the major advances at that time in documenting the specificity of pollination patterns are due to the monumental work of Müller, Thompson, et al. (1883), providing the list of pollinator species for 400 plant species, and Knuth (1898), reporting records for more than 6000 species. Early researchers on plant-seed disperser interactions (Hill, 1883; Beal, 1898; Sernander, 1906) also emphasized the diversity and subtleties of mutual dependencies among the partners and provided well-grounded evidence for mutual coadaptations between them (Fig. 1.1b). Beal provides an analogy with pollination systems, quoting

2

CHAPTER 1

a

b

Figure 1.1. The work by early botanists and zoologists represented the foundations for later studies on mutualistic interactions. Prominent among them was a series of monographs on different types of interactions (pollination, seed dispersal, ant-plants, etc.) appearing between the late 1700s and early 1900s. (a) The beautifully detailed front page of Sprengel, the author of an important monograph on flowers and pollination (Sprengel, 1793); (b) detailed view of one of the plates illustrating Sernander monograph on seed dispersal by ants (Sernander, 1906), showing the anatomical details of elaiosomes (reward tissue) attached to the seeds.

Darwin’s orchid book (Darwin, 1862): The more we study in detail the methods of plant dispersion, the more we shall come to agree with a statement made by Darwin concerning the devices for securing cross-fertilization of flowers, that they “transcend,” in an incomparable degree, the contrivances and adaptations which the most fertile imagination of the most imaginative man could suggest with unlimited time at his disposal (Beal, 1898, p. 88). The complexity that such interactions could take was already recognized by Darwin in the final paragraphs for the first edition of on the Origin: It is interesting to contemplate an entangled bank, clothed with many plants of many kinds, with birds singing on the bushes, with various insects flitting about, and with worms crawling through the damp earth, and to reflect

B I O D I V E R S I T Y A N D P L A N T- A N I M A L C O E VO L U T I O N

3

that these elaborately constructed forms, so different from each other, and dependent on each other in so complex a manner, have all been produced by laws acting around us (Darwin, 1859, p. 498). Similarly, in Chapter III, Struggle for Existence, we can read: I am tempted to give one more instance showing how plants and animals, most remote in the scale of nature, are bound together by a web of complex relations (Darwin, 1859, p. 73). Darwin also envisioned the mutually reciprocal effects involved in the pollination of red clover by “humble-bees” and the potential effects of declines in pollinator abundance. He foresaw the complexity of mutualistic networks, a complexity that precluded a community-wide approach. Mutualism and symbiosis became quickly incorporated into the research agenda after de Bary (1879) coined the term symbiosis to account for interactions among two or more dissimilar entities living in or on one another in intimate contact. These developments of the study of mutualisms were well grounded on the empirical evidence obtained by botanists documenting every detail of the morphological structures of flowers, fruits and seeds (Fig. 1.1) as well as the intricacies of the interactions with animals. Since then, a myriad of scientific papers have described the mutually beneficial (mutualistic) interactions between plants and their animal pollinators or seed dispersers. But the interest of ecologists and evolutionary biologists in mutualistic interactions has been quite variable in emphasis and prevalence during this period of time. Work on mutualism, like the analysis by Pound (1893), remained marginal to dominant views in ecology. Antagonistic interactions were at the core of Clements and Tansley’s views of plant ecology, which dominated the field in the United States and United Kingdom during the early 20th century. This was paradoxical given the rapid discovery of new major symbiotic interactions like mycorrhizae in the 1880s and 1890s (Schneider, 1897). In fact, a few years after the Lotka-Volterra models were developed for antagonistic interactions, Gause and Witt (1935) proposed dynamic models of mutualism based on very similar formulations. However, mutualistic interactions were ignored in the extensive treatment that Volterra and D’Ancona (1935) dedicated to the dynamics of “biological associations” among multiple species. Up to the early 1970s, mutualism was not at the center of ecological thinking (L. E. Gilbert and Raven, 1975), which was more focused on the dynamics of antagonistic interactions such as predation and competition as the major forces driving community dynamics.

4

CHAPTER 1

Most recent textbooks on ecology and evolution just treat mutualisms as iconic representations of amazing interactions among species, lacking a formal conceptual treatment at a similar depth to predation or competition (Sapp, 1994). Boucher (1985a) provides a lucid analysis for the reasons why mutualism had a marginal importance in ecological studies up to the late 1970s and early 1980s, when dynamic and genetic models of mutualistic interactions started to be revisited (May, 1982). Among these reasons, there are the technical difficulties to find stable solutions for dynamic models of mutualism (May, 1973) and the lack of appropriate empirical and theoretical tools to develop a synthesis of the enormous diversity of mutualistic interactions (May, 1976). Also, the association of the idea of mutualism with anarchist thinking related to the 1902 book Mutual Aid by Peter Kropotkin most likely had an influential effect on the demise of mutualism in the early 1900s and its marginal consideration (Boucher, 1985a). Ehrlich and Raven, in their classic paper, emphasized the pivotal role of plant-animal interactions in the generation of biodiversity on Earth (Ehrlich and Raven, 1964). Interestingly enough, insects and flowering plants are among the most diverse groups of living beings, and it is assumed that the appearance of flowering plants opened new niches for insect diversification, which in turn further spurred plant speciation (Farrell, 1998; McKenna, Sequeira, et al., 2009). This scheme has some alternative explanations, such as that one group may have been tracking the previous diversification of the other one without affecting it (Ehrlich and Raven, 1964; Pellmyr, 1992; Ramírez, Eltz, et al., 2011). However, the relevant point is that animal-pollinated angiosperm families are more diverse than their abiotically pollinated sisterclades (Dodd, Silvertown, et al., 1999). Since the seminal paper by Ehrlich and Raven (1964), there has been a flourishing of studies on plant-animal interactions in general and on mutualisms among free-living species in particular. A significant amount of this work stems from recent advances in the study of coevolutionary processes (Thompson, 1994, 1999a) and the recognition of their importance in generating biodiversity on Earth. Fortunately, there is ample fossil evidence of the origin of mutualistic interactions. Thus, the first preliminary adaptations to pollination can already be tracked around the mid-Mesozoic, almost 200 million years ago, and became widely observed from the mid-Cretaceous, more than 100 million years ago (Labandeira, 2002). In relation to seed dispersal, the early evolution of animaldispersed fruits in the upper Carboniferous, together with the diversification of small mammals and birds in the Tertiary, allowed the diversification of plant fruit structures and dispersal devices (Tiffney, 2004). Therefore, multi-specific

B I O D I V E R S I T Y A N D P L A N T- A N I M A L C O E VO L U T I O N

5

interactions among free-living animals and plants have been an important factor in the generation of biodiversity patterns for a very long time. But mutualisms have been important not only in the past. They remain important in the present. Mutualisms among free-living species are one of the main wireframes of ecosystems, simply because extant ecosystems would collapse in absence of animal-mediated pollination or seed dispersal of the higher plants. Effective pollen transfer among individual plants is required by many higher plants for successful fructification, and active seed dispersal by animal vectors is a key demographic stage for maintaining forest regeneration and dynamics. Both processes depend on the provision by plants of some type of food resource that animals can obtain while foraging. These plant resources (nectar, pollen, fleshy pulp, seeds, or oil) are fundamental in different types of ecosystems for the maintenance of animal diversity through their keystone influence on life histories and annual cycles. From a conservation point of view, hunting and habitat loss are driving several species of large seed dispersers toward extinction, and these effects cascade towards a general reduction of biodiversity through reductions in seed dispersal (Dirzo and Miranda, 1990; Kearns, Inouye, et al., 1998; S. J. Wright, 2003). Looking back through time, evidence for these effects comes from the fossil record. Episodes of insect diversity decline, such as the ones during the Middle to Late Pennsylvanian extinction, during the Permian event, and at the Cretaceous/Tertiary boundary, have been followed by major extinctions of flowering plants (Labandeira, 2002; Labandeira, Johnson, et al., 2002). All this evidence already suggests that in conservation we cannot treat these species isolated from each other or consider only pairs of interacting species. Rather, we need to have a network perspective. The first studies on mutualism focused on highly specialized one-to-one interactions between one plant and one animal (Johnson and Steiner, 1997; Nilsson, 1988). Examples of these highly specific pairwise interactions are Darwin’s moth and its orchid (Darwin, 1862; Nilsson, Jonsson, et al., 1987), long-tongued flies and monocot plants (Johnson and Steiner, 1997), fig wasps and figs (Galil, 1977; Wiebes, 1979; J. M. Cook and Rasplus, 2003), and yucca moths and yuccas (Pellmyr, 2003). However, their strong emphasis in evolutionary studies probably reflects more the aesthetics of such almost perfect matching than their frequency in nature (Schemske, 1983; Waser, Chittka, et al., 1996). Motivated by this fact, several authors already advocated a community context to address mutualistic interactions (Heithaus, 1974; Feinsinger, 1978; Janzen, 1980; Herrera, 1982; Jordano, 1987; Fox, 1988; Petanidou and Ellis, 1993; Bronstein, 1995; Waser, Chittka, et al., 1996; Iwao and Rausher, 1997; Inouye and Stinchcombe, 2001).

6

CHAPTER 1

Waser, Chittka, et al. (1996) made the point that generalism is widespread in nature and advanced conceptual reasons based on fitness maximization in highly fluctuating interaction environments. More recently, and as a consequence of this interest in expanding the pairwise paradigm, there has been significant progress in our understanding of how pairwise interactions are shaped within small groups of species across time and space (Thompson and Pellmyr, 1992; Thompson, 1994; Parchman and Benkman, 2002).

A BIT OF NATURAL HISTORY Mutualisms are assumed to be among the most omnipresent type of interaction in terrestrial communities (Janzen, 1985). Beyond the mutualistic interactions among conspecific individuals (i.e., the subject of kin-selection and parentoffspring interactions), most of these interactions are allospecific interactions, involving species, or sets of species, completely unrelated. Multispecific interactions involving mutual benefits among partner species are extremely widespread and involve all the terrestrial vertebrates, plants, and arthropods. Many of these mutualisms involve sets of animal species interacting with plant species. Only five major groups of multispecific mutualisms exist in natural terrestrial ecosystems: (1) pollination and (2) seed-dispersal mutualisms among animals and plants (Jordano, 1987); (3) protective mutualisms among ants (and sometimes other arthropods) that protect plants and homopterans (RicoGray and Oliveira, 2007); (4) harvest mutualisms, including the gut flora and fauna of all vertebrate species and many invertebrates, the root rhizosphere occupants, lichens, decomposers, epiphyllae and some epiphytes, and antplants (ant-feeding plants; L. E. Gilbert and Raven 1975; Janzen 1985; RicoGray and Oliveira 2007). A fifth type of mutualism is the interaction between humans and plants (agriculture) and animal husbandry (Boucher, 1985b), mediated by the domestication process. Facilitative interactions among plants can also be considered as a type of mutualism with beneficial consequences for both partners (Verdú and Valiente-Banuet, 2008), although in many cases the positive effects occur only during specific stages (e.g., facilitation of seedling establishment). In this book we focus on pollination and seed dispersal with brief excursions into protective and ant-plant mutualisms (Fig. 1.2). The reason for this choice is because this is where the majority of research on mutualistic networks has focused and is where our expertise lies. Still, there is no evidence to suggest that the same rules do not apply to other mutualistic networks.

B I O D I V E R S I T Y A N D P L A N T- A N I M A L C O E VO L U T I O N

7

Figure 1.2. Examples of plant-animal mutualisms illustrating interactions among free-living species. These mutualisms typically involve the harvesting of plant resources by animal species with outcomes of fitness gain directly derived from the interaction. Clockwise from top right: Tangara cyanocephala swallowing a Campomanesia (Myrtaceae) berry (Ilha do Cardoso, SE Brazil; photo courtesy of André Guaraldo). Agouti, Dasyprocta aguti, feeding on fallen fruits of a Sapotaceae tree (Amazonia, N Brazil; camera-trap photo courtesy of Wilson Spironello). Eristalis tenax (Syrphidae) visiting an inflorescence of Allium sp. (Sierra de Cazorla, SE Spain; photo by P. Jordano). Ectatomma tuberculatum ants tending the extrafloral nectary of an Inga tree (Gamboa, c Alex Wild, used with permission). Panama; photo:


Typically, we might expect the net outcomes of mutualistic interactions among individuals or among species to fall somewhere along a gradient between antagonism (e.g., parasitism or cheating) and legitimate mutualism (Thompson, 1982). For instance, Rico-Gray and Oliveira (2007) document that ant-plant interactions most likely originated from antagonistic interactions, but the most frequent form of their ecological relationships is mutualistic. And this range can be observed in the interaction of two partner species (variation among individual effects) or when multiple species are involved (variation among species effects). For example, consider the diverse assemblage of insects visiting the flowers of a plant species. The whole range of interactions

8

CHAPTER 1

in a given population of the plant would be the result of the combined interactions of individual plants with individuals of different flower-visiting insects. Some individual plants will be visited by many insect species, whereas others (e.g., plants growing in isolated patches within the population) would be visited by a restricted set of flower-visitors, presumably with lower visitation frequency. If insect species differ in the effectiveness of pollen transfer, we could imagine that some individual plants receive most visits by legitimate pollinators, whereas other might be visited more frequently by noneffective pollinators (e.g., nectar thieves). Individual fitness variation across individual plants would depend on the relative location of each plant along the gradient of effectiveness defined by its flower-visitor assemblage: plants with reduced fruit set most likely had visits by low-efficiency pollinators, and those with higher seed set were most likely visited by legitimate pollinators. The overall interaction pattern for the plant species would be a composite of the visitation pattern to the different individuals in the population. The dynamics of mutualistic interactions are surprisingly robust to the presence of cheaters or antagonists (Bronstein, Wilson, et al., 2003), yet they determine ample temporal and spatial variation in the outcomes. Multispecific mutualisms involving plant-animal interactions are harvest-based mutualisms, mostly through the feeding of one species on the other (Janzen, 1985; Holland, Ness et al., 2005). Plants offer a resource (nectar, pulp, pollen, volatile fragrances, resin material to build nests, corolla parts, or other ancillary structures) that are collected by animals. The mutualistic service by animals directly derives from their foraging and movement patterns, resulting in dispersal of the plant propagules (seeds or pollen) or protection for the plant against herbivores or pathogens. Seed-dispersal mutualisms involve benefits in terms of fitness gain for both the mother plant dispersing its progeny and the progeny itself (individual seeds), largely because the fitness of both the mother plant and the propagule would be lower in the absence of the mutualistic interactions (Janzen, 1983; Jordano, 2000). So, most—if not all—multispecific mutualisms among freeliving plants and animals are resource based, and many involve dispersal events of some type of propagule. Therefore, these multispecific mutualisms play a central role in the population dynamics of plants (Fig. 1.3), where regeneration cycles depend on the successful establishment of new propagules and the successful closing of the dispersal loop (Wang and Smith, 2002). Mutualistic interactions are key at the specific stages (flowering, fruiting) where plant propagules need to be dispersed, and any environmentally driven collapse of such interactions will have far-reaching consequences for the plant population, with a negative feedback on the animal mutualists. Frugivorous

B I O D I V E R S I T Y A N D P L A N T- A N I M A L C O E VO L U T I O N

V

Seed dispersal

IV

Flowering and fruiting

III

Growth

II

Seedlings

I

Seed bank

9

Figure 1.3. The main elements of the population cycle of a higher plant (Harper, 1977). I, the seed bank; II, the early recruitment stage (seedlings); III, the growth phase (saplings); IV, production of flowers and fruits; and V, dispersal of the seeds. Mutualistic animals mediate the dispersal of pollen among flowering individuals and the dispersal of seeds. The entire population demography loop critically depends on the successful outcome of these interactions (Wang and Smith, 2002).

animals can remove large fractions of the fruit crop of maternal plants and move seeds to microsites with high probabilities for progeny establishment. Animals that eat fleshy fruits and disperse seeds have thus potential fitness influences on the plants (Jordano, 2000). Higher fruit-removal rates from mother plants result in more progeny successfully dispersed away from the parent and a lower probability for the seeds to die beneath the canopy of the mother plant. But animal-mediated dispersal also entails delayed effects that depend on the quality of the dissemination site for the survival prospects of newly established individuals. So, ultimately, an efficient disperser is a frugivore able to (1) efficiently consume and remove fruits from the canopy of the fruiting plant and (2) disseminate the seeds to suitable microsites for seed germination and seedling establishment (Schupp, Jordano, et al., 2010). These two aspects determine the quantity and quality components, allowing ample variation across mutualistic species in seed-dispersal effectiveness. Now, consider the potential fitness effects and benefits of the fruiting plant on the animal (Jordano, 2000). Most animals that depend on fruit food for their living rely on multiple plant species. A given fruit species, even those

10

CHAPTER 1

that contribute the basic diet of a frugivore species, frequently represents only a fraction of the whole diet and daily energy income of the frugivore. There are many reasons for this, but two important ones are that (1) the fruit-ripening period of most plant species is limited to certain periods of the year and (2) most fleshy fruits have pulps limited in a specific major nutrient (e.g., lipids, protein, nonstructural carbohydrates, and minerals) or energy content. The main result of these two important limitations is that frugivores relying largely on fruit food need to diversify their diets by interacting with multiple fruit species that, collectively, supply the daily energy and nutrient budget. Thus, there is ample variation across mutualistic plant species in terms of their effectiveness as food sources for animal frugivores. Plant-frugivore mutualisms, therefore, typically show very low specificity of the interactions (Jordano, 1987). Turning now to the second major type of mutualism, pollination also involves multiple types of outcomes, illustrating ample variation between the antagonism-mutualism extremes as well as in the specificity of the interaction (Feinsinger, 1983; Waser and Ollerton, 2006; Schemske, 1983; Waser, Chittka, et al., 1996). As with multispecific seed-dispersal mutualisms, pollination conveys dispersal of plant propagules (pollen) to distinct target sites (stigmas) of conspecific individuals. This constitutes a type of plant movement mediated by animals. Pollinators thus determine both the male and female fitness functions for the plants and, together with animal frugivores, mediate gene flow patterns in zoophilous plant species. Visitation frequency, pollen removal, and type of pollen deposition (e.g., on stigmas of the same plant— geitonogamous crosses—or on stigmas of a different individual—xenogamous crosses) influence the effectiveness of pollinator species. Therefore, the fitness effects of interactions with specific pollinators also depend on both the quantity and quality components of their foraging patterns. Ample variation between the extremes of antagonistic flower visitation and legitimate pollen transfer thus exists in multispecific pollination mutualisms. Despite the fact that seed dispersal and pollination mutualisms share some analogies, they are very different in many aspects (Wheelwright and Orians, 1982; Table 1.1). These differences mainly relate to the foraging patterns and outcomes involved and to the temporal span of the effects derived from the interaction. Although a specific target (conspecific stigmas) is very clear in pollination mutualisms, potentially advantageous targets for dispersed seeds are much less clear and often unpredictable at the time of seed release by maternal plants. For instance, high-quality microsites for survival to postdispersal seed predators and/or germination generally are poor-quality sites for early seedling establishment (Schupp, 1995). Successful pollination often benefits from high specificity of visitation by flower-visiting insects. This assures effective pollen

B I O D I V E R S I T Y A N D P L A N T- A N I M A L C O E VO L U T I O N

11

Table 1.1. Analogies and differences between pollen and seed dispersal. Modified from Wheelwright and Orians, (1982). Pollen Dispersal

Seed Dispersal

Target-site suitability

Stigma of conspecific flower.

Microsite adequate for germination and establishment.

Predictability of target quality

Distinctive. Color, shape, odor apparent at a distance.

Unpredictable.

Temporal suitability

Synchronous with pollen dispersal.

Scarcely predictable; often with delayed effects. Good microsites for seeds are poor for seedlings.

Advantage of interaction specificity

High. Most pollen lost if carried over other species’ flowers.

Low. Negative effects of proximity to conspecific adults.

Ability of directed dispersal to suitable sites

High. Plant can “control” directionality of movement.

Low. Movement influenced by multiple foraging choices.

carryover to multiple conspecific stigmas during short foraging bouts and avoidance of stigma clogging by allospecific pollen (Table 1.1). Color, odor, and display clues often signal the receptivity of flowers to foraging flower visitors. In contrast, most frugivores visit a number of fruiting plant species while foraging and maintain switching behaviors to consume alternative food items to fruits (e.g., insects, leaves, etc.). Therefore, multiple natural history details underpin the outcomes of pollination and seed-dispersal mutualisms, ultimately favoring some degree of generalization by a widespread occurrence of variable outcomes of pairwise interactions. Rather than reductionistic approaches that underscore the exceptions, understanding how these fascinating natural history details influence evolution and coevolution requires tools that incorporate these complexities to identify shared patterns. This tension between reductionistic and synthetic approaches has been a dominant theme in research on plant-animal interactions since the early 1970s, when interest in coevolution of mutualisms reflourished in ecological research (Waser, 2006). Ollerton (1996) describes it as a paradox. On one hand, broad groups of floral and fruit traits seem unequivocally related to visitation patterns by specific subsets of animal mutualists—that is, the interaction syndromes described by Stebbins (1970). Yet looking to the details of interactions, many plant species share a wide array of animal mutualists as pollinators and

12

CHAPTER 1

seed dispersers, conjuring up visions of intense reciprocal effects (Schemske, 1983). Understanding complex networks of interaction will be helpful to assess patterns of evolution in generalization–specialization in mutualisms, a subject ultimately related to the issue of niche variation and community assembly processes.

COEVOLUTION IN MULTISPECIFIC MUTUALISMS Coevolution is evolution of interacting species; therefore, the outcomes of multispecific interactions underpin the evolutionary process of the partner species involved. The coevolutionary process thus involves the joint evolutionary trajectories of two separate gene pools that do not mix (Thompson, 1982, 1994). Coevolution is then one of the many outcomes of plant-animal mutualisms. Strict-sense coevolution (Janzen, 1980) involves specific and reciprocal effects directly derived from the mutualistic interaction. As a consequence, it is difficult to anticipate how coevolution can produce complex webs of interaction involving hundreds of species (Thompson, 2006). Most multispecific interactions in nature are highly diversified, with species that range widely in the net effect of the interaction, from legitimate mutualists to mutualism parasites. This has caused some confusion in the use of the term coevolution, broadly applied to any interaction among species irrespective of the geographic scale, potential outcome, sign and magnitude of the reciprocal effects, and specificity. The catchall term diffuse coevolution has thus been applied to the many circumstances where the complexities of the natural history of mutualisms are well beyond simple pairwise interactions (Herrera, 1982; Fox, 1981, 1988; Strauss, Sahli, et al., 2005). The recent conceptual development of coevolutionary studies has thus been limited by a lack of appropriate frameworks that provide testable hypotheses about how diversified coevolution operates in natural systems. A recent insight to understand the coevolutionary process has been framed by John N. Thompson around the concept of geographic mosaics of coevolution (Thompson, 1994, 1999b, 2005; Gomulkiewicz, Drown, et al., 2007). This has represented a major advance in our understanding of how coevolution proceeds in complex natural settings, such as the spatially— and temporally— variable mutualistic interactions among species. Most mutualisms show marked temporal and spatial changes in their main components: species involved, strength of the interaction, and outcomes of the interaction (Bronstein, 1994; Chamberlain and Holland, 2009). The challenge is to provide robust generalities underlying what may seem a markedly context-dependent process.

B I O D I V E R S I T Y A N D P L A N T- A N I M A L C O E VO L U T I O N

13

A mosaic pattern of coevolving interactions accommodates the characteristic context dependency that multispecific mutualisms show, based on three premises: (1) interactions occur among species that are themselves distributed in populations; (2) the outcomes of interactions vary across populations; and (3) interacting species do not necessarily have matching geographic ranges. In multispecific mutualisms, a given species typically shows local (amongpopulation) variation in the pollinator or seed-disperser assemblage, and the composition of local plant-animal communities interacting can also vary. Therefore, we might expect (1) a selection mosaic across populations, with variation in selection regimes and outcomes leading to different evolutionary trajectories; (2) actual coevolution occurring in a subset of the populations (coevolutionary hotspots), with no change in others; and (3) remixing of traits resulting from gene flow, genetic drift, and local extinction of populations (Thompson, 2005). Multispecific interactions will be structured in local communities also subjected to this mosaic of interaction outcomes. Few traits will be locally favored given the marked local context dependency of the interaction patterns and the low specificity of the interactions (Thompson, 1994). Local variation in the degree of trait matching among interacting species is expected, as well as local variation in the degree of convergence among species in each of the partner groups (animals and/or plants) (Thompson, 2006). The next frontier is to extend these multispecific systems to embrace whole networks, to address how these large assemblages of species are organized by ecological and evolutionary processes, and to infer the consequences of network architecture for the persistence of biodiversity. This book is about this frontier. Our ultimate goal is to understand how diversified mutualisms among animals and plants evolve and coevolve into megadiverse assemblages of species. To this aim, we will be looking at multispecific mutualisms as networks of mutual dependences among species. First, though, we need some tools and concepts derived from the study of other types of networks. Our understanding of community-rich mutualistic interactions has indeed been constrained by the lack of an appropriate conceptual framework. This is a situation that echoes similar limitations in other fields addressing complex systems formed by a large number of different elements interacting among themselves. Traditionally, the reductionistic approach has followed the path of breaking up these complex networks in basic units and studying them in isolation. Given the complexity of these networks, without the appropriate conceptual framework, we could not have advanced our understanding of mutualisms at the community level. The theory of complex networks, indeed, provides the right framework for addressing entire communities. First, network theory provides a way

14

CHAPTER 1

to describe such communities. Second, it brings a set of tools that allows measuring the architecture of these networks. Third, there are dynamical models that exemplify the processes that may be at work in building up these networks. Finally, this framework allows comparing the properties of these ecological networks with those of other networks such as the Internet or protein-regulation networks. Network theory provides a conceptual framework for learning how to tackle mutualisms as integral elements rather than just focusing on their basic components. As with any such conceptual framework, we have to face some trade-offs between realism and complexity. Thus, we have to leave aside (momentarily; see following chapters) differences across species that are now represented as similar nodes. The hope is that this sacrifice of detail will be compensated by our ability to extract meaningful patterns in what was once thought to be a diffuse and intractable set of dependencies. We are convinced that the enormous challenge to understand how extremely diversified mutualisms evolve and coevolve in systems such as tropical rainforests requires an integrative approach. This should combine a solid knowledge of the natural history of plant-animal interactions with robust quantitative tools aimed at visualizing, exploring, and analyzing their complexity.

SUMMARY Plant-animal mutualistic interactions are the wireframe that supports many terrestrial ecosystems. The importance of these interactions and the mutual benefits conveyed to the partner species were recognized since the early times of ecological studies. Yet, the interest in mutualistic interactions and their patterns of evolution and coevolution has been marginal during most of the recent history of ecology, with its central emphasis on antagonistic interactions. A persistent challenge has been to understand how multispecies interactions evolve and coevolve among free-living species. This understanding has been limited by the absence of methodological tools enabling the integrated analysis of the intrinsic complexity of details that make mutualisms so fascinating. Reductionistic approaches have underscored limiting cases such as highly specialized one-on-one interactions, failing to identify the general, shared patterns in multispecies assemblages. This book aims to describe these mutualistic patterns, which can be regarded as the architecture of biodiversity.

CHAPTER TWO

An Introduction to Complex Networks

We saw in the previous chapter that mutualisms can involve dozens, even hundreds, of species and that this complexity has precluded a serious communitywide approach to plant-animal interactions. The most straightforward way to describe such an interacting community is with a network of interactions. In this approach, species are represented as nodes of two types, plants and animals. A link between two such nodes is established if the plant is pollinated or dispersed by the animal. This will allow shedding light and order out of an apparently entangled web, such as the one depicted in Figure 2.1. Indeed, Figure 2.1 illustrates the challenges in addressing community-wide mutualisms. How do we represent them? What do we do with them? What should we measure? What is their meaningful scale? The risk of this type of representation if we do not have an adequate conceptual framework is not seeing the forest for the trees. The approach that we will be outlying in this chapter will dispel this risk, allowing us to tackle the complexity of Figure 2.1. In what follows, we will describe statistical measures that represent different aspects of the complexity of these networks, such as the distribution of the number of interactions per node, the basic building blocks of these networks, or they tendency to be organized in modules. These are just a few examples of several components of these networks that can be objectively quantified. Our goal in this chapter is to provide the reader with the tools needed for a network approach. This will provide a background with which to understand the structure of mutualistic networks. As evocatively written by Cohen (2004), mathematics is the new microscope of biology, only better. One of our microscopes here will be the physics of complex networks. This nicely illustrates the usefulness of network theory to study complex ecological webs. In turn, mutualistic and other complex ecological networks open new areas for conceptual exploration in physics—biology is mathematics’ next physics, only better (Cohen, 2004). First, this “microscope” allows us to visualize their complexity. Second, it provides us with tools to examine its structure. Finally, it allows hypothesis testing by comparing empirical data with competing null models built according to some basic principle. We now turn to a description

Figure 2.1. A complex network of interactions among frugivorous birds and fruits in the Atlantic rainforest of southeastern Brazil. The small spheres represent animal and plant species (nodes of the network) linked by multiple interactions representing mutual dependences: birds eat the fleshy fruits and plants get the seeds dispersed by the animals. Based on Silva, de Marco, et al. (2002).

Plants

Animals

A N I N T RO D U C T I O N T O C O M P L E X N E T W O R K S

17

of network theory and its applications to other types of complex networks. This, however, is not the only microscope we will be using. Others will be provided from the field of island biogeography and the phylogenetically based comparative methods. The idea is to bring together an interdisciplinary approach in the hope that the integration across disciplines will help us to disentangle the Web of Life.

A NETWORK APPROACH TO COMPLEX SYSTEMS Complex networks can be identified as the fourth strike of the sciences of complexity (Newman, Barabási, et al., 2006; Solé and Bascompte, 2006; Bascompte, 2007). First, deterministic chaos proved that simple, nonlinear deterministic models have the potential to generate aperiodic temporal series, destroying the long-standing tenet of Newtonian physics that knowledge implies prediction. Second, spatially explicit models of ecological populations were seen to generate complex spatiotemporal patterns that emerge in otherwise homogeneous landscapes. This suggested that complex spatiotemporal patterns observed in nature do not require complex mechanisms. The sandpile as the iconic representation of self-organized criticality followed, illustrating that some nonequilibrium systems spontaneously evolve toward a critical state. In such a state, small perturbations do not always have small consequences because some perturbations such as the fall of a sand grain can be amplified through an avalanche affecting the entire system. And so to the subject of this chapter. A network is a representation of a system composed of multiple elements connected by links (Fig. 2.2). Networks are everywhere. They are paradigmatic representations of the complexity of multiple structures (Barabási and Albert, 1999; Amaral, Scala, et al., 2000; Albert and Barabási, 2002; Strogatz, 2001; Dorogovtsev and Mendes, 2002; Newman, 2003, 2004; Watts, 2003; Proulx, Promislow, et al., 2005; Montoya, Pimm, et al., 2006; May, 2006; Newman, Barabási, et al., 2006; Solé and Bascompte, 2006). Examples are genes linked by gene regulation; proteins interacting to create a structure; individuals in a society engaged in cooperation; or species in ecological communities linked by predator-prey feeding relationships. These objects have a counterpart in the mathematical realm known as graphs. This, therefore, represents our first detour in the quest to characterize network architecture. The analysis of networks dates back to Leonard Euler (1736), who became interested in the problem known as the Königsberg bridges (Fig. 2.3a). Located on the Pregel river in what is now the city of Kaliningrad (Russia), only two

18

CHAPTER 2

a

b

Figure 2.2. Graph representations of a food web (a) and a mutualistic network (b) as complex networks of interactions (links) among species (nodes) in ecosystems. The food web represented here is Ythan Estuary (Hall and Raffaelli, 1991), with 92 species (taxa) and 382 links. In (b), the graph depicts the interactions among a set of ant species that protect plant species from herbivores (41 species— taxa—with 48 links; Fonseca and Ganade, 1996).

a

b 1 4

6 6

1 4 3

2

3

5

2

5 7

7

Figure 2.3. (a) Schematic map (from 1 km altitude) of the city of Kaliningrad (Russia), the 18th-century Königsberg. Leonard Euler provided a proof in 1736 that no path exists that crosses all seven bridges (indicated by numbers) exactly once each. Five of the bridges were destroyed in World War II, and only three were rebuilt. This proof was based on the representation of the system as a graph (b) where links represented the bridges between land masses (nodes).

of the original bridges remain in place (3, 6 in Fig. 2.3a). The other bridges were destroyed during World War II, although two of them were rebuilt (1, 2) and a third one was replaced (7). The problem consisted of proving that no path exists such that we can cross all bridges only once. To prove this, Euler used a graph representation, where nodes represented the three land areas of the city plus the small island and the edges were the seven bridges (Fig. 2.3b). More generally, Euler proved that to fulfill the preceding condition, a graph

A N I N T RO D U C T I O N T O C O M P L E X N E T W O R K S

19

should have exactly zero or two nodes with an odd degree, therefore showing how the solution of the problem depends on the degree of each node of the graph. More recently, graph theory was developed by Paul Erdös in collaboration with Alfred Rényi. Erdös is considered one of the most prolific mathematicians in history and certainly the most social one (Hoffman, 1998). Without a fixed home or a job, he used to travel around the world with all his belongings in an old piece of luggage. He would unexpectedly show up at a colleague’s home to greet him or her with “my brain is open” in search of a successful collaboration. Among the many lucky colleagues was Rényi, and the two mathematicians jointly published a series of very influential papers on graph theory (Erdös and Rényi, 1959). A random graph is an idealization of a real network. It describes a collection of nodes where two such nodes have a probability p of being connected. It is an interesting representation because it allows us to relate a dynamic process of network formation to a resulting structure. Also, it is simple enough to be amenable to analytical treatment. Erdös and Rényi (1959) described how the size of clusters of interlinked nodes (nodes belonging to a unique component) increased as the probability p of connecting two randomly picked nodes increased. For low p-values, there are several small clusters, all having pretty much the same size. But once a critical point pc is reached, a giant component, to which the majority of nodes belongs, suddenly appears. The probability of finding such a giant component does not show a smooth increase with p but rather an “all-or-nothing” behavior. Below pc , it is almost impossible to find a giant component, whereas beyond such a threshold it is almost certain to find it. Similarly, Erdös and Rényi analytically derived the probability distribution of finding a node with a specific number of links for these random graphs. The number of links per node follows a Poisson distribution. There is a welldefined average connectivity per node, and the probability of finding a node with a larger number of links drops very fast with the number of links. This corresponds to a homogeneous, or democratic, network, where all nodes have approximately the same importance in terms of number of links. Some years went by after this seminal work by Erdös and Rényi setting the foundations of graph theory before physicists and sociologists turned their attention to real networks. Currently, there is a growing interest in the field of complex networks (Strogatz, 2001; Newman, 2003, 2004; Newman, Barabási, et al., 2006; Solé and Bascompte, 2006). As we will see in this chapter, real networks are quite different from the expectations for a random graph. Biological and nonbiological networks have a much more heterogeneous structure that deviates markedly from a homogeneous pattern

20

CHAPTER 2

of links among nodes. Interestingly enough, this structure is recurrent across seemingly disparate systems, which suggests similar dynamic rules of network buildup or constraints on network organization. These findings are also of importance because, as we will see, there is a strong relationship between network structure and robustness to perturbations (Albert, Jeong, et al., 2000). The framework of complex networks has represented an important change in paradigm in several fields. For example, molecular biology used to employ quite a reductionist approach, studying one or a few genes at the time. However, a unique biological function can rarely be ascribed to a single molecule. Recently, a large number of papers have described the network of gene regulation, allowing a better understanding of genetic diseases (see, e.g., Luscombe et al., 2004). More broadly, most biological functions arise from interactions among many component parts, not just as a single effect of a particular gene or protein. One-Mode and Two-Mode Networks Technically, there are two main types of networks, and the distinction is significant in light of what comes up in the following chapters. One-mode networks, represented as unipartite graphs, are those in which there is only one type of node (e.g., servers or Web pages) so that any two such nodes can be connected by a link. In ecology, examples would be food webs (Fig. 2.2a) and spatial networks (as follows). On the other hand, the subject of this book is a clear example of a two-mode network represented as a bipartite graph. There are two sets of nodes, plants and animals, with interactions between— but not within—sets (Figs. 2.1, 2.2b). Other examples of two-mode networks in ecology are those representing the interactions between hosts and their parasitoids (Müller, Adriaanse, et al., 1999; van Veen, Morris, et al., 2006), with some of them even being tripartite networks with interactions among hosts, parasitoids, and superparasitoids (Müller, Adriaanse, et al., 1999). Beyond ecology, an example of two-mode network would be that formed by the CEOs linked to the administrative boards they sit on. The bipartivity in two-mode networks represents a very adequate framework, therefore, for representing coevolutionary interactions between plants and their animal pollinators and seed dispersers (Jordano, Bascompte, et al., 2003; Bascompte, Jordano, et al., 2003). This is the type of representation we will be using through this book. Technically, a bipartite graph is defined by an adjacency matrix whose elements ai j will be 1 if node i of class A and node j of class B interact and zero otherwise (Fig. 2.4b). In comparison, the unipartite graph that illustrates a food web (Fig. 2.4a) has a square matrix representation, whose elements ai j will be 1 if node i and node j interact and

A N I N T RO D U C T I O N T O C O M P L E X N E T W O R K S

a

21

b

A

P 1

A1 A2 A3 A4 A5 A1 A2 A3 A4 A5

0 1 1 0 0

1 0 0 1 0

1 0 0 1 1

0 1 1 0 0

0 0 1 0 0

P1 P2 P3 P4 P5 P6 P7

1

2

4

3

5

A1 A2 A3 A4 A5

1 1 1 1 1

1 1 1 0 1

1 0 1 0 0

0 1 0 1 0

1 0 1 0 0

1 1 0 0 0

1 0 0 0 0

1

2

2

3

3

4

4

5

5

6 7

Figure 2.4. (a) Matrix representation of a unipartite graph illustrating a food web (one-mode network), with all interactions occurring among the five species. (b) Matrix representation of a bipartite graph illustrating a two-mode network of interactions among a set of animal species (A; darker nodes) and a set of plant species (P; lighter nodes). The panels include the graph representations of the two adjacency matrices. Note that both the matrix representations and the corresponding graphs can be qualitative (each cell ai j of the matrix and link in the graph illustrating the presence or absence of interactions) or quantitative (weighted), with matrix cells and graph links illustrating some weighted or quantitative index of the relative intensity of interactions.

zero otherwise, given that all the nodes are of the same class S (e.g., species in an ecosystem). Networks can also be weighted, meaning that links have a measure proportional to the strength of the interaction (see the following section).

MEASURES OF NETWORK STRUCTURE Connectivity Distribution We have already noted that the random graphs have quite homogeneous distribution of connections per node. This property is described by the connectivity distribution, or degree distribution. The connectivity distribution represents the number of links per node (in the x-axis) versus the probability of finding a node with such a number of links (in the y-axis; Fig. 2.5). Technically, we normally represent this plot in a log-log scale for reasons that will be evident shortly. Also, it is customary to represent the cumulative probability, that is, the probability of a node having one or more links, two or more links, three or more links, and so on. There are two reasons for using such cumulative representations. First, it is a way to smooth the noise in real, finite data. Untransformed distributions would contain gaps and ups and downs that would make the analysis more difficult. Second, cumulative probability distributions are more amenable to be fitted by several functions. We will consider specific

22

CHAPTER 2

b

P(k)

a

1

P(k)

0.1 0.01 0.001 0.0001 0.00001 0.000001 0.0000001 0.00000001

P(k) ∝ e–γk 1

P(k) ∝ ck–γ 10

1 100 1000 Number of links/node, k

10

100

1000

Figure 2.5. Connectivity distributions illustrate the frequency of nodes with increasing number of links (k) (top) and their associated cumulative distributions P(k) (bottom). (a) Random networks are characterized by exponential decays of P(k) with increasing k values. Most nodes have a similar degree, and the distributions is well characterized by its scale (i.e., the mean value of k). Well-connected nodes do not exist because the frequency distribution has an exponential cutoff at some low k value. (b) Many complex networks are characterized by connectivity patterns that follow a power-law distribution for P(k) values, that is, a scale-free distribution. Nodes with many links (hubs) are infrequent, but they do now exist.

examples in Appendix B, where interested readers can find information on the statistical process of model fitting and model selection. Albert, Jeong, et al. (2000) pioneered the use of degree distributions to characterize the structure of the Internet. This was an important paper for various reasons. First, it clearly demonstrated that a graph representation of the Internet, as with many other complex networks, is quite different than the random graphs studied by Erdös and Rényi. Although we have already seen that random graphs are quite homogeneous—with most nodes having a similar degree (Fig. 2.5a)—real networks are much more heterogeneous. In this case,

A N I N T RO D U C T I O N T O C O M P L E X N E T W O R K S

23

the bulk of nodes have a few links, but a few nodes are much more connected than what we would expect by chance (Fig. 2.5b). Second, Albert, Jeong, et al. (2000) showed that this affects the robustness of the Internet to attack and failure. These highly connected nodes are hubs; as we will see, they are very important in explaining network robustness to error. To some extent, they act as the glue keeping the network together. In this case, the average number of links per node does not provide any useful information at all. It is not a good descriptor of the network any more, because variance is just too high. Mathematically, this nondemocratic, heterogeneous network is characterized by a degree distribution described by a power-law function: P(k) ∝ k −γ ,

(2.1)

where P(k) is the probability that a node has k links and γ is a critical exponent. The preceding distribution is also called scale free because the relationship between k and P(k) is not defined on a particular scale (Schroeder, 1991). This is the reason we were plotting this degree distribution in a log-log plot: the power law is easily identified by a straight line of slope −γ for all the range of k values. This is not true for any other distribution. For example, an exponential distribution is defined on a particular scale, the one defined by the average number of links per node. In this case, the relationship between log k and log P (k) changes as we move through the x-axis (Schroeder, 1991). Most empirical distributions of the number of connections per node indicate the existence of three classes of networks (Amaral and Ottino, 2004): (1) scalefree networks, characterized by power-law distribution; (2) broad-scale or truncated scale-free networks, characterized by a degree distribution that has a power-law regime followed by a sharp cutoff not due to the finite size of the network; and (3) single-scale networks, characterized by a degree distribution with a fast-decaying tail, such as exponential or Gaussian (see Appendix B for the exact mathematical expressions of these classes of connectivity distributions). Two major classes of power laws are commonly explored in the ecological literature (White, Enquist, et al., 2008). The first class includes bivariate relationships between, for example, the species-area relationship or body-size allometries. The second class is a frequency distribution, where the frequency of some event (e.g., number of species showing a given number of trophic links) is related to its magnitude (e.g., the number of interactions per species). Throughout this book we will refer to the latter class of power laws, as repeatedly studied in the structure of food webs (Solé and Montoya, 2001; Camacho, Guimerà, et al., 2002; Dunne, Williams, et al., 2002a).

24

CHAPTER 2

Strength Distribution Up to now, we have been considering binary networks—that is, networks in which two nodes either interact or do not interact—without measuring the strength of such an interaction. But, of course, quantifying the weight or intensity of these links can provide much more information, and recent methods and definitions have been extended to account for weighted networks. For example, a quantitative extension of the concept of degree, or number of links, is that of a node’s strength, that is, the sum of the weights of all its links (Barrat, Barthelemy, et al., 2004). The edges among nodes in weighted graphs can be associated with any scalar value quantifying the link strength, a cost, or a measure of flow along the edge (Steuer and Zamora-López, 2008). For the network of the worldwide air traffic, for example, we could quantify the total number of passengers flying between two airports. The strength of an airport would then be defined as the sum of the number of passengers going from that airport to any other for which a direct flight exists. This amount would give us the quantitative importance of that airport in the context of the number of passengers using it. Similar to the degree distribution defined in the previous subsection, we could also plot the strength distribution, defined as the number of airports that have a certain volume of passengers. This distribution is also highly skewed (Barrat, Barthelemy, et al., 2004). Interestingly enough, average node strength increases faster than linearly with node degree in this airport traffic network, although this does not seem to be the case for other weighted networks, such as the network of coauthorship among scientists (Barrat, Barthelemy, et al., 2004).

Small-World Networks The sociologist Stanley Milgram performed an interesting experiment in 1967. He was curious about how many acquaintance links separate two given citizens in the United States. For example, if sitting on an airplane next to a complete stranger, we discover that I have a friend who is friend of my traveling partner, then our path length would be two. What is the average path length in a large country such as the United States? Milgram decided to unravel this question using the following scheme. He chose a group of random people in the cities of Wichita and Omaha, in Kansas and Nebraska, respectively, and asked them to write a letter to someone they did not know in Sharon, Massachusetts. They were asked to send the letter to someone they knew that they thought would be closer to this target person (e.g., a cousin visiting Sharon, an oldtime friend who was living in Massachusetts, and so on). This new person was requested, in turn, to do the same until the letter eventually reached

A N I N T RO D U C T I O N T O C O M P L E X N E T W O R K S

25

its addressee. To keep a record of the experiment, each intermediary was requested to send a copy of their letter to Milgram so he could keep track of the process. Interestingly enough, and quite contrary to intuition, the chains varied between 2 and 10 intermediate acquaintances. This interesting result, later on made popular with the name of six degrees of separation supported the expression ‘What a small world’ (or ‘El mundo es un pañuelo’ in our native Spanish). More recently, Watts and Strogatz (1998) formalized the idea of a small world by using a simple model that was aimed at capturing this property (as follows). It also led to a more formal definition. A network shows the smallworld property if it has both a high clustering coefficient (meaning that the friends of my friends are also my friends) and a short path length, as found by Milgram (Fig. 2.6). Many complex networks seem to be scale free and to have a small-world structure. These are the two ingredients any model of a real network should be able to reproduce. Several examples of real networks beyond the social one studied by Milgram have been found to show the small-world property (Watts, 2003). In ecology, Montoya and Solé (2002) were the first looking for this property in food webs. They found that food webs have the small-world property, and so, despite these networks having a large number of species, all their nodes are within a short path length from each other (Montoya and Solé, 2002). Similarly, Olesen, Bascompte, et al. (2006) analyzed the smallworld property of pollination networks, as we will review in the next chapter. Modularity Modularity is another property of complex networks beyond the description provided by the degree distribution. As other measures of network architecture, it describes a deeper level of structure focusing not so much on how many interactions one node has but rather with which other nodes it interacts. A modular network would be one organized in distinct modules in which nodes within a module are very much interrelated but show few interactions with nodes from other modules (Fig. 2.7). These modules are also called communities, compartments, or cliques in the physics, ecological, and social literature, respectively. Modularity has a central relevance in complex networks. It can represent an inherent design principle (Steuer and Zamora-López, 2008). In addition, the identification of modules is a fundamental step in understanding the functioning of complex networks (Guimerà and Amaral, 2005; Guimerà, Sales-Pardo, et al., 2007; Sales-Pardo, Guimerà, et al., 2007). For example, the functionality

a

b

Path length or clustering coefficient

1.0

0.8

0.6

0.4

Average path length Clustering coeffcient

0.2

2–10

2–9

2–8

2–7

2–6

2–5 p

2–4

2–3

2–2

2–1

Figure 2.6. Small-world networks. (a) the Watts and Strogatz (1998) model uses a regular network as the starting point (left) and rewires links at random with probability p. As iterations proceed, the network eventually becomes a random, Erdös-Rényi network as p → 1 (right). Somewhere in the middle, the network is small world. b. Average path length and clustering coefficient as a function of the fraction of nodes reconnected, p. Path length is defined as the number of links in the shortest path between two nodes, averaged over all pairs of nodes. The clustering coefficient reflects the extent to which partners of a given node are also partners of each other. There is a rapid drop in path length corresponding to the onset of the small-world phenomenon; note that during this drop, the clustering coefficient remains almost constant at its value for the regular network, indicating that the transition to a small world is almost undetectable at the local level. Dashed lines indicate the 95% confidence interval for simulated values.

1

A N I N T RO D U C T I O N T O C O M P L E X N E T W O R K S

27

Figure 2.7. Modules are subsets of nodes more densely connected among themselves than with nodes in the rest of the network (three modules indicated by grey ellipses). Different nodes in a modular network have different roles: for example, connectors (black) do not necessarily have many links, but they connect other modules to each other.

of integrated electronic circuits that build up complex technological networks cannot be assessed solely on the basis of individual transistors. Specifically, the identification of functional components such counters, gates, or shift registers is fundamental for their effective design and use (Steuer and ZamoraLópez, 2008). Similarly, in biological networks shaped by evolution, different modules might represent distinct functionalities, and a modular structure can be optimal to cope with constraints by quickly incorporating new functionality (Hartwell, Hopfield, et al., 1999; Kashtan and Alon, 2005). Many of these modules, in fact, group together many lower-level elements or component parts and are themselves part of more complex structures, frequently assembled in hierarchical networks (Sales-Pardo, Guimerà, et al., 2007; Clauset, Moore, et al., 2008). This hierarchical and modular architecture is also observed in many biological and ecological networks (Junker and Schreiber, 2008).

28

CHAPTER 2

In physics, Newman and Girvan (2004) have provided one of the most frequently used module-finding algorithms, implemented through an optimization approach by Guimerà, Sales-Pardo, et al. (2007). This approach, technically named simulated annealing (SA), allows the search for approximate solutions when finding an exact solution is not feasible due to huge computational time. Danon, Diaz-Guilera, et al. (2006) compare different methods of calculating modularity and conclude that the SA algorithm is the best available. Roughly speaking, the modularity function M, which is maximized through the heuristic procedure of simulated annealing, is defined as follows (see Appendix D for details). For each module, we calculate the difference between the observed and expected fraction of links within that module. The modularity function is the sum of such difference across modules. Here, the expected fraction of links is calculated as if two nodes were interacting at random; therefore, the probability of such an interaction was proportional to the degree of both nodes. Thus, for bipartite networks, the modularity function can be written as (Barber, 2007; Guimerà, Sales-Pardo, et al., 2007; Sawardecker, Amundsen, et al., 2009):  n 
ei diP diA − , (2.2) M= L L L i=1 where n is the number of modules, ei is the observed number of interactions within module i, L is the total number of interactions in the network, and diP and diA are the sums of the degrees of nodes in module i for the plant and animal sets, respectively. Similar versions of the preceding function have been used for one-mode networks, such as one-mode and weighted networks (see Appendix D for a full derivation). These measures have been applied to the analysis of a wide range of complex networks such as air-traffic or metabolic networks (Girvan and Newman, 2002; Guimerà and Amaral, 2005; Guimerà, Mossa, et al., 2005). Modularity analysis allows not only the description of the group structure of the network, but also the identification of roles for different nodes, depending on their pattern of connections within and among modules (Fig. 2.7). The concept of modularity in food webs is far from new (see Appendix D). However, despite this interest, there are only a few clear examples of modularity in food webs (Girvan and Newman, 2002; Krause, Frank, et al., 2003; Rezende, Albert, et al., 2009; Guimerà, Stouffer, et al., 2010), and only one study has addressed the suite of life-history traits correlated with these compartments (Rezende, Albert, et al., 2009). In particular, phylogeny, body size, and habitat preference are all significantly correlated to the ascription of a species within a module.

A N I N T RO D U C T I O N T O C O M P L E X N E T W O R K S

29

Similarly, modern algorithms of modularity have been applied to gene flow networks (Fortuna, García, et al., 2008; Fortuna, Albaladejo, et al., 2009) and social networks (Wolf and Trillmich, 2008; Fortuna, Popa-Lisseanu, et al., 2009). Modularity techniques have been applied very recently to mutualistic networks by Olesen, Bascompte, et al. (2007), as expanded in the following chapter.

Motifs and Subgraphs Motifs are small subgraphs or patterns of interconnections among a few nodes that are overrepresented in complex networks when compared to appropriate randomizations of such networks (Milo, Shen-Orr, et al., 2002, Fig. 2.8). These network motifs, therefore, are considered as the basic building blocks of complex networks. Different network motifs have been found in a wide range of networks—both technological and biological (Spirin and Mirny, 2003; Milo, Itzkovitz, et al., 2004; Sporns and Kotter, 2004; Vázquez, Dobrin, et al., 2004; Solé and Valverde, 2006; Schwöbbermeyer, 2008). For example, the feed-forward loop (Fig. 2.8a, number 10), the bifan (Fig. 2.8a, combining two number 1s), or the three-node feedback loop (Fig. 2.8a, number 4) are characteristic of the gene regulatory networks of Escherichia coli and Saccharomyces cerevisiae, the neural network of Caenorhabditis elegans, and the electronic circuits of logic chips (Milo, Shen-Orr, et al., 2002; Schwöbbermeyer, 2008). The concept of network motif was first introduced in ecology by Bascompte and Melián (2005) as a way to assess to what extent particular trophic modules (Fig. 2.8b) studied by theoreticians are widespread in entire food webs. Thus, whereas empirical studies had described static parameters of large empirical food webs, theoretical models had described the dynamics of simple tritrophic food chains. To assess to what degree theory based on simple modules is relevant for entire food webs, we need to know how relevant these trophic modules are in real food webs (Arim and Marquet, 2004; Bascompte and Melián, 2005; Stouffer, Camacho, et al., 2007). Thus, although apparent competition and intraguild predation are overrepresented in empirical food webs, the frequency of omnivory largely varies across communities (Bascompte and Melián, 2005). A quantitative extension of these motifs, including interaction strength, was studied in the context of a detailed Caribbean food web as a way to assess the likelihood of trophic cascades after the overfishing of sharks (Bascompte, Melián, et al., 2005). These motifs also have analogs in the bipartite networks that represent mutualistic ecological interactions (Jordano, 2010, Fig. 2.8c).

30

a

CHAPTER 2

1

2

3

4

5

6

7

8

9

10

11

12

13

b

c

Figure 2.8. Motifs in complex networks. (a) All the 13 possible 3-node subgraphs explored in the Escherichia coli transcriptional network and other biological networks (Milo, Shen-Orr, et al., 2002). The numbers 7, 9, and 10 identify motifs that appear more frequently than expected in both the yeast (Saccharomyces cerevisiae) and Escherichia coli transcriptional networks. (b) Different types of motifs in food webs illustrating simple food chain, omnivory, apparent competition and intraguild predation (Bascompte and Melián, 2005). (c) Potential motifs in bipartite graphs (Jordano, 2010) that can be formed from the interactions between two partner pairs, from the pairwise specialization (upper left subgraph) to the full connectedness among generalized species (lower right).

Whereas the first studies on network motifs were structural—that is, did not consider the dynamic causes or consequences of such motifs—recent papers have addressed their dynamic implications (Prill, Iglesias, et al., 2005). For example, Stouffer and Bascompte (2010) have shown that the modules that are most commonly observed in empirical food webs are precisely those conferring the greatest persistence of the entire network. This clearly suggests dynamic justifications for the observed motifs. Although some papers have

A N I N T RO D U C T I O N T O C O M P L E X N E T W O R K S

31

gone a step further and have invoked natural selection to explain these markedly deviant patterns of motif frequencies from the expectation under a process of random assemblage, emergent properties of the networks (i.e., the process of network buildup) could also cause these deviations (Konagurthu and Lesk, 2008). We next consider some basic models of network assembly and how particular network patterns, such as a scale-free connectivity distribution, arise.

MODELS OF NETWORK BUILDUP Arguably, the first and simplest model of network formation is the recipe that Erdös and Rényi used to construct a random graph. As explained before, any two nodes have a probability p to be connected. This model, however, was not able to reproduce the structure of real complex networks, especially the extremely heterogeneous degree distribution. A second major model of network formation increasing the realism of Erdös-Rényi graphs was developed by Watts and Strogatz (1998). Their pathbreaking approach captured the fact that many networks had a distinct local clustering. They started with a lattice (regular) network, where each node is solely connected to its two nearest neighbors on each side, and extended it so that shortcuts between distant nodes could exist, just by rewiring two distant nodes with probability p (Fig. 2.6a). As p increases, the model becomes closer to an Erdös-Rényi graph. The model thus captures the coexistence of local structure (clustering) and short path lengths in real networks. Barabási and Albert (1999), building on the previous work by Simon (1955) and Price (1965), thought about a simple model of network formation able to generate scale-free networks. The model is known as preferential attachment, and it works as follows. There is a set of initial nodes randomly connected, such as in the Erdös and Rényi model. A new node is then introduced and linked to one of the existing nodes. The key aspect is the process of selecting the node with which it will interact. The probability of interacting with a given node is proportional to the degree of that node. Once this is linked, a new node is introduced, and the process keeps going on for a certain number of iterations (Fig. 2.9). This is a kind of “rich get richer” process, so that already well connected nodes have a higher probability of having yet more interactions. At the end, we will end up with a scale-free network with a few hubs (Barabási and Albert, 1999). The preferential attachment has two obvious merits that have stimulated an enormous amount of research in recent years: First, it provides a sound mechanism for the observation of scale-free distributions in

32

t1

CHAPTER 2

t2

t3

t4

Figure 2.9. The Barabási and Albert (1999) model starts with a small network to which a new node (white) adds at each time step (t1 to t4 ) and connects to an older node (grey) with probability proportional to its degree, k.

many empirical networks. Second, it describes a process for network evolution, and its time dependency, allowing dynamic models to be incorporated. As the reader can notice, the preferential attachment is defined for onemode networks. In forthcoming chapters we will describe an adaptation of this model for two-mode networks, which will help explain a few differences in the structure of mutualistic networks, as opposed to other complex networks. ECOLOGICAL NETWORKS The study of mutualistic networks is by no means unrelated to previous studies of other systems in ecology (Proulx, Promislow, et al., 2005; Bascompte, 2009). Either implicitly or explicitly, networks have been a powerful representation of ecological communities since the 1950s or even before (Egerton, 2007). To help put our results on mutualistic networks into a broader context, in this section we briefly review similar applications of networks to other ecological systems. It is by no means an exhaustive account. We will consider two broad types of ecological networks, namely, spatial networks where nodes represent discrete habitat patches, which may be linked through dispersal or gene flow, and networks of interactions. The latter broad category can be decomposed in species-level interacting networks such as food webs and individual-level interacting networks such as social and epidemiological networks. Spatial Networks Metapopulation theory has become one of the most important areas of research in ecology. As previously pristine habitats become transformed into an archipelago of small habitat patches surrounded by an ocean of unfavorable habitat, we urgently need a theoretical framework describing the persistence

A N I N T RO D U C T I O N T O C O M P L E X N E T W O R K S

33

of such fragmented populations. Metapopulation theory originated with the classic paper by Levins (1969). It described a spatially implicit model of a metapopulation assuming an infinite number of equal patches and a dynamical balance between local extinction and recolonization from other patches. Since then, this field has grown very fast both theoretically and experimentally (Hanski, 1999). From the initial spatially implicit models, ecologists moved toward spatially explicit lattice models (Bascompte and Solé, 1996). This was a first step toward considering explicit space, a kind of regular lattice where each site had an identical number of nearest sites. This representation of a landscape was not realistic. The spatially realistic metapopulation models addressed this issue and described metapopulation processes for real populations in real landscapes (Hanski, 1999). These latter metapopulation models can actually be viewed as spatial networks, although the explicit consideration of graph theory in a spatial context was incorporated by an influential paper by Urban and Keitt (2001). These authors used graph theory and the concept of minimum spanning tree to analyze the connectivity of habitat patches reachable by the Mexican northern spotted owl (Urban and Keitt). This paper spurred a tradition of using network theory to describe spatial ecology (Holland and Hastings, 2008; Dale and Fortin, 2010; Gilarranz and Bascompte, 2012). Examples include the consideration of riverine networks (Fagan, 2002) and patches of ponds to asses amphibian persistence in stochastic environments (Fortuna, GómezRodríguez, et al., 2006). Perhaps the single most distinctive characteristic of spatial networks is that they are able to represent ecological processes, even in cases where there is no further demographic information, such as in the case of metapopulations (Gilarranz and Bascompte, 2012). Spatial networks can be used as a simplified representation of heterogeneous landscapes and a framework to provide robust predictions about their robustness based on topological measures (Fortin and Dale, 2005; Dale and Fortin, 2010). More recent papers have addressed networks of gene flow among populations (Dyer and Nason, 2004; Rozenfeld et al., 2008) or individuals (Fortuna, García, et al., 2008). Particularly relevant to this discussion is the pioneering paper by Dyer and Nason (2004). These authors provided a bright, methodological approach to study genetic network, and a justification of their value. As noted by these authors, previous metrics such as Fst , AMOVA, and Nei’s distance describe genetic structure in terms of averaging statistical measures, which cannot adequately capture the complexity of intrapopulation dependences. Spatial networks of gene flow (population graphs, as these authors call them) describe the statistical relationships of all populations simultaneously rather than in a pairwise fashion (Dyer and Nason). Although

34

CHAPTER 2

traditional statistical approaches such as AMOVA determine if there is a significant variability, the network approach shows how this variability is distributed among populations. Further advantages of this exercise are allowing a visual representation and the application of the battery of tools developed by network theory (Dyer, 2007). As an example of such a tool, Fortuna, Albaladejo, et al. (2009) characterized the modularity of networks of genetic variability across several plant species in a fragmented landscape in southern Spain. These modules—containing groups of patches holding genetically similar populations—can be thought of as evolutionary significant units or management units (Fig. 2.10). The approach by Fortuna, Albaladejo et al. is also able to identify the most relevant patches from the point of view of the cohesion of the entire network. Food Webs Food webs depict who eats whom in ecological communities (May, 1973; Cohen, 1978; Cohen, Briand, et al., 1990; Pimm, 1982). Probably the first food web ever documented was by Lorenzo Camerano in 1880 (Cohen 1994; Egerton 2007; Fig. 2.11). This Italian author was motivated by the need to determine the beneficial and detrimental effects of different species from the human viewpoint using an aggregate representation of the food web, with groups of species in different compartments, rather than an individual representation of each species role (Egerton). Camerano’s analysis is very insightful, introducing concepts such as equilibrium and stability of species abundances and cascading effects through indirect influences among species (Bersier, 2007). Cohen suggests he was inspired and motivated by the earlier diagrams used by Darwin. A more detailed food-web analysis would arrive with the work of Pierce et al. (1912), who were also interested in the control of cotton crop pests to promote bowl weevil eradication by encouraging its predators and parasites. These representations illustrated the complexity of ecological interactions among species, representing the flux of energy through different trophic levels. Later on, pioneering work on food webs was developed by authors such as Lindeman (1942) and Odum (1956). Food webs were used as a pictorial representation of the complexity of ecological communities (Egerton, 2007). The earlier work on food webs was mainly focused on global descriptors such as connectance, compartments (Pimm and Lawton, 1980), fraction of top predators (Cohen, 1978), and invariance (Sugihara, Schoenly, et al., 1989). Similarly, classical work on food webs already addressed the role of interaction strength (Paine, 1992) and experimentally showed the importance of some predators for the whole community. This work served to coin the concept of a keystone species (Paine, 1969).

d

c

Figure 2.10. Modular organization of the spatial genetic variation in three out of four plant species studied in a fragmented landscape in Andalucía, Southern Spain. Nodes indicate habitat patches inhabited by one population of these four species. Their position reflects the geographic location, and their size is proportional to the intrapopulation genetic variability in relation to the total variability. Interactions between populations depict significant genetic similarity (proportional to their thickness) once the genetic similarity to other populations has been accounted for. These networks contain the minimum link set describing the pattern of genetic covariance among populations. Node shading indicates the module it belongs to. Modularity is significant in (a)–(c) but not in (d). Species are Cistus salvifolius (a), Myrtus communis (b), Pistacia lentiscus (c), and Quercus coccifera (d). Based on Fortuna, Albaladejo, et al. (2009).

b

a

36

CHAPTER 2

Figure 2.11. One of the earliest representations of a food web, evidencing the network pattern of interactions among species in an ecosystem is due to Lorenzo Camerano in 1880 (Cohen, 1994; Egerton, 2007). He used an aggregate representation with distinct trophic levels from the “vegetation” to “phytophagous beetles” and “parasitic insects.”

From a theoretical point of view, the classic paper by Robert May on stability and complexity in model ecosystems had a tremendous influence in pointing out that food web structure is related to stability (May, 1972). Since then, myriads of papers have explored this relationship with more sophisticated models or larger data sets. May proved that in randomly built food webs, complexity begets instability. Thus, real food webs need to show structural patterns that counterbalance such a tendency. May already suggested that a compartmentalized food web may be more stable. Pimm (1979) challenged this conclusion on the basis of his study of more realistic food web models. He found that compartments could make such food web models more unstable. Since then, a rich research agenda has explored the existence of compartments in food webs and their effects on population and community dynamics (May, 1972; Pimm, 1979; Pimm and Lawton, 1980; Girvan and Newman, 2002; Krause, Frank, et al., 2003; Rezende, Albert, et al., 2009; Guimerà, Stouffer, et al., 2010; Stouffer and Bascompte, 2011).

A N I N T RO D U C T I O N T O C O M P L E X N E T W O R K S

37

Another direction of theoretical work deals with developing the simplest model able to reproduce the complexity of real food webs. This research is related to the models of network buildup reviewed previously, but adds explicit ingredients from the ecology of species. The first such model, the cascade model by Cohen, Briand, et al. (1990), assigns each species a random number uniformly drawn from the [0, 1] interval. Each species has a certain probability of consuming species with values lower than its own. This simple rule encapsulates the fact that large animals eat smaller animals. This was followed by the niche model, which also assumes that large animals eat smaller ones but constrains animals to eat species that have a niche value within a certain range (Williams and Martinez, 2000). This model was claimed to accurately predict a wide range of empirical properties of real food webs, despite its simplicity. It has been intensively used in the last few years to generate realistic models of food webs. Further models aimed at incorporating phylogenetic signal (Cattin, Bersier, et al., 2004) or synthesizing the relevant elements reproducing realistic food web properties (Stouffer, et al., 2005) have been recently described. In the last few years, a new generation of higher-quality food webs has been analyzed with new tools that have illuminated community organization (Martinez, Hawkins, et al., 1999; Dunne, Williams, et al., 2002a; Belgrano, Scharler, et al., 2005; Bascompte and Melián, 2005; Montoya, Pimm, et al., 2006; Pascual and Dunne, 2006; Otto, Rall, et al., 2007; Berlow et al., 2009; Stouffer and Bascompte, 2010). To some extent, these papers have followed the recent wave of enthusiasm on complex networks. The influential paper by Albert, Jeong, et al. (2000) on the architecture of the Internet has spurred papers looking at the connectivity distribution of food webs and performing node deletion simulations by Albert, Jeong, et al. to explore the fragility of food webs in front of species extinctions (Solé and Montoya, 2001; Dunne, Williams, et al., 2002b; Camacho, Guimerà, et al., 2002; Memmott, Waser, et al., 2004). Host-Parasitoid Networks Parasitoids are a very specific type of predatory arthropods that lay their eggs inside, on top, or near their hosts. The host is later used as food by the feeding larvae. This type of interaction is extremely abundant in nature and has generated very important empirical and theoretical work. Because this predatory interaction is simpler than other ones, host-parasitoid interactions have been very amenable to mathematical modeling (Hassell, 1978). Obviously, this is by nature a predatory interaction, so it should be included the food-web section. However, host-parasitoid interactions have been successfully represented and studied as two-mode networks, in the

38

CHAPTER 2

same way as mutualistic networks. Charles Godfray and colleagues have produced very important contributions using weighted networks displaying several layers of information such as species abundance and intensity of interaction (Memmott, Godfray, et al., 1994; Müller, Adriaanse, et al., 1999). These studies allowed a community-wide quantification of parasitism rates and the role of indirect interactions (van Veen, Morris, et al., 2006; Müller, Adriaanse, et al., 1999). These indirect interactions can be pervasive in their evolutionary effects. For example, Ives and Godfray (2006) report that the interaction pattern between herbivorous caterpillars and their parasitoids and hyperparasitoids is largely driven by the phylogenetic effects of host plants in shaping the interactions with the caterpillars; these phylogenetic effects of host plants on the phytophagous insects scale up with a phylogenetic signal on the interaction pattern between the insects and their parasitoids. Host-parasitoid networks are based on antagonistic interactions, and thus ecologists would expect them to show networks with contrasting architectures to those found in mutualistic networks (Thompson, 2006). Indeed, whereas mutualistic interactions may facilitate the incorporation of new species through convergence and complementarity of traits among interacting species, antagonisic coevolution between predators and prey may favor escalating arms races among groups of interacting species. This is expected to generate compartments with species sharing some reciprocal specialization in defenses and counterdefenses (Thompson, 2006). Epidemiological Networks Epidemiological networks first assumed (as early theoretical ecology) wellmixed populations in which each individual has the same probability of interacting with each other. This type of mean-field approach allowed mathematical tractability and, thus, the finding of simple, straightforward solutions. An important example is the system of ordinary differential equations named SIR (susceptible-infected-recovered). One example of the type of result obtained with these mixed, or homogeneous, models is the concept of eradication threshold. The idea is that it is not necessary to vaccinate the entire population for a disease to disappear. Once a critical fraction of the susceptible population is vaccinated, the disease vanishes (Anderson and May, 1992). This is the same mechanism underlying extinction thresholds in models of metapopulation persistence in fragmented landscapes (Lande, 1987; Nee, 1994; Bascompte and Solé, 1996) or in the eradication of agricultural pests (Liebhold, Bascompte, 2004). Later, as we have seen for the case of spatial networks, epidemiologists used spatially explicit models in which individuals interact with a similar fraction

A N I N T RO D U C T I O N T O C O M P L E X N E T W O R K S

39

of nearest-neighbor individuals. Similarly, different approaches introduced heterogeneity (both spatial and social) and explored how such a heterogeneity would affect the basic reproductive number, a parameter indicating the average number of infections produced by an infected individual within a susceptible population (Anderson and May). This work, partly developed to understand the spread of the human immunodeficiency virus in heterogeneous societies, already showed that heterogeneity increased the basic reproductive number (Anderson and May, 1992). Physicists, in turn, have more recently been interested in other types of viruses: computer ones. Despite the wide use of antiviruses in personal computers, viruses have not been entirely eliminated. This observation motivated research on the dynamics of computer viruses on complex networks (PastorSatorras and Vespignani 2001a, b). As noted before, we already knew that the Internet is very heterogeneous, with a power-law degree distribution. The idea, then, was to study simple epidemiological models on scale-free networks. The infection never disappears from these scale-free networks: the eradication threshold has just vanished (Pastor-Satorras and Vespignani) . This result confirms the previous work finding an increase in the basic reproductive number in epidemiology (Lloyd and May, 2001). The precedings may sound important for technological networks, but it is also relevant for our understanding of sexually transmitted diseases. One paper by Liljeros, Edling, et al. (2001) showed that the network of sexual contacts is extremely heterogeneous, following a power law. Although the bulk of individuals have only one or a few sexual partners, a few individuals have dozens and even hundreds of partners. In this situation, theory predicts that it will be very difficult to eradicate a disease such as AIDS (Anderson and May, 1992). The good news, however, is that such a resistance is very much dependent on the few hubs. Vaccination campaigns focusing on these few hubs would be much cheaper and effective than a random vaccination of a fraction of individuals. This is a nice example proving the importance of network architecture for dynamics. Social and Individual Networks Other models of social networks have emphasized the role of network architecture for the evolution of cooperation. Interestingly enough, the structure of the underlying social network greatly affects the outcome of evolutionary games. For example, although some networks amplify selection, others reduce selection or fix any advantageous mutations (Lieberman, Hauert, et al., 2005). Similarly, an extension of the famous game model of the Prisoner’s Dilemma on networks of different structure showed that heterogeneous networks such

40

CHAPTER 2

as the scale-free ones highly induce the appearance of cooperation (Santos, Rodrigues, et al., 2006). More recently, several papers have extended theory on social networks to empirical data and explored the relationship between network structure and risk of disease spread. To achieve this, different individual organisms are represented as the nodes of the network and the links illustrate interactions among them (e.g., social interactions, mating, sharing of roosts and nest sites, etc.). Guimarães, de Menezes, et al. (2007) applied network theory to describe the social structure and interactions among 58 individual killer whales. Each individual is a node, and a link between two individuals is established if they were recorded at least once in the same group. These authors found that the structure of this social network increases the risk of disease outbreak (Guimarães, de Menezes, et al.). Similarly, Fortuna, Popa-Lisseanu, et al. used a network approach to address the utilization patterns of roosting trees by a population of bats in a city park in Sevilla, Spain. They found three modules corresponding to groups of bats having a similar use of roosting trees. The study illustrated the role of a few trees in linking across the three modules, which would provide managers an objective tool for identifying the more relevant trees for the spreading of a disease through this population. To explore to what degree the structure of this social network affected the pattern and shape of disease spread, Fortuna, Popa-Lisseanu, et al. used a simple susceptible-infected-epidemiological model running on this network and on a population of randomized networks. This provided a way to correlate measures of network structure and dynamics. The modular structure of this system explains that the time for disease to spread through the entire population was significantly lower than what would be expected by similar random networks missing such a modular structure. However, this rate of infection very much depends on the tree of initial spreading (Fortuna, Popa-Lisseanu, et al., 2009). Wolf and Trillmich have detected nested modules in the social network of the Galápagos sea lion Zallophus wollebacki. The network is partitioned into large modules roughly corresponding to spatial areas. Inside each such module, there are smaller cliques representing an individual closer social neighborhood. As a consequence of phylopatry, genetic structure is formed within these cliques (Wolf and Trillmich, 2008). Along this line, individual-based networks have been used by Fortuna, García, et al. (2008) to examine the mating networks among individual trees in a population, showing a marked pattern of assortative mating depending on the spatial locations of the trees. Using genetic markers, they built a spatially explicit network of pollination events between any two trees in a population of Prunus mahaleb, an insect-pollinated plant.

A N I N T RO D U C T I O N T O C O M P L E X N E T W O R K S

41

Then, they used complex network tools to characterize a marked structure of such a mating network with well-defined modules or compartments formed by distinct groups of mother trees and their shared pollen donors. Similarly, individual-based networks focusing on plant-mating events mediated by pollinators have been used recently by Gómez, Jordano, et al. (2011) to explore the functional consequences of generalized insect pollination in eight populations of Erysimum mediohispanicum (Brassicaceae). Plants in these individual-based networks are now connected to other conspecifics with links defined by their sharing of pollinators. Their data reveal a consistent relationship between topology and function of the networks, because variation across populations in the average per-capita production of juvenile plants was positively and significantly related with network nestedness, connectivity, and clustering. In other words, the way individual plants mate through sharing of pollinators influences population-level reproductive success. Ryder, McDonald, et al. explored how social network structure affects the individual behavior of a population of wire-tailed manakins (Pipra filicauda). This species shows a conspicuous lek mating. While some males perform their mating displays in isolation, other males coordinate such displays. An interaction between two males means that these two males have been collectively displaying. The authors measured properties of this social network, such as degree distribution, and related them to measures of individual performance such as reproductive success. They found that the number of interactions of a male is strongly correlated with its ability to become a territory holder. The heterogeneity of this network is compatible with the conditions for the emergence of cooperation, as predicted by previous theoretical models (Ryder, McDonald, et al., 2008). SUMMARY A large number of complex systems can be represented as networks. In the last few years, there has been a growing body of work characterizing the architecture of such complex networks. The underlying rationale is that network structure highly affects network robustness. Recent findings have underlined the existence of universal network architectures across different systems. This may reflect the existence of common mechanisms of network buildup and similar constraints for network persistence. Here we review some of the measures of network structure, basic models of network formation, and previous work on networks in other ecological systems. This chapter thus provides the tools and concepts for characterizing mutualistic networks and placing them into a broad context.

CHAPTER THREE

The Structure of Plant-Animal Mutualistic Networks

In the last chapter we provided the basic concepts from network theory. They allow us the visualization and analysis of networks of interactions in highly diverse communities. These concepts also provide ways to statistically compare network patterns across communities. The first comparative study looking at mutualistic interactions from a network perspective is arguably by Jordano (1987). This paper anticipated the most recent interest in mutualistic networks and already applied ideas from food webs to mutualisms. Apparently nothing happened for awhile; only in the last decade or so has there been a significant advance in our understanding of mutualistic networks (Fonseca and Ganade 1996; Memmott 1999; Memmott and Waser 2002; Bascompte, Jordano, et al. 2003, 2006a; Jordano, Bascompte, et al. 2003; D. P. Vázquez and Aizen 2004; Thompson 2006; Olesen, Bascompte, et al. 2006, 2007). These first papers have described the structure of mutualistic networks. We will briefly summarize the results on network structure in this chapter. (Appendix A lists the main statistical indices used in the analysis of mutualistic networks.) In subsequent chapters we will discuss the potential mechanisms leading to such network patterns and the implications of network architecture for coevolution, community ecology, and conservation.

DEGREE DISTRIBUTION Degree distributions are usually represented as either the frequency distributions of the number of interactions per species (Fig. 3.1a) or the cumulative frequency of species with at least k interactions (Fig. 3.1b). Jordano, Bascompte, et al. explored the connectivity distribution of 29 plant-pollinator and 24 plant-frugivore networks. Most networks exhibited power-law-type fits to the degree distribution of the number of interactions per species, but with some

T H E S T RU C T U R E O F P L A N T- A N I M A L M U T UA L I S T I C NETWORKS

a

b

700

1.000 0.500

600 500 400

P(k)

Frequency

43

300 200

0.050

0.005

100 0

0.001 0

10

20

30

40

1

k

2

5

10

20

k

Figure 3.1. The frequency distribution (degree distribution) of the number of interactions per plant species k in a plant-animal assemblage (a) and the corresponding cumulative distribution (b), that is, the probability P(k) of finding a plant species with at least k interactions with pollinator species. Lines in (b) indicate the best fits to a power-law function (black line), truncated power law (dashes), and exponential (short dashes). Blank dots indicate the empirical data, corresponding to a Mediterranean shrubland plant-pollinator network in Greece (Petanidou and Ellis, 1993). An analogous degree distribution could be constructed for the animals.

significant deviations (see Appendix B). Their results illustrated the generality of degree distributions with a power-law regime but decaying as marked cutoff, that is, truncated, power laws or broad-scale networks. This type of distribution is often called a Zipf’s distribution when discrete data are involved (such as in our case, where we analyze the distribution of the number of interactions per species), or Pareto’s distributions for the continuos case. About 65.6% of the connectivity distributions both for plants and animals followed this broadscale distribution, whereas 22.2% of the cases were best fit by a scale-free distribution, and the rest were best described by an exponential distribution or had no fit at all (Jordano, Bascompte, et al., 2003). Let us consider in more detail the shape of a truncated power law that can be described by the following equation: P(k) ∝ k −γ e−k/kc ,

(3.1)

where the new term in relation to equation (2.1) is e−k/kc , which defines the exponential truncation. This exponential cutoff means that as the number of interactions reaches the critical kc value, the probability of finding more connected species drops faster than expected for a power law (Fig. 3.1b). There is a limit to the degree of generalization of the most connected species. Mutualistic networks are still very heterogeneous, that is, the number of

44

CHAPTER 3

interactions per species varies much more that expected by chance across species. Although the bulk of species have a few interactions, a few species have a very large number of interactions (although not as many as for a scale-free network). The truncation of degree distributions has been generally attributed to the action of constraints on the addition of links per node as the network grows (Amaral and Ottino, 2004). We will address these constraints in our search for mechanisms of network assembly in Chapter 4. Interestingly for our search of general, invariant patterns, both plantpollinator and plant-seed disperser networks show the same degree distribution. More than that, regardless of the differences in ecosystem productivity, species composition, and latitude, mutualistic networks have a common and well-defined connectivity distribution (Jordano, Bascompte, et al., 2003). To look for invariant properties, physicists tend to use scaling functions where distributions across several systems can be overimposed through appropriate normalizations (Solé and Bascompte, 2006). For example, in our case we can scale the plots of connectivity distributions by showing the scaled cumulative distributions of links per species, k −γ P(k), versus the scaled number of links per species, k/kc (Fig. 3.2). This allows for comparisons across networks with different shapes and sizes. Jordano, Bascompte, et al. (2003), following a suggestion by Bartolo Luque, showed that different communities collapse to a simple scaling function. Despite the intrinsic variation of the empirical mutualistic networks examined, this is an indication of a shared pattern of internal topology independent of scaling considerations (Sugihara, Schoenly, et al., 1989; Bersier et al., 1999). This is particularly relevant in our search for generalities beyond obvious differences at other levels. It represents the type of approach that a physicist would use. Similarly, Stouffer, et al. (2005) have used scaling functions to unravel universal properties in food-web structure. In this case, the authors plotted the cumulative distribution of the scaled number of prey per predator species. All the data for 11 communities collapse onto the same curve, indicating a common, underlying architecture regardless of biological differences in number of species, connectivity, and other properties.

NESTEDNESS The concept of nestedness, applied to network structure, does not have its roots in physics as other measures. It was first introduced in the field of island biogeography and has been widely used since then to characterize the patterns of species presences across islands (Patterson and Atmar, 1986; Atmar and Patterson, 1993).

T H E S T RU C T U R E O F P L A N T- A N I M A L M U T UA L I S T I C NETWORKS

Pollination

45

Seed dispersal

Animals

100

10–2

kγ P(k)

10–4

10–6

Plants 100

10–2

10–4

10–6 10–2

10–1

100

101

102

10–2

10–1

100

101

102

k/kc Figure 3.2. Data collapse to a scaling function when plotting the scaled cumulative distributions of links per species, k γ P(k) versus the scaled links per species (k/kc ), where kc is the cutoff value of each distribution (the k-value where the distribution departs from the power-law fit). Panels represent 21 pollination networks (left) and 11 seed dispersal networks (right). A collapse suggests invariant topologies of web organization despite variation in size and number of links. Based on Jordano, Bascompte, et al. (2003).

In the previous section, we were looking at the number of interactions per species, without noticing the identity of the partners. Now we will consider species identities and will look at whether the set of animals interacting with one plant species, for example, overlaps with the set of animals interacting with another plant species (Fig. 3.3a, d), such that the partners interacting with a given species form a proper subset of those interacting with more generalized species.

!

b 5

30

25

20

15

10

Heteroptera

Hymenoptera

Diptera

c

5

1 2 3 4 5 6 7 8 9 10

10 35

40 45

d

1 2 3 4 5 6 7 8 9 10

Animals

50

55

60

65

70

75

Animals

30

1 2 3 4 5 6 7 8 9 10

25

Animals

20

1 2 3 4 5 6 7 8 9 10

15

Figure 3.3. Graph (a) and matrix representation (b) of an arctic plant-pollinator interaction network (Olesen, Bascompte, et al., 2008). The interaction pattern in (b) shows a significant nestedness, with most pairwise interactions located in the matrix to the left of an isocline of perfect nestedness, as in (c), which depicts a perfectly nested matrix. The empirical pattern markedly deviates from a random pattern of interactions, as in (d). In a nested pattern, the most generalized species in a number of interactions tend to interact among themselves, forming a core in the interaction matrix [inset in (c)] whereas a species with few interactions, more specialized, tend to interact with species in the core. The nested pattern is visually appreciable in (a), given that nodes have been sorted in decreasing number of interactions for each pollinator group of nodes: the density of links within the elliptical graph is not homogeneous, with virtually no links among species with few interactions. Links occur predominantly among generalists and among specialists and generalists. Modified from Bascompte and Jordano (2007).

PLANTS

Lepidoptera Plants

a

Plants

T H E S T RU C T U R E O F P L A N T- A N I M A L M U T UA L I S T I C NETWORKS

47

Bascompte, Jordano, et al. (2003) first introduced this concept in interaction networks as a macroscopic description of the interaction patterns between plants and their animal mutualists. These authors analyzed 25 plant-pollinator and 27 plant-frugivore networks, concluding that these networks are neither randomly organized nor assembled in compartments arising from tight, reciprocal specialization. Plant-animal mutualistic networks were found to be highly nested, with specialists (e.g., species with low degree) interacting with species that form perfect subsets of the species generalists interact with (Fig. 3.3b). In other words, if we rank plants from the most specialists to the most generalists, the set of animals the ith-most generalist plant interacts with is contained in the larger set the (i − 1)th-most generalist plant interacts with. This gives rise to an organization such as in Chinese boxes, with smaller boxes within larger ones (Bascompte, Jordano, et al., see Appendix C for the explicit measures of nestedness). A nested matrix has two significant implications that are potentially relevant for our understanding of mutualistic networks (Fig. 3.3b). On one hand, nested matrices have a core of generalist plants and animals interacting among themselves. A few species control a very high number of interactions, which may imply that there is functional redundancy and the possibility for alternative routes for system persistence in the face of the disappearance of some of these interactions. This core of generalists has also been shown empirically to be more resistant to extinction (Aizen, Sabatino, et al., 2012). A few species contained in this core have the potential to drive the selective forces experienced by other, more specialist species that are attached to this core. This is a network structure with the potential to act as a coevolutionary vortex sensu Thompson (2005). As mutualistic networks are nested, this indicates that the importance across species in driving such a coevolution can be highly variable, with species in the core most likely playing a major role. The second important implication of a nested network is the existence of asymmetries at the level of specialization. Nested matrices characteristically have no interactions between species with few interactions (specialists), conferring a distinctly asymmetric pattern in the specificity of the interactions (Fig. 3.3). Specialists in a nested network tend to interact with the most generalist species (Fig. 3.3b). This asymmetric specialization was independently reported by D. P. Vázquez and Aizen (2004) and further confirmed by Joppa, Bascompte, et al. (2009) using a more conservative null model. The finding of this asymmetric specialization perfectly illustrates the need to tackle a network approach to mutualism. From the knowledge of isolated species—for example, knowing how many interactions each has—we could not have been able to anticipate this property of the network. It has been a long

48

CHAPTER 3

way from the first studies assuming mutualisms are reciprocally specialized pairwise interactions. As described in the historical overview of Chapter 1, Schemske (1983) and Waser, Chittka, et al. (1996) already emphasized the generalization level of these mutualisms, that is, the fact that some species have a very large number of interactions. The network approach, in turn, has advanced this discussion by focusing on how a wide group of species with contrasting generalization levels interact among themselves to form functional networks of mutual dependences. We can assume that generalist species tend to be more abundant and less fluctuating than specialists because they rely on so many other species. Therefore, this asymmetric structure may provide pathways for the persistence of rare, specialist species (Bascompte, Jordano, et al., 2003). Note however, that in this section we deal exclusively with a description of structural patterns and have no information on dynamics. This will have to wait to Chapter 6. In summary, as it happened for the connectivity distribution, the bulk of mutualistic networks, regardless of the type of mutualism and other biological details, have a common, general architectural structure.

SMALL WORLD Olesen, Bascompte, et al. (2006) analyzed the small-world property in a dataset of 22 pollination networks. As already noted in the previous chapter, a short path length means that we need only a few steps to connect any two species in the network, whereas a high clustering coefficient means that the species interacting with a given species have a strong tendency to also interact among themselves. Before proceeding to analyze these pollination networks, Olesen, Bascompte, et al. transformed the original two-mode networks into one-mode networks. Therefore, we now have two one-mode networks for each original two-mode network. One of them depicts the pattern of shared plants among pollinators, with pollinator species linked whenever they share two plant species for visitation. The other one-mode network depicts the shared pattern of pollinator services among plants: two plant species will have a link among them whenever they share a given pollinator species. This transformation was historically first made by Joel Cohen when depicting food webs as networks of trophic similarity linking two species if they shared the same resource (Cohen, 1978). Cohen used this transformation to test whether there are discontinuities in the overlap of prey use among all the species in a community, something now called niche intervality (Cohen, 1978; Sugihara, 1983; Cattin, Bersier, et al., 2004;

T H E S T RU C T U R E O F P L A N T- A N I M A L M U T UA L I S T I C NETWORKS

49

Stouffer, Camacho, et al., 2006). More recently, Guillaume and Latapy (2006) and Kondoh and Kato (2010) have used a similar approach for the inverse case: obtaining the bipartite representation of any one-mode network. There is a strong transformation, therefore from the original bipartite network to the two transformed unipartite networks. The procedure is simple, because given an adjacency matrix M representing the links between A animal species (rows) and P plant species (columns), the unipartite projection for the rows is obtained by the product of M by its transpose, M T . To get the projection for the plants, we first transpose M to get the plants as rows and then multiply this by its transpose. In both cases we obtain a squarematrix representative of the unipartite networks for each group of species (Fig. 3.4). Before proceeding, and to assess to what degree the analysis of the transformed networks is informative about the original networks, Olesen, Bascompte, et al. (2006) calculated the correlation between the values of several metrics for the original and transformed networks. These metrics included the total number of interactions and the linkage density (average number of interactions per species) and can be measured in both types of networks. In the majority of comparisons, there was a significant correlation between the values of the metrics of the original and the transformed networks. Thus, this is an indication that, at least to some degree, information on the transformed network is useful to characterize the untransformed one. The transformed one-mode pollinator and plant networks have a marked small-world property. Specifically, the mean average path lengths of plants and animals were, respectively, 1.47 and 1.73. There is a strong correlation between these two values for each pair of transformed networks, meaning that both the transformed plant network and the transformed animal network behave similarly. The average clustering coefficients for the plants and the animals were, respectively, 0.84 and 0.85. In this case, and in opposition to path length, there was no correlation between the pair of clustering coefficients coming from the same untransformed network. Combining the values for the path length and the clustering coefficient, we can conclude that pollination networks have very strong small-world properties. The most direct implication of this finding is that each pair of species in these mutualistic networks is very close to each other, which means that a perturbation in any species can easily and quickly affect any other species in the entire network (Olesen, Bascompte, et al., 2006). At the same time, interactions are localized within neighborhoods, as identified by the relatively high clustering coefficient. To look in more detail to this localized structure, we turn now to explore modularity.

50

CHAPTER 3

a 1

2

3

4 Animals

Plants A

B

C

D

E

F

b Plants

Animals

B

2 D

4

1

A

E

C

F 3

Figure 3.4. Plant-animal mutualistic networks are bipartite graphs showing two distinct sets of nodes, animals and plants, with interactions between (but not within) sets. From this bipartite graph, one can construct two unipartite projections (b) of each set (Guillaume and Latapy, 2006). These unipartite graphs represent the shared pattern of partners among the species in each set.

MODULARITY As described in the previous chapter, modularity is the tendency of a network to be organized in distinct clusters, where species within a module tend to interact with a much higher frequency among them than they do with species from other modules (Fig. 3.5; see Appendix D for a quantitative definition). Olesen, Bascompte, et al. (2007) studied the modular organization of pollination networks using an expanded data set of 51 networks, including almost 10,000 species and twice as many interactions.

Figure 3.5. Modular organization of a seed-dispersal network in the Brazilian Pantanal. Circles, animal species; squares, plant species. Circle size refers to animal body mass (large circles representing species with body mass ±4.5 kg), whereas the size of squares refers to fruit diameter (large squares representing species with fruit diameter ±9,5 cm). The network includes five modules (different shades of grey): two bird modules (top), each with 22 species; two mammal-dominated modules (bottom) with 25 and 18 species, respectively; and a fish module with 4 species. Based on Donatti, Grüber, et al. (2011).

52

CHAPTER 3

The majority of large networks were significantly modular. Each one of these modules includes a few species with convergent morphological traits. To some extent, modularity analysis provides a quantitative, bottom-up way to detect potential coevolutionary units. Similar to the study of network motifs in complex networks (Milo, Shen-Orr, et al., 2002) and food webs (Bascompte and Melián, 2005; Stouffer, Camacho, et al., 2007), these groups of strongly interacting species can be regarded as the basic building blocks of mutualistic networks (Olesen, Bascompte, et al., 2007; Jordano, 2010). Another interesting aspect of the modularity analysis is that we can classify the roles of each species within the network. The original paper by Guimerà and Amaral (2005) classified nodes in relation to two variables, the withinmodule degree and the participation coefficient. The first one ranks species in terms of how well connected they are within their module. The second, on the other hand, measures the connectivity of a node form the point of view of the entire network, measuring how well spread the links of one node are among all the other modules. In pollination networks, only about 11% of the species (including beetles, flies, and small to medium-sized bees) were important from the point of view of connecting several different modules (Olesen, Bascompte, et al.; Fig. 3.6b). Their importance stems from the fact that the disappearance of these species can highly alter the structure of the entire network by losing the connection among modules and moving toward a situation where these modules are more isolated among themselves. This research is important because, in comparison with the previous analysis of network structure, it identifies the role of each species in the network. Indeed, it provides an objective criteria for assessing the role of individual species for the maintenance of the entire network of dependencies among species (Olesen, Bascompte, et al., 2007). The previous measures of network structure are not necessarily independent of each other and may be more or less related. It is no surprise that the first studies of network structure focused on one of these measures in isolation, but later studies started looking at how different measures may relate among themselves (Lewinsohn, Prado, et al., 2006; Olesen, Bascompte, et al., 2007) and to advocate the claim that finding evidence for a pattern does not overrule the possibility for alternative patterns (Lewinsohn, Prado, et al.). For example, understanding how modules are glued among themselves to form the entire network is important in advancing our understanding on how modularity and nestedness are related. Thus, we could conclude that a significantly nested network and a significantly modular one are opposite concepts, but this does not seem to be the case (Olesen, Bascompte, et al., 2007; Fortuna,

a

b

Figure 3.6. The matrices represent the same network; in (a) species are ordered to maximize nestedness and in (b) they are ordered to maximize similarity in patterns of interaction and module affiliation (to seven distinct modules with different shades of grey). Small squares indicate observed pairwise interactions; four species (three animals, in rows, and one plant, in columns) are highlighted as bullets to illustrate species that have interactions with species in different modules. These species are hubs, and in the nested representation of the network are part of the core. A nested architecture thus requires modules as its basic building blocks but also the glueing effect of interactions that the hub taxa provide by connecting different modules. Based on Olesen, Bascompte, et al. (2007).

54

CHAPTER 3

Stouffer, et al., 2010). The particular way by which modules are connected may increase the nestedness of the network (Fig. 3.6). This points toward a picture in which mutualistic networks combine basic building blocks in ways that may enhance cohesion. Most interestingly, this cohesion is provided largely by interactions of species within the core of the nested network. In other words, although modules can be seen as the basic building blocks of complex mutualistic networks, interactions with generalists provide the glue needed to attain the cohesion reflected in a nested architecture. More work is necessary to fully explore the relationships among all these network measures and to generate methods to look simultaneously to several such metrics and weighting, which is the one that gives a more accurate description of the entire network (Allesina, Alonso, et al., 2008; Fortuna, Stouffer, et al., 2010).

WEIGHTED NETWORKS Up to this point we have described the structure of mutualistic networks based on binary, qualitative data. That is, our data described whether two species interact or not but did not give us information on the strength of such an interaction. We already knew from the study of food webs that interaction strengths between predators and their prey are highly variable (Ulanowicz and Wolff, 1991; Paine, 1992; Fagan and Hurd, 1994; Wootton, 1997; Bascompte, Melián, et al., 2005). The previous results using just qualitative (presenceabsence) interactions might not be informative enough; in the worst-case scenario, they could be an artifact of mapping variable interactions into a one-or-nothing scheme. This motivated the extension of network analysis to weighted networks, in a similar way than physicists were also doing for other complex networks such as the scientific coauthorsip or the worldwide air transportation network (Yook, Jeong, et al., 2001; Barrat, Barthelemy, et al., 2004; Newman, 2004) and other ecologists had been doing for food webs and host parasitoid networks (van Veen, Morris, et al., 2006; Banašek-Richter, Bersier, et al., 2009). Ideally, the weight of an interaction would describe the per-capita effect of one species on its partner. From a practical point of view, this information is very difficult to obtain for a large number of species. An alternative Figure 3.7. The strength of interactions depicted in qualitative, bipartite graphs (a) can be determined with surrogates such as visitation frequency, consumption rates, or total effect of pollination or seed-dispersal services. In this way, each qualitative link can be decomposed into two dependencies (b): the dependence diPj

a Plants

b

Animals

P

dij A

dji

c 0.6 0.4 0.2 0

Probability

0.6 0.4 0.2 0 0.6 0.4 0.2 0

0.2 0.4 0.6 0.8

1

0.2 0.4 0.6 0.8

1

0.2

0.4 0.6 0.8

Dependence Figure 3.7. (continued) of plant species i on animal species j (e.g., the fraction of all animal visits coming from this particular animal species) and the dependence d jiA of animal species j on plant species i (i.e., the fraction of all visits by this animal species going to this particular plant species). These parameters are frequently obtained during field work where focal observations of animals and/or plants are accumulated. A very general pattern (c) is a strong skew in the frequency distributions of dependence values across species; i.e., most interactions actually involve weak dependencies. In black, plant dependency values; in grey, animal dependency values. Each panel represents an empirical mutualistic network. Based on Bascompte, Jordano, et al. (2006a).

1

56

CHAPTER 3

measure of interaction weight, or dependence of one species on another, is provided by the relative frequency of floral visits or relative frequency of fruits consumed (Jordano 1987; Bascompte, Jordano, et al. 2006a; Fig. 3.7). Thus, the dependence of a plant on a frugivorous bird, for example, is estimated as the fraction of fruits consumed by that bird species in relation to the total number of fruits consumed by all the animal species. Similarly, the dependence of the bird on the plant is estimated as the relative fraction of fruits consumed from the animal that come from that particular plant. Obviously, this is a simplification of the mutualistic effect of one species on another because we are missing the quality component (Schupp, Jordano, et al., 2010)—that is, the fact that one animal can better disperse the seeds of the plant than others. Some evolutionary biologists have seen this as a difficulty in using mutualistic networks to address coevolutionary questions. For example, a plant might provide in its fruit pulp some essential element (e.g., high protein, N, or Ca content) that makes frugivorous animals highly dependent on it irrespective of the visitation frequency to the plant. Interestingly enough, however, Jordano (1987) used different estimates of dependence, incorporating just ingestion rates of fruits for frugivores, yield of nitrogen or energy content, and so on, and found highly consistent results in relation to the estimation of dependence. To further address to what extent the frequency of interaction is a good surrogate for per-capita, or mutualistic, effect, Vázquez, Morris, et al. compared data on frequency of interactions with data on per capita effects of animals on plants for a few cases in which this information was available. They concluded that the frequency of interaction is a good surrogate for percapita effects. The reason is that although there are certainly differences in the quality of the services, the number of visits is so variable across species that this overcomes differences in quality. Besides the evidence from these case studies, D. P. Vázquez, Morris et al. were also able to provide a mathematical justification for this point. Thus, with the data at hand, we can use weighted mutualistic networks to address ecological and coevolutionary questions on the dependencies of one species on others. Pairwise frequency of interaction seems a good surrogate for the total effects of a pairwise interaction in most networks (D. P. Vázquez, Morris et al., 2005).

Dependence and Asymmetry Reinforced by the previous result, we can go back with more confidence to describe weighted patterns in mutualistic networks. As noted in the examples plotted in Fig. 3.7c, the frequency distribution of dependence values is highly

T H E S T RU C T U R E O F P L A N T- A N I M A L M U T UA L I S T I C NETWORKS

57

skewed, with lots of weak values and a few strong dependences (Jordano, 1987; Bascompte, Jordano, et al., 2006a). Up to this point, we have described only the frequency distribution of interaction strengths. Although this is very informative, as noted before, recent studies have moved beyond the distribution of interaction strengths to address the patterning of such interaction strengths. For example, Neutel et al. (2002) have found that weak interactions in long loops contribute highly to food-web persistence. Similarly, Bascompte, Melián, et al. (2005) have shown that a marine food web has fewer combinations of strong interaction strengths in two consecutive levels of a trophic chain in comparison to random expectations. This, as shown by a bioenergetic model, reduces the magnitude of a trophic cascade following the overfishing of top predators (Bascompte, Melián, et al.). Thus, it seems a natural next step to look at how two dependence values are combined within a plant-animal pairwise interaction (Fig. 3.7b). The difference between two pairwise dependence values, measured in absolute value and normalized, constitutes an index of asymmetry in reciprocal dependence values between each animal-plant pair (see Appendix A for an exact formulation). These pairs of values are highly asymmetric (Jordano 1987; Bascompte, Jordano, et al.), as noted in Figure. 3.8. However, we always have to put this type of result in a proper context. We have to be compare it to a random expectation. One way to do this is to shuffle all dependence values between each animal-plant pair and to compare observed asymmetry values against this distribution of new asymmetry values. Is, then, the real network more asymmetric than the random expectation? The answer, taking all data, is no in general. Mutualistic networks are asymmetric but not more asymmetric than expected based on the skewed distribution of dependence values (Bascompte, Jordano, et al., 2006a). As noted previously, although the bulk of dependence values are small, a few are quite strong. If we now repeat the comparison between observed asymmetry values and their randomizations using only these pairwise interactions in which a member of the pair is strong, we then find a significant pattern of pairwise dependence combinations. Thus, in the few cases in which a plant species, for example, is highly dependent on an animal species, that animal tends to rely very little on the plant (Bascompte, Jordano, et al., 2006a). Figure 3.8 illustrates this pattern when pooled data from a number of wellstudied communities is considered. Weak, asymmetric interactions may help mutualistic partners to coexist, because if both plant and animal would depend strongly on each other, a decrease in plant abundance would be followed by a similar decrease in the animal, which abundance, in turn, would feedback

58

CHAPTER 3

Dependence of the plant

a

b

Pollination

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

Seed dispersal

0

0 0

0.2

0.4

0.6

0.8

1.0

0

0.2

0.4

0.6

0.8

1.0

Dependence of the animal Figure 3.8. Interaction asymmetries arising from skewed patterns of dependencies among species in natural mutualistic assemblages. Each dot represents the combined value of dependence in a pairwise interaction. Most pairwise interactions are weak, falling close to the lower-left corner of the panels. Stronger interactions are very often asymmetric, yielding the higher densities of points along the axes in (a) and (b). Pairwise interactions involving strong mutual dependencies [expected to lie close to the upper right corners in (a) and (b)] are extremely rare. Data correspond to several networks of pollination (a) and seed dispersal (b). Based on Bascompte, Jordano, et al. (2006b).

on its partner. This kind of downward loop would be less common in uneven relationships because the plant could recover by relying on a generalist partner that depends on many other species (see Sec. 6.2). Species Strength In the previous section, we looked at pieces of the network, namely, individual links and pairs of links. Of course, we are interested in understanding weak and asymmetric dependences in the context of the whole network. The goal now is to scale up to address how this weighted pattern explains the shape of the entire network. To address this aim, we will now consider the weighted equivalent of the degree distribution. As defined in the previous chapter, a quantitative extension of species degree, species strength, can be defined as the sum of dependences of the animals on a specific plant, or the sum of dependences of the plants on a specific animal. It is a measure of the quantitative importance of a species from the point of view of the other set. Degree just counts the number of links; strength sums the weight of each interaction.

T H E S T RU C T U R E O F P L A N T- A N I M A L M U T UA L I S T I C NETWORKS

a

b

4

c

59

8

10 2

Species strength

0

4

5

0

5

10

15

0

20

d 80

0

10

20

30

0

40

f

e 30

0

5

0

10

0

5

10

15

20

25

40

20 40

20 10

0

g

0 0

50

100

0

h

6

10

20

30

0

40

i

8

20

30

40

20

4 4

10

2 0

0

5

10

15

0

0

5

10

15

20

0

10

15

20

25

Species degree Figure 3.9. Relationship between the number of interactions of a species (degree) and its quantitative extension, species strength. Each panel depicts one set (animal or plant) of a mutualistic network. Based on Bascompte, Jordano, et al. (2006a).

Not surprisingly, there is a positive relationship between species degree and strength. More interesting, this relationship is quadratic, which means that species strength increases faster than species degree (Bascompte, Jordano, et al. 2006a; Fig. 3.9). This means that mutualistic networks are even more heterogeneous when using weighted information. The preceding higher-than-expected strength of generalists can be explained from our understanding of previous levels of network structure. For example, remember that nestedness imposes the requirement that specialist species interact with generalists. Thus, as the number of interactions of a species grows, this species will tend to attract specialist species. Because, by definition, specialists interact exclusively with these generalists, they contribute to a high increase in their strength. Before, we noted the asymmetry in specialization, that is, the fact that specialist species tend to interact with generalist species. In this section, we have focused on asymmetry at the level of links, which certainly builds on the previous asymmetric specialization. Further work should quantify what component of asymmetry at the link level is explained by asymmetry at the species level.

60

CHAPTER 3

COMPARISONS WITH OTHER ECOLOGICAL NETWORKS We may now ask whether the previous network structure is unique to mutualistic networks or similar to other ecological networks. This is important for putting these results in a broad context and to start exploring whether similar mechanisms of network buildup or similar constraints for system functionality and persistence exist. We would like to know, for example, whether mutualistic networks differ from interactions between predators and prey in connectivity, nestedness, and other network features. The broad-scale distribution of connectivities is similar to the distributions found in related ecological networks such as food webs. Dunne, Williams, et al. (2002a) have tackled this question by analyzing the connectivity distributions of a data set of 16 food webs. This general analysis followed the earlier papers by Solé and Montoya (2001) and Camacho, Guimerà, et al. (2002) focusing on only a few food webs. Although Solé and Montoya tested the fit to a power law, Camacho, Guimerà, et al. found stronger evidence of an exponential distribution. The more general study by Dunne, Williams, et al. concluded that, although a few communities are better described by either a power-law or a truncated power-law distribution, the majority of food webs are best fit by an exponential decay. This contrasts with the findings reported here for mutualistic networks described by broad-scale connectivity distributions in the majority of cases. This difference in the connectivity distribution between mutualistic networks and food webs raises the question of what is the ultimate cause for this difference, either biological attributes or differences in sampling intensity, which, in turn, affect network size and connectivity. As we will see in Chapter 4, the bipartite character of mutualistic networks already imposes some differences in the connectivity distribution when there are differences in the size of the two sets. On the other hand, it is certainly possible that a difference in resolution level between food webs and mutualistic networks can also influence this result. Although food webs have a considerable degree of lumping (e.g., several species sharing the same prey or predators are lumped into a single trophic species; Martinez, Hawkins, et al. 1999), mutualistic networks are generally resolved at the level of species—that is, each node corresponds to a taxonomic species. To assess whether differences in aggregation can explain this difference in connectivity distribution, we can perform a computer simulation. Guimarães and Bascompte (unpublished work) used a theoretical food web with a well-defined scale-free distribution and proceeded by aggregating either taxonomically (species from the same general become lumped into a single species) or ecologically (species sharing

T H E S T RU C T U R E O F P L A N T- A N I M A L M U T UA L I S T I C NETWORKS

61

the same prey become a single species). As aggregation proceeded, there was a transition from a power-law to a truncated power-law and to an exponential distribution. This suggests that differences in lumping can certainly affect the differences observed between food webs and mutualistic networks. The finding of nestedness in mutualistic networks has spurred the search for this community pattern in other types of mutualisms and interaction types (Ollerton, Johnson, et al., 2003; Guimarães, Sazima, et al., 2007; Lafferty, Dobson, et al., 2006; Selva and Fortuna, 2007; Sazima, Jordano, et al., 2012). Regarding other types of mutualism besides pollination and seed dispersal, Guimarães, Rico-Gray, et al. (2006) analyzed extrafloral nectary-bearing plants and their partner ants, finding that the values of nestedness for these networks were quite similar to the values previously reported by Bascompte, Jordano, et al. (2003) for pollination and seed dispersal. Similarly, high and significant values of nestedness have also been found in fish parasites (Poulin and Valtonen, 2001), scavenger communities (Selva and Fortuna, 2007), and marine cleaning mutualisms (Guimarães, Sazima, et al.). Interestingly enough, the level of nestedness is lower for symbiotic mutualisms (Guimarães, Machado, et al., 2007). Regarding food webs, if we consider two-mode versions of the original network, such as the ones formed by plants and their herbivores or insect hosts and their parasitoids, were found to be significantly less nested than mutualistic networks (Bascompte, Jordano, et al., 2003). This agrees with the main wisdom of a larger propensity for compartments in plant-herbivore networks (Lewinsohn, Prado, et al. 2006; Thébault and Fontaine 2010; see, however, Thébault and Fontaine 2008; Kondoh and Kato 2010). Another interesting extension of nestedness into food-web studies has been done by Lafferty, Dobson, et al. (2006) in their search of the effects of parasites in food webs. These authors have found that the consideration of the oftentimes neglected parasites increases the level of nestedness of the food web. This, in turn, may increase the cohesion of the whole network and, as we will see in Chapter 6, affect its size and robustness (Lafferty, Dobson, et al.). Nestedness has been also evidenced in the plant world, specifically when analyzing plant-plant facilitative interactions (Verdú and Valiente-Banuet, 2008) and the interactions between epiphytes and their hosts (Burns and Zotz, 2010). The dominance of weak dependences is similar to what has been found in food webs, where a few strong interactions are embedded in a matrix of weak interactions (Ulanowicz and Wolff, 1991; Paine, 1980, 1992; Fagan and Hurd, 1994; Raffaelli and Hall, 1995; Wootton, 1997; Bascompte, Melián, et al., 2005). This pattern of frequency distribution of interaction strengths is potentially very interesting because there is a mounting evidence suggesting

62

CHAPTER 3

that the dominance of weak interactions in food webs promotes community persistence and stability (May, 1973; McCann, Hastings, et al., 1998; Kokkoris, Troumbis, et al., 1999; Bascompte and Melián, 2005). Intriguingly, this pattern of a few strong and many weak interactions seems to become stronger in less structurally complex habitats, where generalist consumers can more easily locate their preferred prey (Tylianakis, Tscharntke, et al., 2007). Regarding the small-world property, mutualistic networks do not seem to differ substantially from food webs (Montoya and Solé, 2002; Williams, Berlow, et al., 2002) and other nonecological networks (Amaral, Scala, et al., 2000) at a qualitative level. That is, all these types of networks show the small-world property. However, if we look at more detail, quantitative differences exist between mutualistic networks and food webs. As a matter of fact, mutualistic networks have the strongest small-world property of any network studied so far: mutualistic species are more clustered and have a shorter distance than traditional food webs. Also, whereas in food webs a low clustering coefficient tends to be observed in small food webs, the reverse happens in pollination networks: the smallest networks have the highest clustering coefficient. We, should, however, be cautious with this claim because as mentioned before, these networks were transformed to one-mode networks prior to these analyses. The view of cohesive networks identified by the nestedness analysis is coherent with recent work on food webs that uses two related concepts, that of k-subwebs, and that of connectivity correlation showing that generalists also interact with generalists (Melián and Bascompte 2002a; see Appendix D for details) . This property also speaks of a cohesive structure in food webs, as has been described for mutualistic networks.

SUMMARY Despite their apparent complexity, mutualistic networks show repeated, universal structural patterns independent of species composition, size, and other ecological details. First, mutualistic networks are very heterogeneous: whereas the majority of species have only one or a few interactions, a few species are much more connected than expected by chance. Second, mutualistic networks are highly nested, that is, specialists interact with well-defined subsets of the species generalists interact with. These networks are thus neither randomly assembled nor organized in compartments arising from tight, parallel specialization. Third, mutualistic networks are built on weak, asymmetric interactions

T H E S T RU C T U R E O F P L A N T- A N I M A L M U T UA L I S T I C NETWORKS

63

among species. When looking at the weighted extension of species degree, that of species strength, mutualistic networks are also quite heterogeneous: the majority of species have little dependence with other species, but a few species are extraordinarily important because other species depend largely on them. Fourth, mutualistic networks have a strong small-world property; that is, they simultaneously have a short path length among any pair of species and a high clustering coefficient. Finally, these networks are also significantly modular— that is, there are small groups of species with morphological convergence of traits that interact strongly among themselves and more loosely with species from other modules. These modules can be regarded as the basic building blocks of mutualistic networks and their coevolutionary units.

CHAPTER FOUR

Ecological and Evolutionary Mechanisms

In the previous chapter we described the architecture of mutualistic networks. This has uncovered repeated patterns of structure that emerge in these complex networks of multispecies interactions. They include broad-scale or scale-free distributions of the number of interactions per species, a nested arrangement of interactions such as in Chinese boxes, and a modular organization with multiple modules that act as the basic blocks of the complex network. So far, however, we have used few biological concepts, because this has been mainly a statistical process describing a pattern. A persistent challenge in evolutionary biology has been understanding how coevolution has produced complex webs of interacting species (Thompson, 2005, 2006). In this chapter, we turn to investigating the suite of ecological, evolutionary, and coevolutionary mechanisms responsible for generating such network patterns. In previous chapters we saw that physicists have studied preferential attachment as a toy model for generating networks with a scale-free structure similar to the one observed in real networks. One avenue in here is to explore modifications of these basic mechanisms to produce most of network patterns. In fact, there is ample evidence that large-scale networks of interacting mutualists show similar patterns of connectivity, nestedness, and modularity to other biological and nonbiological networks. Which are the biological causes behind preferential attachment mechanisms in these networks? Consider, for example, one of the most pervasive patterns observed in empirical networks, namely, truncated power-law connectivity distributions (Jordano, Bascompte, et al.). The first explanation for such statistical patterns would be small size effects. According to this, if the network is small, it cannot accommodate the hubs or super generalist species; as a consequence there is a cutoff in the shape of the degree distribution (Keitt and Stanley, 1998; Mossa, Barthélémy, et al., 2002; Jordano, Bascompte, et al., 2003; Guimarães,

E C O L O G I C A L A N D E VO L U T I O NA RY M E C H A N I S M S

65

de Aguiar, et al., 2005). However, Jordano, Bascompte, and Olesen found that there was not a major frequency of truncated power laws among the smallest communities they examined, which suggests that there may be other explanations for such a truncation. Physicists have coined the term information filtering to describe a simple modification of the preferential attachment rule that produces such truncations. The idea is to assume that when a new node enters the network and tries to interact preferentially with the most connected node, it does not have a complete knowledge of the network; for example, it samples only a fraction of nodes. An alternative explanation for the broad-scale patterns in connectivity distributions arises from knowledge of the species involved in these interactions. It is based on the concept of forbidden links (Jordano, Bascompte, and Olesen, 2003, 2006), or biological constraints that prevent the occurrence of certain pairwise interactions among those possible in the plant-animal assemblage. Forbidden links can operate as constraints of the preferential attachment mechanism, in an analogous way to the proposed information filtering mechanisms involved as general explanations for truncation of degree distributions in a wide variety of nonbiological networks. Other mechanisms can explain truncations without such constraints. One example is provided when the initial core of species over which the preferential attachment operates is large enough, in which case power laws become truncated (Guimarães, de Aguiar, et al., 2005). Also, the two-mode nature of these networks implies the existence of truncations that would not be observed in one-mode networks in specific cases, such as if one of the sets (plants or animals) is much larger than the other (Guimarães, Machado et al., 2007). We also need to explain how these basic mechanisms of network buildup are mediated in ecological networks. For example, the mechanism of preferential attachment illustrated in Chapter 2 may be at work, but perhaps it does not operate through species degree. If degree is correlated with any other ecological property, such as local abundance, phenophase length, or geographic distribution range size, a new species may tend to become attached to the most abundant or widely distributed species. This would be the ecological mechanism behind a process that would look equivalent to preferential attachment. When considering complex ecological networks, we may think that the ultimate network structure arises from variable patterns of addition of interactions: some webs might result from a quick addition of links once a species is added to the community, whereas other networks might result from a slow-paced addition of more specialized interactions.

66

a

CHAPTER 4

c

b

Figure 4.1. How species interact within complex networks can be affected by their evolutionary history (phenotypic traits). For example, closely related species might have a similar pattern of interaction, simply because of niche conservatism (a). Alternatively, (b) describes a scenario without phylogenetic signal; (c) represents a real network and corresponding phylogenies. Modified from Rezende, Lavabre, et al. (2007).

Similarly, when addressing the biological attributes of species in terms of explaining their position in the network of interactions, we have to keep in mind that species are not independent entities. Rather, they are the product of a common evolutionary history and, therefore, linked in a hierarchical fashion (Fig. 4.1). Thus, we first have to determine the magnitude of the phylogenetic signal in a species role in the network. We now need to look at the species-specific traits that drive the interaction outcomes. In this way, the evolution of complex networks of interaction among diversified sets of free living species would be directly mediated by the evolutionary histories of the two groups of species. The basic rationale of our approach here is outlined in Appendix E (Fig. E.1). Next, we introduce these biological concepts on top of the mutualistic networks, starting from the earlier work introducing a single ecological trait and moving toward an integrative framework, where several ecological correlates and phylogenies are simultaneously accounted for.

E C O L O G I C A L A N D E VO L U T I O NA RY M E C H A N I S M S

67

Table 4.1. Types of links in an interaction matrix. (See Fig. 4.4 for an example.) Link Type

Formulation

Definition

Potential links

AP

Observed links

I

Unobserved links Forbidden links

AP − I

Size of matrix; A and P are numbers of interacting animal and plant species, respectively. Total number of observed links in the network for a sufficiently strong sampling effort. Number of zeros in matrix.

Missing links

F = AP − I −M

M

Number of links that remain unobserved because of linkage constraints and irrespectively of a sufficiently strong sampling effort. Number of links that do exist in nature but need more sampling effort to be observed.

SINGLE ECOLOGICAL TRAITS Forbidden Links Forbidden links refer to the fact that, in contrast to other types of networks, some connections are not biologically possible because of constraints underpinned by the natural history details of species. Well-sampled interaction matrices still can have a substantial number of unobserved interactions, and a sizeable fraction of them can be accounted for as forbidden links (Jordano, Bascompte, and Olesen; Olesen, Bascompte, et al.; Table 4.1). Thus, many zeros in interaction matrices should be considered as structural zeros and not necessarily artifacts of undersampling or passive sampling (Jordano, Bascompte, et al., 2003; Blüthgen, Menzel, et al., 2006; Olesen, Bascompte, et al., 2008; Chacoff, Vázquez, et al., 2012). This is especially evident in robustly sampled datasets, as we will see in Chapter 5. Outside ecology, missing links receive a lot of attention, for example; in social networks (Liben-Nowell and Kleinberg, 2007; Clauset, Moore, et al., 2008). Here, a long recognized problem is the boundary specification problem, which concerns decisions about inclusion or exclusion of nodes and links at the borders of the study network, because network closure most often is an artifact of research design. Although rarely addressed (Martinez, Hawkins, et al., 1999), this is also a problem in ecology. Typically, empirical mutualistic networks (Fig. 4.2) provide snapshots of plant-animal interactions

NCH

HR

Figure 4.2. The topology of the interaction networks in two plant-frugivore networks in the highlands (NCH) and the lowlands (HR) of southern Spain after an ordination by multidimensional scaling. Circles are plant species; squares are frugivore species. Size of the nodes is proportional to the eigenvector centrality (eigc), a measure dependent on both the number of links of a species and that of its partners.

E C O L O G I C A L A N D E VO L U T I O NA RY M E C H A N I S M S

69

occurring, for example; throughout a year and within a delimited region. For example, Jordano and Bascompte (in preparation) have analyzed two plant-frugivore networks robustly sampled in southern Spain (Fig. 4.2) to assess the natural history details that underpin these complex interaction patterns. Two examples of ecological factors contributing to forbidden links in natural communities are given in Figures 4.3 and 4.4. Temporal uncoupling (Fig. 4.3), for example, takes place when phenophases of potentially interacting species do not overlap (Olesen, Bascompte, et al., 2010; Fabina, Abbott, et al., 2010; Encinas-Viso, Revilla, et al., 2012). Thus, a plant species would not interact with a pollinator that arrives at the community after the flowering period. For example, the spring and autumn passage periods of migratory Garden warblers (Fig. 4.3b) do not match the fruit-ripening period of several late-autumn- and winter-ripening plant species (Fig. 4.3a), resulting in phenological mismatches. This concept is related to the general match/mismatch hypothesis, which has deep roots in ecology (Elton, 1927) and beyond (Bloch and Ritter, 1977). A second example of forbidden link is provided by size mismatching, which has been found to play a major role in plant-hummingbird networks (Snow and Snow, 1972; Jordano, Bascompte, et al., 2006), plant-insect networks (Stang, Klinkhamer, et al., 2006; Stang et al., 2007; Cocucci, Moré, et al., 2009), and plant-frugivore networks (Jordano, Vázquez, et al., 2009, Fig. 4.4). Additional reasons explaining unobserved links are varied and include habitat selection, reward mismatching, foraging constraints, and physiological-biochemical constraints (Jordano, 1987; Nilsson, 1988; Borrell, 2005; Jordano, Bascompte, et al., 2006; Blüthgen, Fründ, et al., 2008; Olesen, Bascompte, et al., 2008). In one of the communities studied by Jordano, Bascompte, and Olesen et al. (2003; Fig. 4.2), 51% and 24% of the nonobserved interactions were due to phenological uncoupling and size restrictions, respectively (Fig. 4.5). These figures are confirmed in the analysis of subsequent mutualistic networks. Thus, for a few pollination and seed dispersal networks recently studied (Fig. 4.6), 48% to 80% of all links were forbidden and 20% to 30% were so because of phenological uncoupling (Olesen, Bascompte, et al., 2010). Phenological differences alone seem to be an important source of mismatching between the activity periods of plant and animal partners, causing unobserved interactions. To sum up, these forbidden links represent biological constraints on species interactions, that is, factors causing a given species to sample a reduced range of the partners available for interaction. This reduction of the potential

70

CHAPTER 4

a

18 17

Fruiting sequence

16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Number of birds (/10 ha)

b

Yr 1

Yr 2

Yr 3

20 10

J

M

M

J

S

N

J

M

M

J

S

N

J

Figure 4.3. Examples of ecological explanations for forbidden links in natural communities. (a) The fruiting phenology of fleshy-fruited plant species in a lowland Mediterranean shrubland in SW Spain (HR, Fig. 4.2), during two successive fruiting seasons. Lines indicate the fruiting phenophases (with ripe fruit available), periods with maximum availability (thick bars), and dates of peak availability (vertical bars) for 18 plant species. The bottom panel (b) shows the local abundance of Garden warblers Sylvia borin (blank dots, dotted lines; an autumn passage migrant) and Blackcaps Sylvia atricapilla (dots, continuous lines; a wintering warbler). Both species are strongly furgivorous in late summer–early autumn (Garden) and late autumn–winter (Blackcap), yet they miss the fruits of some species because of nonoverlapping phenophases. Data from Jordano (1988).

range of realized interactions helps explain the generalized truncations in the degree distributions of plant-animal mutualistic networks discussed in previous sections.

M

E C O L O G I C A L A N D E VO L U T I O NA RY M E C H A N I S M S

71

4

Number of species

3

2

1

0 5

10

15

20

25

Fruit diameter or gape width (mm) Figure 4.4. Examples of ecological explanations for forbidden links in natural networks. Frequency distribution of fruit sizes in the Nava de las Correhuelas highland community (see Fig. 4.2; filled histogram) and mean gape widths of frugivorous birds in the area (lines with dots below the histogram). The large dot on the abscissa indicates the mean fruit size in this community (10.6 mm). Size restrictions limit most of the frugivores (16 out of 33 species) to handle and consume the five fruit species above this mean size. Frugivorous birds rarely efficiently handle fruits above their gape size. Data from Jordano and Bascompte (in preparation).

Species Abundance Species abundance plays certainly an important role in structuring mutualistic networks (Jordano, 1987; Olesen, Eskildsen, et al., 2002; Jordano, Bascompte, et al., 2003; D. P. Vázquez and Aizen, 2004). Importantly, however, the magnitude and even direction of the correlation between abundance and species degree may change across species and communities (Blüthgen, Menzel, et al.). Also, the simultaneous consideration of several mechanisms

Vulpes vulpes Martes foina Meles meles

Figure 4.5. The distribution of forbidden links in the Nava de las Correhuelas community (NCH in Fig. 4.2). The interaction matrix highlights the observed interactions (cells with dots) as well as the main factors associated to the unobserved interactions (greyed): phenological uncoupling of the partner species (filled), size mismatches between gape width and fruit diameter (dark grey), accessibility constraints of the fruit display (medium grey). Light grey indicate missing links due to unknown causes. Data from Jordano and Bascompte (in preparation).

Daphne laureola

Sorbus aria

Juniperus sabina

Juniperus communis

Hedera helix

Taxus baccata

Rosa canina

Rubus ulmifolius

Lonicera arborea

Juniperus phoenicea

Berberis vulgaris

Crataegus monogyna

Prunus mahaleb Turdus merula Turdus viscivorus Erithacus rubecula Turdus iliacus Turdus torquatus Phoenicurus ochruros Fringilla coelebs Parus ater Sylvia atricapilla Coccothraustes coccothraustes Columba palumbus Corvus corone Parus major Turdus pilaris Garrulus glandarius Parus caeruleus Phoenicurus phoenicurus Sitta europaea Sylvia communis Turdus philomelos Emberiza cia Ficedula hypoleuca Loxia curvirostra Parus cristatus Pica pica Sylvia conspicillata Corvus corax Dendrocopos major Serinus citrinella Serinus serinus Sylvia borin Sylvia cantillans Sylvia melanocephala

Percent Observed Phenology

Size

Unk.

Access.

100

80

60

40

20

0

Polygonatum odoratum

Paeonia officinalis

Lonicera etrusca

Juniperus oxycedrus

Prunus prostrata

Lonicera splendida

Cotoneaster granatensis

Arum italicum

Rubia peregrina

Rhamnus saxatilis

Rhamnus myrtifolius

Amelanchier ovalis

74

CHAPTER 4

Observed Phenological uncoupling Other constraints 100 54%

75

23%

53%

25%

50

52% 22%

30% 25

25% 16% 0

ZACK Pollination

HR

Figure 4.6. The relative frequencies (percent) of different types of forbidden links in pollination (Zackenberg network, Greenland) and seed-dispersal networks (Hato Ratón and Nava de las Correhuelas, southern Spain). Figures indicate the percentage of observed interactions (black) and the percentage of the total potential interactions A × P not recorded due to phenological mismatches (nonoverlapping phenophases of plants and animals; dark grey) and other constraints such as size differences, microhabitat differences, accessibility constraints, and so on (light grey). Data from Olesen, Bascompte, et al. (2010).

NCH

Seed dispersal

(e.g., morphological constraints and abundance) can show a much higher explanatory power than considering only one of those (Jordano, Bascompte, and Olesen, 2003; Stang, Klinkhamer, et al., 2006; Blüthgen, Menzel, et al., 2006; Stang et al., 2007; Krishna, Guimarães, et al., 2008). For example, a species with long flowering or fruiting season could compensate lower local abundance by extending its availability for interactions over a longer period each season. This would increase the range of partner species with which it could interact, irrespective of abundance. Given that abundance might have pervasive consequences for the structuring of interactions in local communities, a central issue is to assess how robust the sampling of interactions is, independent of abundance variation among interacting partners. We consider this in more detail in Chapter 5, when we analyze the temporal and spatial components of interaction networks. Briefly, even in well-sampled communities, interaction networks are characterized by a high number of infrequent interactions, many unobserved pairwise interactions, and a few species that concentrate most interactions (see Chapter 5). Therefore, we consider unwarranted recent claims (see, e.g., Blüthgen, Fründ, et al., 2008; Blüthgen, 2010) that network patterns are largely based on artifacts of passive sampling. The overemphasis on the role of abundance

E C O L O G I C A L A N D E VO L U T I O NA RY M E C H A N I S M S

75

may stem on the use of inadequate null models, beyond the obvious logistic limitations of sampling that pervade any ecological study. Two limitations are nonindependent estimates of species abundance and overestimation of rare interactions (see Appendix F for details). Indeed, sampling biases have been acknowledged and accounted for since the early work of Jordano (1987) and Olesen and Jordano (2002). Despite these potential biases, well-sampled networks show characteristic general patterns of missing pairwise interactions (see Chap. 5). Some interactions are rare for analogous reasons to why some species are rare. We, therefore, should not be surprised to find interaction singletons in well-sampled interaction matrices in the same way as we find species singletons in well-sampled communities. The major hypotheses about the emergence of nestedness in interaction networks involve differences in abundance of interacting species (Jordano, 1987; Lewinsohn, Prado, et al., 2006), higher extinction rates for specialists that interact with other specialists (Ollerton, Johnson, et al., 2003), or the convergence and complementarity of traits among species in a set (Thompson, 2005; Guimarães, Rico-Gray et al., 2006; Santamaría and Rodríguez-Gironés, 2007). Therefore, if the interaction pattern itself (i.e., the probability of interaction by interspecific encounter) is marginally associated to variation in local abundance of mutualists, then we might expect abundance per se to explain a significant fraction of the interaction pattern (Jordano, 1987). For example, it is not unexpected that local abundance relates to the variation in degree among species in a community, yet the magnitude of this effect can vary among communities. Basic natural history informs us that other life history traits, such as body mass, phenology, ecomorphology of body parts, home range size, territorial behavior, vision and digestive physiology, can determine generalization-specialization patterns or interaction strength beyond the simple effect of abundance variation. Local abundance therefore, is an important and central biological property of the interacting species, not a confounding factor or noise-generating property that has to be controlled for a proper assessment of interaction patterns. This is especially true if we are interested in understanding the ecological bases that underpin coevolved interactions in megadiversified communities. When abundance variation across species has been adequately incorporated in null models, it has been shown that it by itself cannot explain the diversity of interactions patterns. Stang, Klinkhamer, et al. (2006) studied simultaneously the effects of local abundance and flower morphology constraints on insect visitation. They reported that morphological limitations accounted for 71% of the variation in the number of visitor species and that abundance alone was a poor predictor of interaction patterns at the community level. Krishna,

76

CHAPTER 4

Proportion of nestedness explained

0.80

0.70

0.60

0.50

0.40

Nava Correhuelas

Hato Ratón

Figure 4.7. Quantitative estimates of the relative contribution of abundance and frugivory-intensity to nestedness in two real plant-frugivore interaction networks (see Fig. 4.2). Black bars indicate the fraction of nestedness explained by abundance; grey bars indicate the fraction of nestedness explained by frugivory– intensity-weighted abundance (i.e., abundance data after controlling for speciesspecific levels of dependence on fruit food); and white bars indicate the average fraction of nestedness explained by frugivory–intensity-weighted abundance for randomized networks in which frugivory was randomly shuffled among species. Lines indicate the standard errors for the randomized networks. Data from Krishna, Guimarães, et al. (2008).

Guimarães, et al. (2008) applied neutral theory to mutualistic networks by generating a relative species-abundance distribution for both the plants and the animals and proceeded by assuming that the probability of interaction between one such plant and animal species is proportional to their respective abundances. Such neutral processes explain about 60% to 70% of the empirical nested patterns in two mutualistic communities (Fig. 4.7). Nestedness is better explained if the model incorporates species-specific traits such as forbidden links (Krishna, Guimarães, et al., 2008). More recently, D. P. Vázquez, Chacoff, et al. (2009) showed that information on relative abundance and phenology suffices to predict several aggregate properties of a pollination network in Argentina (connectance, nestedness, interaction evenness, and interaction asymmetry). However, such information was insufficient to predict the detailed network structure (the frequency of pairwise interactions), leaving a large amount of variation unexplained. Santamaría and Rodríguez-Gironés (2007) came up with three different reasons by which strict neutrality should be rejected as the most parsimonious explanation of the observed network structure (in contrast to the role of forbidden links). First, it is debatable whether generalist species are generalists because they are more abundant or the other way around

E C O L O G I C A L A N D E VO L U T I O NA RY M E C H A N I S M S

77

(Stang et al., 2007). Second, the assumption that abundance determines the frequency of interaction has recently been challenged (Blüthgen, Menzel, et al., 2006). Finally, the ample empirical evidence of forbidden links (Jordano, Bascompte, and Olesen, 2003) contradicts the assumption of neutral theory that phenotypic characteristics of interacting species are irrelevant for network patterns.

PHYLOGENETIC EFFECTS If we want to understand the structure of mutualistic networks from a biological perspective, we need to integrate several sources of information. We cannot just estimate a correlation between a species position in the network and any ecological variable. First, several such ecological attributes may be correlated, and we need to disentangle such correlations to asses the relative contribution of any of them. As in any similar comparative approach, each of the ecological variables aimed to explain network patterns can have a phylogenetic signal, defined as the tendency of species closer in the phylogeny to be more or less similar in network properties than expected from chance (Freckleton, Harvey, et al. (2002); Lewinsohn, Novotny, et al. (2005); Blomberg, Garland, et al. (2003); Garland, Bennett, et al. (2005); Ives and Godfray (2006); Fig. 4.1a, c). The existence of a significant phylogenetic signal would be an indication that past evolutionary history helps to understand network architecture. This would point out the limitation of explanations based exclusively on ultimate ecological factors (Ives and Godfray, 2006; Rezende, Jordano, et al., 2007; Rezende, Lavabre, et al., 2007; Jordano and Bascompte, in preparation) and ask for an integrative approach, where several ecological mechanisms are simultaneously considered while controlling for their phylogenetic signal. There are two main ways in which phylogenies can influence network patterns. First, the shared phylogenetic history among species in one set (e.g., the animals in Fig. 4.1a, b) might determine strong similarities among species in their pattern of interaction, that is, their number of interactions or with whom they interact. This would be an analogue to phylogenetic niche conservatism (Harvey and Pagel, 1991): closely related species would tend to show similar interaction patterns (Fig. 4.1a). Quantitative methods to assess phylogenetic signal are discussed in Appendix E. Second, the interaction pattern itself—that is, the values in the interaction matrix—can be influenced by the two phylogenies (Appendix E; Fig. E.1; Ives and Godfray, 2006). This results in different scenarios (Fig. 4.8) of how the

78

a

CHAPTER 4

*

ns

*

*

ns

*

c

b

d

ns

ns

Figure 4.8. The interaction pattern itself is subjected to the effect of both the animal and plant phylogenetic histories, with four potential outcomes, depending on the relative influences of the animal or plant phylogenetic histories (marked as significant with an asterisk, versus nonsignificant, ns: (a) with a marked trend for the interactions to match the phylogenetic history of the two groups; one of them [(b) the plants; (c) the animals] or none [(d) with a random assortment of interactions relative to the phylogenetic histories of the two groups]. Grayed spheres indicate observed interactions (1s in the interaction matrix); 0s are indicated as blurred circles. Modified from (Jordano, 2010).

whole interaction pattern can be driven by either phylogeny, both phylogenies, or none of them (Jordano, 2010; also see Appendix E). In the most simple (and unlikely) scenario, the interaction partners of any species in one set are drawn randomly from the phylogeny of the other set of mutualists, so that no phylogenetic signal can be traced (Fig. 4.8d).

E C O L O G I C A L A N D E VO L U T I O NA RY M E C H A N I S M S

79

The other extreme scenario would be that closely related animals interact with distinct and closely related subsets of the plants and vice versa, so that the whole interaction matrix shows a distinct signal and structure that reflects the topologies of both phylogenies, most likely resulting in a highly modular network (Fig. 4.8a). An intermediate scenario would result, for example, when any given animal species interacts with closely related plant species and closely related plant species interact with similar animal species that, indeed, need not to be closely related phylogenetically (Fig. 4.8b, c). The characteristic feature of this scenario is that each plant species is sampling a much-wider phylogenetic diversity of the animal partners, whereas the animals interact with distinct, and closely related, subsets of the plants (Fig. 4.8b). To some extent, the phylogeny of the plants would “drive” this scenario, because animals would track only limited subsets of the plant phylogenetic diversity. Therefore, the signal of evolutionary history, when assessed from this interaction perspective, can have two main influences: first, by determining the individual species’ positions in the network; second, by determining how the overall interaction pattern builds up under the influence of both animal and plant phylogenetic histories. Phylogenies and Species Positions The species positions within large, complex networks of ecological interactions (e.g., Fig. 4.1c and 4.2) are ultimately related to their evolutionary history, which in turn shapes the life-history traits mediating the interactions. A tendency of closely related species to interact with the same set of species seems ubiquitous among the ecological interactions such as those between plants and pollinators (Rezende, Lavabre, et al., 2007), hosts and parasites (Ives and Godfray, 2006), predators and prey (Bersier and Kehrli, 2008; Rezende, Albert, et al., 2009) plants and pathogens (Gilbert and Webb, 2007; Vacher, Piou, et al., 2008) and plants and herbivores (Novotny, Basset, et al., 2002; Weiblen, Webb, et al., 2006). Evolutionarily similar species or taxa tend to show striking similarities in their modes of interaction, and this seems a pervasive pattern throughout the history of life (Gómez, Verdú, et al., 2010). In relation to phylogenetic signal on a species position in the network, Rezende, Lavabre, et al. (2007) compiled the phylogenies for each plant and animal assemblage in a dataset comprising 36 plant-pollinator and 23 plantfrugivore mutualistic networks (Fig. 4.1c). About half of the largest phylogenies showed a significant phylogenetic signal in the number of interactions per species (i.e., degree). Its quantitative extension—species strength—was found to be significant in only 1 of the 38 phylogenies corresponding to weighted

80

CHAPTER 4

a

5

0.8

c

Phylogenetic signal

Animals

0.5

0.4 0.1

0.2

b

0.0 1 0.5

Plants 0.1

10

100

Number of species

–0.2

Signal degree – signal strength

0.6

1

–0.4

400

Phylogenies

Figure 4.9. Phylogenetic signal on the degree [number of interactions per species, (a)] and strength (b) of mutualistic interactions in relation to species richness. Each point represents a phylogeny: grey circles for plants and grey squares for animals. Solid symbols indicate statistically significant phylogenetic signals. (c) Relative magnitudes of phylogenetic signal for degree and strength in these mutualistic networks. Grey and dashed grey bars indicate plants and animals, respectively. Phylogenetic signal values were significantly higher for degree. Based on Rezende, Lavabre, et al. (2007).

networks. As a possible explanation for this difference between degree and strength, we could argue that species strength is very much affected by species abundance, the latter probably responding to more proximate, local factors. Second, Rezende, Jordano, et al. (2007) studied with whom each species interacts as a second property of a species position in the network. This is at the core of the concept of nestedness. To do this, they measured the correlation between two matrices of similarity across species (Fig. 4.9). The first was a matrix of phylogenetic similarity given by how close each pair of species was in the phylogeny. The second matrix represented the ecological proximity between pairs of species so that, for example, two plant species are very similar if pollinated by the same species of insects. Rezende, Lavabre, et al. found a significant correlation between ecological and phylogenetic distance in about half of the 103 available phylogenies. Also, animals showed a stronger correlation between ecological and phylogenetic distances than plants did (more than half the correlations were significant for animals, whereas onethird were significant for plants; Fig. 4.9c). One potential explanation for this

E C O L O G I C A L A N D E VO L U T I O NA RY M E C H A N I S M S

81

difference would be evolutionary differences linked to mobility. Thus, animals can search and select with whom to interact, promoting selection for specific floral or fruit phenotypes (Rezende, Lavabre, et al., 2007), whereas plants may have a more passive role. An alternative explanation would be related to the existence of differences in the level of resolution between the plant and the animal phylogenies. To conclude this section, there seems to be significant phylogenetic signal in both the number of interactions per species and with whom they interact. The phylogenetic signal on interaction patterns is ultimately determined by biological constraints from species-specific traits that limit the richness and range of mutualistic partners that each species can have. Some pairwise interactions are simply not possible to be realized in nature and, therefore, to be recorded. From a conceptual point of view, this implies that besides current ecological factors, past evolutionary history also explains network patterns. From a methodological point of view, the incorporation of phylogenetic methods is also important when searching for ecological correlates of network patterns such as species abundance. Ecological Correlates of Species Positions The species positions within a complex network of interactions (see, e.g., Fig. 4.2) are ultimately related to variation in life-history traits that mediate these interactions. Some species are more central in the network because they interact with a higher number of partners; indeed, these partners interact with many others. Some species are more peripheral because they tend to interact with a few partners, and some of those are indeed peripheral. We may ask whether species in the core of the network share similar ecological traits that differentiate them from more peripheral species. Jordano and Bascompte (in preparation) performed phylogenetically independent contrasts correlations between a species’ ecological traits and a couple of measures of the position of such a species in the network: its degree k and eigenvector centrality eigc. They used two southern Spanish plant-frugivore communities (Fig. 4.2) with highly detailed information and interaction records. The final objective was assessing to what extent body or fruit size, local abundance, phenological spread, and geographic range all correlate with variation in degree and centrality across species once phylogenetic effects are accounted for. Compared to plants, frugivores showed more significant contrast correlations between k and eigc, on one hand, and the ecological variables, on the other hand (Table 4.2). For plants, only phenological spread, size of the geographic range, and local abundance showed significant trends, but not

82

CHAPTER 4

Table 4.2. Pearson correlations between phylogenetic independent contrasts of interaction statistics and ecological characteristics of species in two plant-frugivore networks (NCH and HR; see Fig. 4.2). Variables: size represents gape width (frugivores) or fruit diameter (plants); fruit diet, percent of fruits in diet; local abundance as number of birds km−1 for frugivores; number of fruits ha−1 for plants); phenology indicates the phenological spread (for frugivores, number of months present in the area; for plants, number of months with ripe fruit available); k, degree; eigc, eigenvector centrality. Frugivores

Plants

NCH Body mass Size Fruit diet Abundance Geographic range Phenology

k −0.09 NS 0.45 * 0.07 NS 0.70 *** 0.48 ** 0.80 ***

eigc −0.16 NS 0.47 * 0.13 NS 0.66 *** 0.53 ** 0.63 ***

k — 0.03 NS — 0.11 NS 0.64 *** 0.37 *

eigc — 0.06 NS — 0.20 NS 0.76 *** 0.47 **

HR Body mass Size Fruit diet Abundance Geographic range Phenology

k 0.41 ∼ 0.29 NS 0.43 ∼ 0.84 *** 0.43 * 0.54 *

eigc 0.36 * 0.25 NS 0.51 ∼ 0.19 ** 0.46 * 0.46 *

k — −0.24 NS — 0.84 *** 0.07 NS 0.18 NS

eigc — −0.26 NS — 0.60 * 0.04 NS 0.14 NS

***, P < 0.0001; **, P < 0.001; *, P < 0.05; ∼, P < 0.1; NS, nonsignificant. See Jordano and Bascompte (in preparation).

consistently in the two communities. For the frugivores, both degree and centrality showed positive and significant correlated evolution with body size (either indicated by body mass or gape size), local abundance, size of geographic range, and phenological spread. This supports the results indicating a stronger phylogenetic effect of species-specific traits on the position of the frugivore species in the webs. This is not surprising given that frugivore traits closely related to the interaction outcomes—in terms of how profitable a given fruit species is to a consumer species—have, themselves, a strong phylogenetic component. In the case of plants, the limited species richness of these communities (25 and 16 plant species, respectively) probably limits the power to detect phylogenetic signal (Rezende, Lavabre, et al., 2007). Besides, fleshy-fruited local plant communities in southern Mediterranean Spain are characterized by a relatively rich representation of higher taxa (e.g., families) and few congeneric species. Thus, differences between the phylogeny of plants and animals in terms of their internal topologies (i.e., diversification patterns) might also contribute to these differences.

E C O L O G I C A L A N D E VO L U T I O NA RY M E C H A N I S M S

a

b

ANIMALS −3.0

−1.0

1.0

3.0

PLANTS −1.5

5.0

k eigc w g

83

f

a geog ph

−0.5

0.5

1.5

2.5

3.5

k eigc

g

a geog ph

Figure 4.10. Phylogenetic patterns of ecological traits and topological properties of frugivore (a) and plant (b) species in the Nava de las Correhuelas community (NCH in Fig. 4.2). Squares represent the scaled trait values for each species. Filled and blank squares illustrate positive and negative deviations, respectively, from the community mean values and their relation to the phylogenetic pattern. k, degree; eigc, eigenvector centrality; w, body mass; g, gape width (animals) or fruit diameter (plants); a, local abundance (number of individuals per unit area, animals; number of ripe fruits per unit area, plants); f , degree of frugivory (percentage of fruit food, by volume); geog, size of the geographic range [estimated from distribution maps; from Cramp (1988) for animals and Jalas and Suominen (1985) for plants]; and ph, phenophase length in number of weeks with presence in the area (animals) or number of weeks with ripe fruits available (plants).

Visually, such patterns mean that species-specific values of both traits and node-specific network parameters such as degree and centrality appear clustered at certain subclades of the phylogenies (Fig. 4.10). For instance, the Turdus subclade in Figure 4.10 is characterized by degree and centrality values well above the mean of the frugivore assemblage, matching relatively high

84

CHAPTER 4

values in life-history traits such as intensity of frugivory, gape width, local abundance, and geographic range. Central species tend to be those frugivores with medium-to-large body sizes, intensively relying on fruit food or yearround residents or with long presence periods (e.g., overwintering). Peripheral species are generally those with more mixed diets, where fruit is a relatively minor component, those that are short-term residents in the area (i.e., transient migrants), or species that otherwise show a low abundance. Among the plants, the core of the network in the two communities is invariably composed of species that show staggered fruiting seasons, extending between midsummer and early spring. These tend to be plants with high cover in the area and sizeable fruit productions, with relatively long fruiting seasons. In contrast, peripheral plant species include those with low cover, short or episodic fruiting seasons, limited fruit crop sizes, and relatively large fruit size, which restricts access to a subset of the frugivores. In sum, central species combine a suite of traits that make it highly likely that they interact with a wide array of mutualistic partners, whereas peripheral species have specific traits that constraint their interactions to the more-central partners. So far, we have studied to what degree these species-specific constraints affect the position of a species in the network of interactions, for example, the phylogenetic signal in plants and in animals. Next, we will expand these phylogenetic methods to determine to what degree species-specific constraints also determine global network patterns, namely, how past evolutionary history of the two mutualistc sets simultaneously affect the whole network. Phylogenies and Global Network Patterns Shifting now to considering the phylogenetic effects on the entire network of interactions, Jordano and Bascompte (in preparation) explored the effect of plant and animal phylogenies in explaining the global structure of interactions using the statistical methods recently developed by Ives and Godfray (2006). To study the phylogenetic structure of the interaction network, it is necessary to obtain (1) a measure of the interaction strength between animals and plants and (2) the phylogenetic trees of both the animal and plant species. Briefly, Ives and Godfray (2006) developed a test of whether closely related fruit (or flower) species tend to be consumed by similar frugivores (or pollinators) and whether closely related animal species tend to interact with similar plant species (see Appendix E for details). Phylogenetic covariation patterns explained a significant fraction of the total variance of the interaction pattern at the whole-network level in the two frugivore communities described in the previos section (Jordano and

E C O L O G I C A L A N D E VO L U T I O NA RY M E C H A N I S M S

85

Bascompte, in preparation,; see Fig. 4.2). In this case, and as opposed to what was found for a species position in the network, there is a more marked effect of the plant’s phylogeny. Jordano and Bascompte (in preparation) present evidence that a model incorporating phylogenetic effects best explained the observed data against the alternatives assuming a star phylogeny (absence of phylogenetic effects) or a random Brownian model of character evolution. The phylogenetic signal of the plant phylogenies, d P , in both the highland and lowland communities was significantly higher than the phylogenetic signals from the frugivore phylogenies. Note also that the confidence intervals of the d P estimates do not include zero, whereas those of the animals include zero. This means a significant signal only on the plant side. Therefore, among the four possible scenarios for the evolutionary histories of animals and plants influencing the overall interact pattern (Fig. 4.8), it appears that the plant phylogeny more strongly determines interactions (scenario b in Fig. 4.8). These results reveal that frugivore species tend to use phylogenetically related plant species (the plant’s phylogeny significantly explains the scatter of interactions for the two communities); however, plant species include a broad range of frugivore species in their assemblages, so that the whole interaction pattern is much less determined by the frugivore phylogeny. This suggests the overall pattern of interactions is markedly influenced by the evolutionary history of the plants and is more labile when mapped against the frugivores. Recent analyses incorporating phylogenetic information have also found a larger effect of the plant phylogeny on other plant-animal interaction systems, including host-parasitoid interactions (Ives and Godfray, 2006), pollination networks (Vázquez, Chacoff, et al., 2009), and tree-pathogen interactions (Vacher, Piou, et al., 2008). Overall, therefore, several ecological correlates and past evolutionary history jointly shape the structure of plant-animal mutualistic networks. This provides an account on the suite of mechanisms generating the network patterns described in the previous chapter. These network patterns, although described so far decoupled from a temporal and spatial dimension, change spatially and temporally in nontrivial ways, a subject we deal with in the next chapter.

SUMMARY This chapter addresses the suite of ecological and evolutionary mechanisms shaping the network patterns described in the previous chapter. Although

86

CHAPTER 4

some predictors of a species position in a network of interactions have been studied—mainly species abundance and forbidden links—we are moving toward an integrative approach, where several variables are tested simultaneously within the framework of phylogenetically independent contrasts. This can help us quantify the relative contribution of different factors given that they are often correlated. Body size, phenological spread, regional abundance, and species abundance are significantly correlated with a species position in the network of interactions in the frugivore set. There is a phylogenetic signal in the position of a species in the network (e.g., its degree). Beyond this, the phylogenetic patterns of shared ancestry also play a key role in explaining the overall pattern of mutualistic associations between the two sets of species. This pattern of interaction is influenced mainly by the evolutionary history of the plants.

CHAPTER FIVE

Mutualistic Networks in Time and Space

Mutualistic networks, as well as food webs and other ecological networks, are mainly static descriptions. Some studies span only a single season; others are cumulative results spanning several seasons but contain no information on temporal dynamics. Similarly, these studies are mainly based on a single locality, sometimes an area of a few square kilometers. In other instances, such as in the composite network by Charles Robertson (1929), later on analyzed by Memmott, Craze, et al. (2007), this spans a huge geographic area. But again, it is a composite snapshot. There is no information on how these networks change through a spatial gradient. On the other hand, it is well known that ecological communities are dynamic, spatially extended entities. The question remains to what extent network structure changes through time and space (Fig. 5.1). The need for space in ecological network research parallels a similar situation in population ecology a decade ago, when space was considered the “last frontier in ecology” (Tilman and Kareiva, 1997; Hanski and Gilpin, 1997; Bascompte and Solé, 1998). An impressive research effort was aimed at building a spatiotemporal description of population dynamics, exemplified by the successful approach of metapopulations (Hanski, 1999). In coevolutionary studies, a similar effort was done by John N. Thompson, who brought the spatial domain in his geographic mosaic theory of coevolution (Thompson, 2005; see Chap. 1) Building from both the Geographich Mosaic and the early metapopulation theory, metacommunity theory has also explored the spatial distribution of small sets of ecologically interacting species (Liebhold, Bascompte, 2004). The first studies on metacommunities were mainly theoretical descriptions of trophic modules, with very few studies describing an entire, realistic ecological network (but see the following). An increasing number of studies progressively filled these gaps by providing both improved spatially explicit approaches (Holt, 2002; Devoto and Montaldo, 2005; K.S. McCann, Rasmussen, et al., 2005; Rooney, ArnaudHaond, et al., 2008) and empirically exhaustive descriptions of the temporal dynamics of networks (Winemiller, 1990; Petanidou, Kallimanis, et al., 2008;

88

CHAPTER 5

a

b

Figure 5.1. Spatial and temporal slices of complex networks. (a) Mutualistic networks may aggregate interactions occurring in distinct spatial patches of complex landscapes. (b) Similarly, these networks may reveal the aggregation of seasonally occurring interactions.

Lundgren and Olesen, 2005; Basilio, Medan, et al., 2006; Neutel, Heesterbeek, et al., 2007; Alarcón, Waser, et al., 2008; Olesen, Bascompte, et al., 2008; Carnicer, Jordano, et al., 2009; Laliberté and Tylianakis, 2010). Here we will review this combination of empirical and theoretical work describing the dynamics of mutualistic networks in time and space, and we will address what components of these networks are time and space invariant.

NETWORK DYNAMICS Daily Dynamics: Network Assembly Many networks in seasonal climates reassemble each year, and we can study the process of assembly. Each day we can register a network of daily plantanimal interactions, so that the entire season could be seen as a sequence of temporal slices, where nodes show distinct persistence times (Fig. 5.2). Depending on their phenophase lengths, some nodes will persist for a longer time than others. Eventually, a species could persist throughout

M U T UA L I S T I C N E T W O R K S I N T I M E A N D S PAC E

t1

t2

t3

89

t4

Figure 5.2. A temporal sequence of interactions in a bipartite network. Dark nodes represent animal frugivore species and light nodes represent plant species; white nodes are species absent in a given time slice but participating in the complex network. The complex network is actually an aggregate overlay of distinct temporal slices (t1 , ..., t4 ) that represent the interactions (dark continuous lines) occurring during short time intervals of the season. The whole season can be a whole year, a flowering season or a fruiting season. Different species (nodes) persist for variable time lengths (phenophases) and “age” as the season progresses, indicated as dashed lines joining nodes between successive time slices.

the whole reproductive season if the blooming or ripening season is very extended, but most species would be available for interactions only during short time spans. The same happens with the activity or residence periods of animals in a given area: variations in the emergence/diapause timing for insects and the local movements (e.g., altitudinal shifts) or migratory habits (e.g., transcontinental migration) of vertebrates will create variable temporal opportunities for interaction with plant partners. In addition to a few recent analyses of temporal variation (Petanidou, Kallimanis, et al., 2008; Lundgren and Olesen, 2005; Basilio, Medan, et al., 2006; Alarcón, Waser, et al., 2008; Olesen, Bascompte, et al., 2008; Carnicer, Jordano, et al., 2009; Baldock et al., 2011), the only study of which we are aware at this daily temporal scale was performed in an Arctic community in the Zackenberg Research Station in Northeastern Greenland, where Jens M. Olesen and collaborators recorded the assembly of the mutualistic network in two consecutive seasons. This locality has an advantage in relation to other sites. Because it is covered by ice and snow

90

CHAPTER 5

almost 9 months of the year, we can register the assembly of the network from day zero when the snow melts. Olesen, Bascompte, et al. recorded the plant and pollinator species every day from day zero to the last day of the season, when the first snowfalls erase any trace of plants or insects. This is an extraordinary occasion to link structure and dynamics, to test mechanisms of network assembly. The reader may remember that in Chapter 3 we described the structure of mutualistic networks in terms of the connectivity distribution. The majority of communities were best described by a broad-scale distribution (a truncated power law in more technical terms). This was a statistical description of a static pattern. As already noted, physicists have come up with the simplest model of network assembly that can generate scale-free networks, the preferential attachment (Barabási and Albert, 1999). Recall from Chapter 2 that this is a kind of rich get richer, in which the probability of interacting with a node is proportional to the number of links this node already has. Also, physicists have been able to follow in time the assembly of some real networks, such as the network of scientific coauthorship. In some of these cases, scientists could see a preferential attachment mechanism at work. This allowed linking the process of network formation with the final structure of this network. However, we had no single dataset in ecology where we could follow the dynamical process, which is a drawback because we have to be careful when inferring process from pattern. The Zackenberg pollination network is a wonderful exception where we can check wether similar mechanisms of network assembly are at work. A cartoon of daily interaction matrices from 6 days in the first season (Fig. 5.3) illustrates the marked seasonality of the interaction dynamics. The complexity of the interactions increases throughout the season over a maximum peak (days 20–25) that roughly corresponds to the peaks of flowering and insect activity, and then abruptly collapses (day 33) at the end of the season. This temporal variation in interaction complexity was generated from the temporal turnover of phenophases, varying considerably in their extent. Olesen, Bascompte, et al. (2008) noted the day a new species entered the community and with which species it interacted. Then, they looked at the number of interactions of these species. Using all day-to-day transitions, we can, therefore, calculate to what degree the probability of establishing a new interaction with a previous species depends on its number of links.

1

2

3

4

5

6

7

day 3 8

9

Day

day 15

day 26

day 33

day 38

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

day 7

Plants

Figure 5.3. Daily interaction matrix slices and pollinator and plant phenophase distributions from a season of the Zackenberg pollination network (day 1 is June 21). (a) Six daily time slice matrices from different days during the season. All species present during the season are included (A and P are the pollinator and plant communities, respectively). For each time-slice t, only links observed at least on day t are shown as black squares. These links may also be present earlier and/or later. (b),(c) The phenophase diagrams show the recorded temporal span of insect flower visitation and flowering of each species. Each bar represents a species. Species are sorted according to the starting date of flowering or foraging, and then according to the end date. Based on Olesen, Bascompte, et al. (2008)

c

b

a

Animals

92

CHAPTER 5

For each time step and each old species, they calculated the probability P(ki ) that pollinator species i with degree ki gets linked to the new plant species blooming. For all the pollinator species, the cumulative attachment probability of species with or fewer ki interactions, P (≤ ki ), is used to study how the attachment dynamics progresses (Jeong et al., 2003): P(ki ) ∝ ki α ,
P(≤ ki ) ∝

ki α dk ∝ kiα+1 ,

(5.1) (5.2)

where α is a scaling exponent determining the type of attachment dynamics and the subsequent distribution of number of links per species. For each transition ti → ti+1 , ln(P(≤ ki )) is linearly regressed against ln(ki ) to estimate α. If α = 0, the attachment probability is constant, that is, independent of k, and this uniform attachment leads to an exponential cumulative degree distribution of k (Barabási and Albert, 1999). If α < 1, P(≥ k) is either exponential or follows a truncated power law. If α = 1, P(≥ k) becomes scale free, and if α > 1, we obtain a network in which one or a few supergeneralists connect to almost all other species in the network, that is, a star-shaped network (Krapivsky et al., 2000). Olesen, Bascompte, et al. (2008) found that the attachment probability was intermediate between preferential and random (when all species have the same probability of interacting regardless of their number of interactions; Fig. 5.4). The cumulative interaction level of species i at day t, ki (t), summarizes the influence of all factors on the attachment of new species to species i during the first t days. Thus, we can expect ki (t) to be a strong predictor of the attachment success, P(ki ), of any new species to i during the subsequent time step from t to t + 1. Because α values were lower than predicted according to a preferential attachment model (α = 0.63 − 0.84), Olesen, Bascompte, et al. suggested that additional species-specific constraints, besides phenology, limited the attachment probability. That is, species entering the community as the season progresses tend to interact with already well-connected species, but with deviations from this trend. These constraints (forbidden links, as discussed in Chap. 4) also push the mode of attachment away from being preferential. Olesen, Bascompte, et al. interpret that the deviation from a power-law connectivity distribution and the fact that α < 1 are due to various constraints operating in the network, limiting high-linkage-level species in attaching preferentially to new species. This is totally compatible with the structural pattern previously found. A broad scale or truncated power-law connectivity distribution suggests that the assembly process could be preferential attachment with some noise or

M U T UA L I S T I C N E T W O R K S I N T I M E A N D S PAC E

93

100

Cumulative probability

Preferential α = 1

10–1 Random α = 0

10–2

100

Number of interactions per species, ki

101

Figure 5.4. Cumulative probability P(≤ ki ) of a new plant species becoming linked to a pollinator species already active with ki links, using the time step from day 9 to day 11 in the study by Olesen, Bascompte, et al. (2008) as an example. Each dot is a species; note the log-log scale. The black continuous line is the regression of P(≤ ki ) on ki across species. The greyed area spans between an upper “preferential” threshold based on a preferential attachment model with α = 1.0 and a lower “random” threshold based on random attachment α = 0.0 (greyed lines). The area above the preferential threshold corresponds to preferential attachment with α > 1.0. The area below the random threshold corresponds to the region of antipreferential attachment.

constraints imposed on it. This is a link between structure and dynamics that can be assessed only when considering temporal dynamics in detail. Seasonal Dynamics and Phenological Variation It would be difficult to compile a reliable daily snapshot of network structure in many tropical species-rich communities. Think of a tropical rainforest where some species can be found in flower or fruit throughout the year. The daily

94

CHAPTER 5

temporal slices used in the previous section could be broadened so that they encompass periods when the relative abundance of animal mutualists or the availability of flowers or fruits remains relatively constant. Examples would be weekly or biweekly periods in which we can consider that nodes do not “age,” representing actual snapshots of the interaction patterns not affected by seasonal trends. The seasonal, year-round dynamics of complex networks are a result of the matching patterns of phenophases of the animals and the plants. A typical flowering or fruiting phenophase for a plant species (see, e.g., Fig. 4.3, top panel) starts with a few individuals producing a few flowers or fruits. This is followed by a period of maximum resource production resulting from anthesis or fruit ripening until a peak availability is reached, which a finally leads to a decline. For most species, these phenophases are characterized by a steep increase until the peak date and then a longer period of decay. When we consider the animal community (see, e.g., Fig. 4.3 bottom), a similar pattern emerges: different species have activity or presence periods with variable overlap among them and with the plant phenophases along the season. A longer phenophase of a species frequently translates into more days of potential attachment to new partners and an increased linkage level. Thus, the shape of the phenophase frequency distribution can pervasively influence the degree distribution among species (Olesen, Bascompte, et al., 2008; Jordano and Bascompte, in preparation), because the overall phenology of the network is just the succession of all these species-specific phenophases. A detailed recent analysis of the seasonal variation in a plant-frugivore network was carried out for the plant-frugivore community of the lowland scrublands in southwest Spain (Carnicer, Jordano, et al., 2009; see Fig. 4.2, HR network). The authors analyzed biweekly time slices of the plant-frugivore interactions (Fig. 5.5) and related temporal changes of network patterns to variations in resource tracking by the frugivore community and the differential ability of frugivores to switch seasonally between arthropod and fruit-based diets. Theory shows that the presence of behavioral switching between alternative resources can contribute to coexistence when competitors differ in trophic-related traits. Interestingly enough, switching can generate Figure 5.5. Temporal (biweekly) changes in the Hato Ratón lowland frugivory network during the 1981–1982 fruiting season (see Fig. 4.3). Dark nodes (left column in each graph) are frugivorous birds; light nodes are plant species. Sizes of the nodes is proportional to their actual local abundance during each time interval. Based on Carnicer, Jordano, et al. (2009).

1-15 July

16-31 July

1-15 August

16-31 August

1-15 September 16-30 September

1-15 October

1-15 November

16-30 November 16-31 December

1-15 January

16-28 February

96

CHAPTER 5

disruptive selection on such traits in a low-diversity community, promoting biodiversity. Carnicer, Abrams et al. (2008) and Carnicer, Jordano, et al. (2009) presented evidence for widespread switching behavior for this Mediterranean scrubland frugivore community, resulting in a succession of temporal networks underpinned by variation in frugivore and fruit abundance. Note the analogies between the daily sequence of the arctic pollination network (Fig. 5.3) and this seasonal trend in the Mediterranean plant-frugivore network (Fig. 5.5). In both cases, the season starts with highly simplified versions of the interaction patterns, peaks in complexity in the middle of the season, matching the peak flowering or fruiting of the plant community, and then decreases in complexity at the end of the season. Carnicer, Jordano, et al. (2009) found that frugivore species differ in beak characters related to their relative use of arthropod and fruit resource channels and that the timing of switching is correlated with the relative use of resources. Hence, the temporal sequence of network buildup is caused by preference changes among species, with a few species at the core of the network contributing most of the trend due to their switching behavior and tracking of the plant resources. Fast switchers in this lowland network tended to be central species in the network, heavily relying on fruits, with large gapes and the ability to efficiently handle a wide array of fruit species. In contrast, more peripheral species (see Fig. 4.2, bottom) tended to use fruits infrequently, have narrow beaks, and slowly respond only to the fruiting of the dominant species in the plant community. The results demonstrate that temporal variation in consumer-resource-based mutualistic networks can be explained by short-term variation in plant resources (e.g., nectar, fruits) and the animal abundances and seasonal dietary switches between alternative resources (e.g., fleshy fruits and arthropods in frugivores). These results confirm the important role of foraging adaptations in determining network patterns, as we discussed in the previous chapter. Year-to-Year Variability A third scale of network dynamics is the year-to-year variability. Recently, several papers have simultaneously analyzed temporal dynamics on the structure of pollination networks (Petanidou, Kallimanis, et al., 2008; Olesen, Bascompte, et al., 2008; Alarcón, Waser, et al., 2008; Olesen, Stefanescu et al., 2011). Both Petanidou, Kallimanis, et al. and Olesen, Bascompte, et al. have found converging results, further confirmed by Alarcón, Waser, et al. All three papers show that there is an important variability component when we look at either the number of species or the number of interactions. For

M U T UA L I S T I C N E T W O R K S I N T I M E A N D S PAC E

97

example, Olesen, Bascompte, et al. found that about 20% of the pollinator species seen one year were not observed the following year. Regarding interactions, these figures go up to more than 60% of nonrepeated interactions. The species involved in this between-year turnover were those with more specificity of interactions, often rare, and whose interactions make up the tails of the nested matrix. Thus, the dynamics of the interaction pattern were confined mainly to the tails of the nested network. Petanidou and Potts (2006) had already noticed a similar, very dramatic turnover in pollinator species number throughout the season, indicating that strong temporal dynamics may be a general property of pollination networks. In a more detailed analysis of yearly variations, Petanidou, Kallimanis, et al. (2008) found that only 53% of the plant species, 21% of the pollinator species, and around 5% of the interactions were present through the 4-year period in a natural reserve within a few kilometers from Athens, Greece. Interestingly enough, these authors were able to assess the mechanism by which interactions were not observed from one year to another. It could be due to the fact that one or the two species were not present or to a lack of interaction despite both species being present. These authors found that approximately 30% of the turnover in interactions between any pair of years was due to species shift in interactions given that both species were present. Other studies have also suggested these shifts in interactions (Lundgren and Olesen, 2005; Basilio, Medan, et al., 2006; Medan et al., 2006; Olesen, Stefanescu et al., 2011). Despite the preceding variability in species and interactions across years, there is a remarkable constancy in some overall network properties, such as connectance and nestedness. These basic network properties remain relatively invariant, either with no variation in species composition among years (Olesen, Bascompte, et al., 2008), or with substantial turnover in species identities (Petanidou, Kallimanis, et al., 2008). For example, Petanidou, Kallimanis, et al. explored degree centralization, connectance, nestedness, average distance, and network diameter and found a robust pattern of network structure. They found that connectance varied between 0.026 and 0.03, degree centralization ranged between 0.09 and 0.15, and nestedness varied between 0.97 and 0.98. The robustness of nestedness through time is in agreement with a previous result on the relationship between this measure of network structure and sampling effort (Nielsen and Bascompte; 2007; see the following). These temporal shifts are also evident in plant-frugivore communities, where despite statistically consistent phenological patterns across species (e.g., Fig. 4.3 for the plant-fruiting seasons), variations exist in the timing, phenophase length, and peak fruiting dates between years (Jordano, 1988). Either for flowering or fruiting, among-year and supra-annual variations in

98

CHAPTER 5

plant reproductive output are frequently associated with mast-flowering and mast-fruiting events (Kelly, 1994; Kelly and Sork, 2002). These supra-annual variations would cause some species to be absent from the network in a particular year. The overall picture, therefore, is one of dramatic changes in the composition of interacting assemblages, with marked seasonality in a given year and broad year-to-year variation in the species identity and interaction patterns. Note, however, that most analyses of yearly variations in network patterns to date deal with qualitative networks. Although quantitative changes can be substantial, we expect species at the core of the network to be more resilient to yearly changes and those at the nestedness tails to be more prone to betweenyear changes in their interaction patterns.

SPATIAL MOSAICS Mutualistic interactions are shaped through space, as beautifully addressed by the geographic mosaic theory of coevolution (see Chap. 1). This approach, led by John N. Thompson, has made the point that the strength and sign of a coevolutionary interaction largely changes through the landscape (Thompson, 2005). Just as the metapopulation approach identified populations distributed in patchy habitats, the geographic mosaic theory of coevolution identifies interactions distributed across disjoint patches. For example, the case study of Greya politella and its plant host Litophragma parviflorum identifies regions where the interaction is mutually beneficial and other regions where it is antagonistic. Whether the net effect is positive or negative depends on the presence of other pollinators. If they are not present, the moth acts as a pollinator even when the developing pupa will eat a small fraction of the ovules. If, on the other hand, there are other pollinators, the interactions are detrimental for the plant (Thompson and Cunningham, 2002). This ecological scenario could be extended to more complex webs of interaction (e.g., Fig. 5.6) where multiple species change over space. In Figure 5.6, a metanetwork of three animal mutualists and three plant species, in analogy to the metacommunity concept, changes in space with four distinct local networks composed of distinct species subsets. What remains to be done is to explore how larger networks are structured in the landscape. This exercise could help bridging the two major approaches to mutualism in species-rich communities, the geographic mosaic and the network approach. For example, we could study how species in the core of a nested network are distributed in space. Because nestedness is a concept

Figure 5.6. A geographic mosaic view of complex networks. Patches within an extensive region can support different subsets of nodes and interactions of a larger, complex network. The ongoing coevolutionary dynamics should be analogous to other geographic mosaic coevolution scenarios, with local differentiation among network topologies balanced by exchange of partners and interactions among localities (arrows).

100

CHAPTER 5

originally proposed in island biogeography, the problem could be phrased in terms of looking at the intersection between two nested distributions: the regional distribution of species across patches and the local distribution of species with the network of interactions. If widely distributed species tend to occupy the core of a nested network of interactions, then we could conclude that geographic range is a predictor of position in the network of interactions, something recently confirmed by Aizen, Sabatino, et al. (2012). In relation to simulations of species extinction (Solé and Montoya, 2001; Dunne, Williams, et al., 2002b; Petchey, Eklof, et al., 2008; Srinivasan et al., 2007; Rezende, Lavabre, et al., 2007), this would provide additional arguments for the assumed ranking by which the first species to be deleted are the least connected ones, since these would also be the species found in a smaller number of localities, and, therefore, more prone to extinction (Davies et al., 2004; Srinivasan et al., 2007). A first step in addressing the spatial distribution of networks across a landscape is provided by Fortuna, Krishna, et al. (2013) analyzing a lattice model of a mutualistic network. The authors used 20 real mutualistic networks as the skeleton of a metacommunity model in which species present in a patch interact (animals can persist only if at least one of its plant partners is in the same patch) and disperse to neighboring patches (a plant can be dispersed only if there is at least one of its mutualistic animals in that same patch). At the beginning of the simulations, we introduce all species in each patch, and the dynamic rules of colonization/extinction take place until the system reaches a stochastic steady state. In this type of setting, we can follow the local network at each patch and construct the cumulative network as more patches are sampled (see Fig. 5.1a). We can then look at how the number of interactions scales with the number of sampled sites. This curve is highly nonlinear (Fig. 5.7), with a fast increase in the number of interactions followed by a leveling off. Also, we can quantify, for a suite of different networks, how many sites have to be sampled to account for all the interactions. This is shown in Figure 5.8. As noted, regardless of the size of the network, there is a similar number of sites that need to be sampled in order to record all the interactions. For the parameter combination used in these simulations, this number is around 10 sites. One interesting aspect that will be discussed in Chapter 6 is how this picture changes when a fraction of the habitat is destroyed. This will allow exploring the robustness of these mutualistic networks to habitat loss. Similarly, Jabot and Bascompte (2012) have taken a step further into merging network and metacommunity studies (Morales and Vázquez, 2008; Filotas et al., 2010; K.S. McCann, Rasmussen, et al., 2005; Gravel et al., 2011). They

M U T UA L I S T I C N E T W O R K S I N T I M E A N D S PAC E

Seed-dispersal networks

Pollination networks

25

60 50

20

Number of interactions recorded

101

40 15 30 10 5

20 0

10

20

30

40

250

10

0

10

20

30

40

0

10

20

30

40

150

200 100

150 100 50

0

10

20

30

40

50

Number of patches sampled Figure 5.7. The number of interactions recorded as the number of sampled patches increases in a spatially explicit metacommunity model of a mutualistic network. Each panel corresponds to a real mutualistic network used as the skeleton of the metacommunity model. Based on Fortuna, Krishna, et al. (2013).

integrated a model of bipartite interactions between plants and their animal pollinators (also extended to antagonistic interactions) in a spatial context. Next, they used a novel approach based on approximate Bayesian computation to parametrize the model with real-world networks. Overall, the results for the parametrized model suggest that plant-animal interactions generally lead to an increase of plant and animal spatial heterogeneity by decreasing local species richness while increasing β diversity. Interestingly enough, the net effect of mutualistic interactions on species richness is achieved as a balance between two opposite forces. On one hand, mutualistic interactions induce fitness differences among individuals when they act differentially among different species. Imagine, for example, a scenario where pollinators preferentially pollinate flowers with short corollas. In this case, flowers with long corollas would tend to produce fewer seeds and would be progressively filtered out of the community. Thus, mutualistic interactions act as an environmental filter for some species if these species benefit less from mutualists than their competitors. On the other hand, mutualistic interactions stabilize coexistence if different species partition biotic resources (mutualists) by having different

102

CHAPTER 5

Number of patches sampled

102

101

102

0

50

100

150

200

250

Number of interactions for each community Figure 5.8. The number of patches that have to be sampled to record all the interactions of a spatially explicit metacommunity model parametrized with the skeleton of a real mutualistic network. Each point corresponds to a real network. Independently of the size of the network, the number of patches that have to be sampled is around 10. Based on Fortuna, Krishna, et al. (2013).

specialized interactions. As shown by Jabot and Bascompte (2012), the balance between these filtering and stabilizing factors is determined by network structure, strength of interactions, and dispersal rate. For the limit of low dispersal and weak interactions, Jabot and Bascompte recover the result by Bastolla, Fortuna, et al. (2009)—i.e., the increase in species richness in nested mutualistic networks (see Chap. 6).

SAMPLING AND ROBUSTNESS The preceding result on how the number of interactions scale with the fraction of sampled sites touches an important methodological question, that is, to what degree the structure of mutualistic networks is robust to sampling effort. This is a frequently adduced criticism of empirical network studies containing no information on the sampling effort (both in time and space) used to compile the network. We turn now to this methodological question. Complex networks are frequently sparse. Not all potential links among species in ecological networks can be recorded during sampling. Forbidden

M U T UA L I S T I C N E T W O R K S I N T I M E A N D S PAC E

103

links, as discussed in the previous chapter, are nonoccurrences of pairwise interactions that can be accounted for by biological constraints, such as spatiotemporal uncoupling or size or reward mismatching (Jordano, 1987; Nilsson, 1988; Borrell, 2005; Jordano, Bascompte, et al., 2006; Blüthgen, Fründ, et al., 2008; Olesen, Bascompte, et al., 2008; see Table 4.1). Therefore, the potential links that can actually be observed in an interaction matrix of A and P animal and plant species is well below the total size of the matrix, AP. Monitoring links is analogous to any biodiversity sampling (i.e., species inventory; Jordano, Vázquez, et al., 2009) and is subject to similar methodological shortcomings, especially undersampling (Coddington et al., 2009). When we study mutualistic networks, our goal is to inventory pairwise interactions. Rather than sampling individuals that add species to our inventory, we observe and record feeding observations, visitation, occupancy, presence in pollen loads or in fecal samples, and so on to accumulate pairwise interactions. Our goal is to reduce the number of missing links in our interaction inventories and to explain the forbidden ones. Interaction accumulation curves (IAC)— analogous to species accumulation curves (SAC)—can be used to assess the robustness of interaction sampling for the studied plant-animal community datasets (Jordano, Vázquez, et al., 2009; Olesen, Bascompte, et al., 2010). For instance, a random accumulator function (library vegan in the R Package; R Development Core Team, 2010), which finds the mean IAC and its standard deviation from random permutations of the data, or subsampling without replacement (Gotelli and Colwell, 2001) can be used to estimate the expected number of pairwise interactions included in a given sampling of records. We start with a vectorized interaction matrix (Jordano, Vázquez, et al.) representing the pairwise interactions (rows) recorded during a cumulative number of censuses or sampling periods (columns), in a way analogous to a biodiversity sampling matrix with species as rows and sampling units (e.g., quadrets) as columns. This procedure plots the accumulation curve for the expected number of pairwise interactions recorded with increasing sampling effort (Fig. 5.9). Links between extremely rare species in a network can be extremely difficult to detect because of an intrinsically low species-encounter rate in the field. According to our link classification in the previous chapter (Table 4.1), these links are missing—that is, not accounted for by any implied biological constraint. However, very low abundance may also limit actual interactions to occur if the probability of interspecific interaction is constrained by the intrinsically low probability of interspecific encounter (MacArthur, 1972). The extent of phenological coupling or overlap is a temporal equivalent to encounter rate. If two species overlap phenologically by one or a few days,

104

CHAPTER 5

a link is possible but extremely unlikely. This is even further enforced by an intrinsically low abundance of the interacting species at the onset and end of their phenophases. Thus, besides the unobservable links accounted for by forbidden interactions, many links could remain missing simply because of an intrinsically low probability of interspecific encounter. This underscores the crucial point of assessing the robustness of the sampling of interactions. Sampling effort also influences the robustness of parameter estimation in network analyses. As noted earlier in this chapter, there is a high turnover in the number of species and interactions through years. Therefore, the robustness of the indices of network structure is a relevant question given the noise and sampling errors. The first paper tackling this question in relation to nestedness was by Nielsen and Bascompte (2007). These authors designed an experimental approach in a pollination site near Siljan in southeast Norway. Sampling effort was controlled both in time and space. Twelve study sites of 20 × 20 meters were evenly distributed among forest stands of different maturity. First, results were quite equivalent for spatial and temporal sampling. Although the number of species and interactions kept growing, absolute nestedness seems to be much more stable. Because absolute nestedness depends on the number of species and interactions, Nielsen and Bascompte also calculated relative nestedness (the value of nestedness for a given matrix compared with the average value for a population of randomizations of such a matrix; see Appendix C). Relative nestedness increases with sampling effort but reaches an asymptotic level well within the extent of the sampling effort considered. That is, nestedness would remain quite constant and no changes would be observed if we were sampling more days or plots beyond the asymptotic value. Of course, the exact level of spatial and temporal sampling that is enough to characterize nestedness will depend on the study system, but the point is that nestedness is much more convergent than number of species and interactions.

Figure 5.9. Interaction accumulation curves for three robustly sampled mutualistic networks. (a) and (b) Two plant-frugivore interaction networks in southern Spain. The interaction sampling in (a) was based on direct observations at focal plants and during transects; in (b) interactions were recorded from the analysis of fecal samples of animal frugivores. (c) A plant-pollinator interaction network in the arctic Greenland; interactions were sampled from focal observations at plants combined with analysis of pollen loads of the flower visitors. Box plots indicate the confidence intervals estimated from bootstrapped samples. See Olesen, Bascompte, et al. (2010) for details.

a 150 + +

100

50 +

++

+ ++ + ++ +

+ ++ ++ + + ++++++

++ ++

+ ++

+ ++ ++ ++ + +++++ +++

+ 0

Number of distinct pairwise interactions

0

10

20

30

40

50

b + + + ++ ++ + ++ +++

80

60 ++ ++ + ++ ++++ +++ ++ ++ +

40

20

+ ++ +++ +++ +

++ + ++ +++

0 0

10

20

30

40

c + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

250 200 150

+ + +

+ + + + + + + + + + + + + + + + + +

100 50 0

0

5

10

15

20

Cumulative samples (days, censuses, etc.)

25

106

CHAPTER 5

Focusing on sampling interactions is a way to monitor biodiversity beyond the simple enumeration of component species and to develop efficient and robust inventories of functional interactions. However, treating missing interactions as the expected unique result of sampling bias would miss important components to understand how mutualisms coevolve within complex webs of interdependence among species.

SUMMARY This chapter has reviewed the sparse examples addressing how mutualistic networks change in time and space. On a daily basis, network assembly is intermediate between preferential and random attachment. This process is compatible with the statistical connectivity distribution addressed in Chapter 3 for aggregated networks. In a year-to-year scale, there is a very high turnover in species and interactions across years, and yet the global structure of the network is quite constant. Across space, theoretical models predict that plantanimal interactions increase spatial heterogeneity. The spatial and temporal dimensions are also interesting from a more methodological point of view to assess the effect of sampling effort. Interaction accumulation curves are the equivalent of species accumulation curves used in biodiversity monitoring and can be used to assess the role of sampling effort. But important natural history details explain a fraction of the nonobserved links. Therefore, treating missing interactions as the expected unique result of sampling bias would miss important components of the ecological and (co)evolutionary basis of mutualistic networks.

CHAPTER SIX

Consequences of Network Structure

In Chapter 3 we described the structure of mutalistic networks. This essentially has been like taking a picture or snapshot. Our motivation was that the structure of ecological networks has a great potential to affect network dynamics and stability (May, 1973; Pimm, 1979; Sugihara, 1983; May, 2006; Pascual and Dunne, 2006). In this chapter we will review the consequences of network structure. In the first sections we will focus on consequences for community dynamics; later on we will review consequences in the face of several drivers of global change such as habitat loss and biological invasions. Finally, we will address the implications of mutualistic network structure for coevolution in highly diversified plant-animal mutualisms. As in Chapter 5, here we have to rely on theoretical approaches because it is quite difficult to perform experiments in the field or have long enough temporal series to address dynamics. A recent number of studies, however, have started to experimentally assess the consequences of, for example, exotic plants, pollinators, or frugivores on native networks (Olesen, Eskildsen, et al., 2002; Memmott and Waser, 2002; Morales and Aizen, 2006; Bartomeus, Vilà, et al., 2008; Aizen, et al., 2008; Kaiser-Bunbury, Muff, et al., 2010). There are two approximations to network robustness, topological and dynamic, both relying on either analytic models or computer simulations. Topological stability is based on node-deletion simulations and the subsequent effects on network fragmentation. This follows the influential paper by Albert, Jeong, et al. (2000) on the Internet, although species-deletion stability had already been explored in ecology by Pimm (1979) using simulations and by Paine (1992) and others using experimental approaches. A second type of stability widely explored in theoretical ecology is demographic stability, for example, the study of local stability in the proximity of a steady state. It describes whether small fluctuations around this steady state will amplify or die out through time (May, 1973; Solé and Bascompte, 2006).

108

CHAPTER 6

a

e

b

f

c

g

d

h

Figure 6.1. The robustness of mutualistic networks to species extinctions. The figure shows the progressive loss of network structure as nodes are progressively deleted from the least connected ones [(a)–(h)]. Based on Bascompte and Jordano (2007).

COEXTINCTION CASCADES The architecture of mutualistic networks may have profound implications for their robustness (Figs. 6.1, 6.2). We have already mentioned that the work by Albert, Jeong, et al. (2000) illustrated the relationship between connectivity distribution and network robustness to error and attack. Random networks with exponential degree distributions are very fragile because the network becomes suddenly fragmented after a small fraction of nodes are removed. On the other hand, a network with a scale-free degree distribution is very robust to the random loss of nodes. The reason is that the low-connected,

C O N S E Q U E N C E S O F N E T W O R K S T RU C T U R E

a’

e’

b’

f’

c’

g’

d’

h’

109

Figure 6.2. The fragility of mutualistic networks to species extinctions. This is as it was Fig. 6.1, but the order of species driven extinct goes from the most connected ones [(a
)–(h
)]. In contrast to the previous case, the network collapses for a small fraction of species deletions. A few highly connected nodes have a strong influence in gluing the entire network, and their removal leads to a cascade of coextinctions of species left without resources. Based on Bascompte and Jordano (2007).

less-frequent nodes are the ones first going extinct. The few highly connected nodes act as the backbone connecting the whole network. The network becomes fragmented only after a high fraction of nodes have been deleted. However, the scenario is very different if nodes are not randomly deleted but we target the hubs. If we delete the most-connected node, then the secondmost-connected node, and so on, the network is very fragile: it collapses very soon. Thus, scale-free networks are very robust to random species extinctions,

110

CHAPTER 6

b

Percent plant species visited

100 80 60

c

40 20

40 80 100 20 60 0 Percent pollinator species removed

1.0

Cumulative prop. secondary extinctions

a

0.8 0.6 0.4 0.2 1.0 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1.0

Prop. species removed Figure 6.3. (a) Extinction patterns for the pollination networks of Clements and Long (1923) and Robertson (1929). Solid lines, moving from the most to least linked pollinator species; dashed line, least to most; dotted line, random extinctions. (b) and (c) Proportion of species lost to secondary extinctions as a function of the proportion of species removed (primary extinction), either from the plant or animal species sets, from Clements and Long and Robertson’s datasets at top and bottom, respectively. The diagonal dashed lines connect points at which all species in the network are lost. Triangles show species removal starting from the most connected species; squares, least connected removed first; diamonds, random removal. Extinction of the strongest interactors always leads to fast-paced loss of partners. Based on Memmott, Waser, et al. (2004).

but very fragile to the extinction of the most generalist species (Albert, Jeong, et al., 2000). Similar species-deletion experiments in food webs have reported on the fragility of ecological networks (Pimm, 1979; Solé and Montoya, 2001; Dunne, Williams, et al., 2002a). In the context of plant-animal mutualistic networks, Memmott, Waser, et al. (2004) repeated this node-deletion experiment with two large plant-pollinator networks. In this case, the authors simulated the progressive extinction of pollinators and explored the cumulative secondary extinction on the plants depending on them (Fig. 6.3). This probably reflects the most endangered status of pollinators. Also, Memmott, Waser, et al. analyzed several extinction sequences, including one never explored before, that is, less-connected pollinators being the first to go extinct. The rationale for this is the correlation between degree and abundance reported previously and the fact that specialists are also more fluctuating and, therefore, have a higher risk of extinction.

C O N S E Q U E N C E S O F N E T W O R K S T RU C T U R E

111

Memmott, Waser, et al. concluded that mutualistic networks are very robust to all extinction orders, especially when the less linked species are the first going extinct. These authors adduced both to the truncated power-law degree distribution and to the nested structure as explanations for such robustness (Morris, 2003; Memmott, Waser, et al.; Jordano, Bascompte, et al. 2006). This relative robustness of mutualistic networks contrasts with the fragility found on food webs. However, Memmott, et al. acknowledged that they were simulating the extinction of species only within one trophic level, whereas in food webs several trophic levels were considered. In the case in which both plant and animals could be driven extinct, starting from the most connected ones, the collapse of the network is similar to the one reported previously for food webs. Burgos, Ceva, et al. (2007) further explored the consequences of the nested structure in the face of species extinction, building on the paper by Memmott, Waser, et al. (2004). Burgos, Ceva, et al. analytically explored a measure of network robustness defined as the cumulative fraction of secondary extinctions as the fraction of species driven extinct increases. They proved that if specialist species have the highest extinction probability (something for which there is solid evidence; see Aizen, Sabatino, et al., 2012), then a nested structure is the most robust one. This confirms arguments based on the properties of a nested network, namely, a cohesive core and asymmetries in degree, as noted in Chapter 3. Recent approaches to coextinction cascades include more realistic assumptions, and we are just starting to grasp their complexities. One example would be the consideration of interdependent networks (Buldyrev, Parshani, et al., 2010). For instance, different networks of plant-animal interactions (plantpollinator, plant-disperser, plant-herbivores, plant-pathogens) actually describe subsets of the whole web of interactions at the ecosystem scale (Melián, Bascompte, et al., 2009; Fontaine, Guimarães, et al., 2011; Pocock, Evans, et al., 2012). A second example would be the possibility for persisting species to reconnect with other species (Kaiser-Bunbury, Muff, et al., 2010). This foraging adaptation was already shown to stabilize model food webs, reversing the predicted effects of complexity on stability (Kondoh, 2003). Mutualistic organisms (either plants or animals) are highly flexible in their interactions, and it is quite likely that the absence of a partner species causes a switch to an alternative partner. This would result in an scenario of “rewiring” the links after a partner disappears and, thus, a more resilient response to loss of mutualists. Kaiser-Bunbury, Muff, et al. accounted for pollinator behavior by including potential links into temporal snapshots (12 consecutive 2-week networks) to reflect mutualists’ ability to switch interaction partners. Simulations of node loss (Fig. 6.4) suggested a linear or slower-than-linear secondary extinction

112

CHAPTER 6

Qualitative (species richness) Animals (% species)

Plants (% species)

100

50 Strongest Weakest Random

0

Quantitative (interaction strength) Animals (% NA)

Plants (% NA)

100

50

0

0

50

Animal extinction (% species)

100

0

50

100

Plant extinction (% species)

Figure 6.4. Extinction trends following systematic removal from the strongest (black solid line) or weakest interactor (grey solid line) and random removal (dashed line). Top, simulations run on qualitative datasets, that is, presenceabsence interaction matrices; bottom, same simulations on quantitative datasets, that is, interaction matrices with interaction strength. Thick lines (restored site) and thin lines (unrestored site) show the distinct extinction trends of the different restoration schemes. The left panels display the decline of plant species (top) and interaction strength (bottom) following the removal of animal species. The right panels display the decline of animal species and interaction strength following the removal of plant species. Based on Kaiser-Bunbury, Muff, et al. (2010).

in terms of species richness, whereas quantitative data showed a sigmoidal decline of plant-interaction strength upon removal of the strongest interactor. The temporal snapshots (see Sect. 5.1) indicated greater stability of rewired networks over static interactions. Any driver of anthropogenic disturbance, however, that promotes the extinction of the strongest interactors might induce a sudden collapse of pollination networks. All the previous work focuses exclusively on the size of the coextinction cascade, that is, on how many species are going coextinct. It does not look at the identity of such species. Of course, not all species are the same, so we may wonder whether there is a bias in the choice of the species that are driven extinct. One way to address this issue is to add phylogenetic information into network structure, as reviewed in Chapter 4.

C O N S E Q U E N C E S O F N E T W O R K S T RU C T U R E

113

Node-deletion experiments were performed in the 10 largest networks with taxonomic affiliation studied by Rezende, Lavabre, et al. (2007). Following the paper by Memmott, Waser, et al. (2004), extinctions started from the most specialist species to the most generalist ones. Because phylogenetically similar species tend to play similar roles in the network, coextinction cascades involve phylogenetically related species. This results in a higher loss of taxonomic diversity and a biased pruning of the evolutionary tree. In conclusion, the interaction between network and phylogenetic structures can ultimately result in nonrandom coextinction cascades. Therefore, although the structure of mutualistic networks makes them more robust to random extinction losses, the phylogenetic signal of network patterns induces a higher loss of taxonomic diversity for a given coextinction size (Rezende, Lavabre, et al.). Similarly, Petchey, Eklof, et al. (2008) showed that coextinction cascades in food web models tend to involve trophically unique species, that is, species that have a set of prey/predators not shared with the rest of the species in the food web. As a consequence, trophic diversity tends to decrease faster than expected. Future simulation work on the effects of extinctions should incorporate not only the losses of species (network nodes), but also the losses of the interactions themselves (links). This would reflect biologically realistic scenarios where the ecological value of species is lost well before they are driven to extinction, just when they become so rare that their interactions remain nonfunctional.

DYNAMIC STABILITY Asymmetry and Stability Up to now we have explored one aspect of network stability, topological stability. A second type of stability widely explored in theoretical ecology is demographic stability. There are different mathematical types of such a broad concept of dynamic stability. One widely used definition in theoretical ecology is that of local (or linearized) stability in the proximity of a steady state. It describes whether small fluctuations around this steady state will amplify or die out through time (May, 1973; Bronstein, Dieckmann, et al., 2004; Solé and Bascompte, 2006). Thus, this approach relies on two points. First, we find a feasible solution of the system, meaning a solution with positive and nonzero abundances of the populations. Second, we study its local stability. In order to analyze the stability consequences of the patterns observed in weighted mutualistic networks, Bascompte, Jordano, et al. (2006a) studied a simple model of multispecies facultative mutualisms. By solving a simplified

114

CHAPTER 6

version of the model, we can find a feasible steady state. For being feasible and stable, this solution has to fulfill the following (Bascompte, Jordano, et al.): ST αβ < , (6.1) mn where αβ is the average product of pairwise mutualistic effects, S and T are the average density-dependence terms for plants and animals, respectively, and n and m are the number of plants and animals, respectively. This suggests that as network size (mn) increases, the average product of pairwise mutualistic effects αβ has to become smaller for the system to remain stable. This product can be small if (1) both terms are small or (2) if one term is large, the accompanying term is very small. This corresponds to two network patterns reported in Chapter 3, namely, the dominance of small dependence values and their asymmetry in the few cases when one dependence is large. Once again, we assume that dependence is a good surrogate for percapita effect, which is justified both in mathematical terms and as observed in empirical studies (Vázquez, Morris, et al., 2005a; see Chap. 4). Thus, this basic analytic result suggests that network architecture increases species coexistence. However, as with any analytic results, a number of strong assumptions are made to be able to end up with such a clear, straightforward expression (Holland, Okuyama, et al., 2006; Bascompte, Jordano, et al., 2006b). To begin with, expression (6.1) is derived from a simple model, which leads to unbounded oscillations beyond certain mutualistic strength (Wright, 1989). Second, the model assumed equivalence in parameter values among species. Holland, Okuyama, et al. proposed a more complex mutualistic model with Holling type II handling times, something already suggested by D. H. Wright (1989). We will consider this aspect of model structure in a little while. However, the main simplification behind result (6.1) is that we assume a mean field, fully connected network by which all plants and all animals interact with each other. That is, the model does not incorporate the observed network topology. This is the price to pay to obtain analytic solutions. A first step in introducing network structure in dynamical models was done by Fortuna and Bascompte (2006) by studying a metacommunity model in which species interact exactly as in two real mutualistic networks. However, this relies on numerical simulations, and we cannot get general, straightforward insight. One exception to obtaining analytical results from structured networks was made by Bastolla, Fortuna, et al. (2009). They started from a general model incorporating all ingredients adduced to be important in mutualistic models such as nonlinear saturating terms (D. H. Wright, 1989; Holland, Okuyama,

C O N S E Q U E N C E S O F N E T W O R K S T RU C T U R E

115

et al., 2006) and more relevant ones such as interspecific competition between species within a set. This model generalizes previous models as particular cases and allows the reconciliation of previous results on particular cases, as we will see shortly. The model is described by the following system of differential equations:
(P) (P) (P)
γik(P) Ni(P) Nk(A) dNi(P) βi j Ni N j + , (6.2) = αi(P) Ni(P) −  (P) (A) (P) dt l∈A γil Nl j∈P k∈A 1 + h where superindices (P) and (A) denote plant and animal, respectively, Ni represents the number of individuals of species i, and P and A indicate the sets of plant and animal species, respectively. The parameter αi represents the intrinsic growth rate in the absence of mutualism, and βi j represents the direct interspecific competition coefficient for resources between species i and j (e.g., light and nutrients in the case of plants; breeding sites in the case of animals). The last term describes the mutualistic interaction through nonlinear functional responses representing a saturation of consumers as the resources increase. The parameter γik defines the per-capita mutualistic strength of animal k on plant i, and h can be interpreted as a handling time. The equations for animal populations can be written in a symmetric form interchanging superindices (A) and (P). Network Structure and Species Richness The approach by Bastolla, Fortuna, et al. (2009) was based on previous theory for competitive systems that predicts the maximum number of coexisting species for a given value of environment variability (Bastolla, Lassig, et al., 2005). This approach is based on two interrelated concepts (see Appendix G for a more detailed development of the conceptual analysis presented here). The first concept is that of structural stability measuring the volume in parameter space compatible with a solution. This contrasts with the most common approach in theoretical ecology based on local stability, measuring whether a local solution is stable in front of small perturbations. The second concept is that of effective competitive matrix, which describes the competitive effect of one species on another. These effects can be both direct competition for common resources, such as light and nutrients for the case of plants (represented by parameters β in eq. 6.2), and indirect effects mediated by the sharing of pollinators. The key to this development is that the sign and magnitude of these indirect effects will depend on network structure. On one hand, plants compete for pollinators (negative effect); on the other

116

CHAPTER 6

hand, there is a facilitation (positive) effect: if a plant does well, this may increase the abundance of its pollinators, which in turn may benefit another plant species. The largest eigenvalue of the effective competitive matrix limits the maximum number of coexisting species. The larger the number of species, the lower the volume in parameter space compatible with a solution of equation (6.2) with all species persisting. Because environmental stochasticity imposes a minimum region of parameter space, this sets up a limit on the total number of coexisting species. The situation considering exclusively competitive interactions will serve as a baseline limit to biodiversity in the absence of mutualistic interactions. Then, we can incorporate mutualistic interactions and still use the analytical approach based on the (now) effective competition matrix. Interestingly enough, we can obtain analytic expressions for the case in which we do not consider a fully connected network, as is customarily done in theoretical ecology, but actually incorporate the structure of the network. We can generalize the condition for stability of the equilibrium given by equation (6.1) for the case where interspecific competition is present (see Appendix G). This condition for persistence was not found by Holland, Okuyama, et al. (2006) because their model was not considering interspecific competition; therefore, the issue of structural stability did not arise in their model. To be more explicit, the elements of the effective competition matrix Ci(P) j can be written as   1 1 (P) (P) (P) (P) ni n j − ni j , (6.3) Ci j = δi j + (P) + R (A) S S (A) + S where δi j is Kronecker’s delta (1 if i = j, 0 otherwise), R is the mutualism-to(P) competition ratio (see Appendix G), and n i(P) , n (P) j , and n i j are the number of interactions of plant species i, the number of interactions of plant species j, and the shared interactions between both species, respectively. The important point to notice here is that the right-hand side of equation (6.3) decreases with the nestedness of the plant set, defined analytically as in equation (C.4) in Appendix C. As a consequence, we can easily see that nestedness reduces the effective interspecific competition for a given parameter combination and number of interactions across plant species. This result can be understood on the basis of nestedness maximizing the facilitative component over the competition component. Because the predicted maximum number of plant species increases

C O N S E Q U E N C E S O F N E T W O R K S T RU C T U R E

117

with decreasing effective competition, the results of Bastolla, Fortuna, et al. (2009) predict that the number of coexisting species increases with the level of nestedness of the network. This can be quantified analytically, and in Appendix G we show the equation providing the increase in biodiversity (from the baseline of an exclusively competitive system) due to the nested architecture of mutualistic networks. This analytical expression is based on the derivative of the predicted maximum number of plant species with respect to the mutualism-to-competition ratio. This theoretical prediction of the increase in biodiversity given the structure of the mutualistic network is shown in Figure 6.5 for the model incorporating the real structure of a dataset of mutualistic networks. As can be seen, more nested real communities show higher increases in biodiversity. Similar results have been provided by Thébault and Fontaine (2010), who showed that the patterns conferring higher persistence in mutualistic and antagonistic networks are nestedness and modularity, respectively. Okuyama and Holland (2008) found through numerical simulations that nested mutualistic interactions favor the resilience of model communities. This numerical result is complementary, showing that, even when the interspecific competition and structural stability studied by Bastolla, Fortuna, et al. (2009) were not limiting biodiversity, nested mutualistic interactions enhance the recovery of model ecosystems from dynamic perturbations. At any rate, it seems, therefore, that the structure of mutualistic networks not only affects the stability of a given network (Bascompte, Jordano, et al., 2006a; Okuyama and Holland, 2008; see, however, Holland, Okuyama, et al., 2006), but also determine the size of the network itself. Combined with the previous results on topological stability addressed in the previous subsections, this confirms a clear link between the structure of the mutualistic networks and their robustness and size. While higher nestedness allows more species to stably coexist in the network, it is unclear whether this benefit is distributed among each participant species in proportions commensurate with their contributions. So far, both nestedness and its implications in terms of structural stability were addressed at the network level. To quantify both the contributions to—and benefits from—such network patterns, Saavedra, Stouffer, et al. (2011) calculated how nestedness contributions vary across species within a network. To measure the individual nestedness contribution of a given species, Saavedra, Stouffer, et al. (2011) quantified the degree to which the interactions of such species are consistent with the network’s overall nestedness. Briefly, they were assessing to what degree the actual nestedness of the network compares

a 103

Biodiversity increase (real networks)

102

101

100 0.1

1 Nestedness

b 103

102

101

100 100

101

102

103

Biodiversity increase (randomizations) Figure 6.5. The nested architecture of mutualistic networks increases species richness by minimizing interspecific competition. (a) Increase in predicted maximum biodiversity as a function of the nestedness of the real network. (b) Relationship between the predicted maximum biodiversity for networks with real structure versus randomizations of such networks. Each symbol corresponds to the skeleton of a real network, solid ones indicating a significant value of nestedness. As noted, all significantly nested networks show a higher increase in biodiversity. Based on Bastolla, Fortuna, et al. (2009).

C O N S E Q U E N C E S O F N E T W O R K S T RU C T U R E

a

b

Plants

Animals

Animals

Plants

119

c

Relative frequency

0.2

0.1

0.0 –4

–2

0 2 4 6 Individual nestedness contribution

8

Figure 6.6. Species-specific contributions to overall network nestedness. (a) and (b) illustrate how the individual nestedness contribution is estimated. Specifically, the contribution of the plant species whose interactions are highlighted is measured as the degree to which the nestedness of the network compares with the equivalent value when this species’ interactions are randomized. (c) shows the empirical value of this measure across all the species of a large data set of mutualistic networks. Based on Saavedra, Stouffer, et al. (2011).

with the same value when randomizing just the interactions of the target species. As shown by these authors, although some species contribute comparatively little to the nested structure of the network, a few species contribute considerably more (Fig. 6.6). Ironically, within a range of mutualistic strength, these strong contributors to network persistence do not gain individual survival

10

120

CHAPTER 6

benefits but are in fact the nodes most vulnerable to extinction (Saavedra et al.). As the old saying says, only the good die young.

GLOBAL CHANGE AND MUTUALISTIC NETWORKS It is well known that global change has insidious consequences for species physiology, abundance, and geographic distribution. However, its influence on species interactions has remained elusive so far, partly as a consequence of difficulties in quantifying changes in interactions compared with changes in species richness. Yet, changes in biotic interactions can have profound influences in the structure and functioning of ecosystems. Tylianakis, Tscharntke, et al. (2008) tackled this issue by synthesizing and analyzing data from 688 published studies involving CO2 enrichment, nitrogen deposition, climate, biotic invasions, and land use. This study concluded that interactions are very sensitive to global change because they depend largely on the phenology, relative abundance, behavior, and physiology of multiple species. Global change increases intensity of pathogen infection, weakens plant mutualisms, and enhances herbivory. However, the majority of these studies focus on pairwise interactions involving a couple or a few species. One of the biggest challenges in global change research, therefore, is to understand the unanticipated community-wide consequences as we scale from pairwise interactions all the way to entire networks. We will turn our attention to this issue in this section. Memmott, Craze, et al. (2007) were the first to simulate climate change effects on the disruption of pollination interactions through the phenological shifts that can be expected under the predictions of global warming. They first compiled information on the projections of global warming with a doubling of atmospheric CO2 in an area of the United States. This was followed by analyzing phenological records of the insects depicting the dates of onset of activity and previously reported responses of flowering phenology due to recent climate warming. Finally, they considered the network structure of a large, composite pollination network. The idea was to scale changes in phenology that will lead to the loss of an interaction and determine how the loss of such an interaction will scale all the way up the network. The simulations carried out by these authors showed that between 17% and 50% of all pollinator species will suffer disruption in food supply, which is translated into a number of extinctions and potential subsequent coextinctions. This work clearly suggests that the expected shifts in phenology induced by climate change will translate into the disruption of temporal overlap between pollinators and their resources.

C O N S E Q U E N C E S O F N E T W O R K S T RU C T U R E

121

Other drivers of global change that have been studied in the context of mutualistic networks are habitat loss and biological invasions, which are briefly reviewed next. Habitat Loss and the Disruption of Mutualistic Networks In previous sections we have addressed the consequences of network structure for species extinctions, community stability in front of perturbations, or number of coexisting species. These consequences were analyzed at the scale of a single community, without any type of spatial consideration. But we have already seen in Chapter 5 that networks are also structured in space. Also, habitat destruction is the major force in the biodiversity crisis. Therefore, we will start here by considering a model of mutualistic network extended in space, as introduced in the previous chapter, and will proceed by simulating the destruction of an increasing fraction of the habitat. First, let us consider a spatially implicit metacommunity model composed of an infinite number of patches in which a subset of plants and animals can coexist and disperse randomly to other patches. Fortuna and Bascompte (2006) numerically analyzed the dynamics of a metacommunity model in which species interact exactly as described in two real mutualistic networks. That is, this model describes a real network of interactions and compares its dynamics with a similar model build on a randomization of such a network. Therefore, the dynamic differences between both models will reflect the effect of network structure. This spatial setting allows us to explore the response of these model networks to habitat destruction, here simulated by an increasing fraction of patches permanently destroyed, as was previously done for single species (Lande, 1987; Bascompte and Solé, 1996; Hanski, 1999) and simple trophic modules (Melián and Bascompte, 2002b). The simulated metacommunity shows that in real, heterogeneous networks some species with few interactions will go extinct soon, but a few extremely well-connected generalists will survive for high destruction values. Second, the core of species in a nested network has a high density of links; even if a few patches remain, these species will have some interactions (Fortuna and Bascompte, 2006). Because the combination of two randomizations allows to control for a different number of network patterns, the overall results identify both the degree distribution and nested structure as determinants of this higher robustness of real networks. The preceding result, however, has limitations because it is based on a spatially implicit model; so, we cannot look at how interactions are distributed in space. One relevant question is to determine how local networks, that is,

122

CHAPTER 6

networks at a local patch, change across patches, or what is the distribution of the number of interactions per patch. This is important because as already noted before, extinctions of interactions can be more detrimental than extinctions of species, and are probably more susceptible to global change. To address the preceding question, Fortuna, Krishna, et al. (2013) have analyzed a spatially extended version of the preceding metacommunity model. As before, these authors used a set of mutualistic networks as the skeleton of the metacommunity model. This model follows simple rules of local extinction and colonization, similar to the rules of the spatially implicit model (see Chap. 5). One important difference is that we can now keep track of the interactions at each lattice site and, therefore, see how they are distributed across the landscape. Figure 6.7 displays the collapse of the mutualistic network as habitat is progressively destroyed. The spatial distribution of interactions is now quite heterogeneous (Fortuna, Krishna, et al., 2013). When habitat is pristine, the number of interactions per patch follows a Gaussian distribution, meaning that the community is quite homogeneously distributed. In this scenario, by sampling only one or a few patches, we get a good representation of the landscape. However, near the brink of extinction, at which interactions start going extinct, the situation changes. In this case, the distribution becomes very skewed, with fat tails (Fig. 6.8). This means that in the majority of patches there are only a few interactions, but a few patches have a large representation of the regional network (Fortuna, Krishna, et al.). Therefore, sampling only a few patches at the brink of extinction provides an inaccurate description of the global network. Interestingly enough, an increase in variance or skewness in a dynamical system has been shown to be an early-warning indicator of an ecosystem shift (Scheffer, Bascompte, et al., 2009; Guttal and Jayaprakash, 2008). This type of exploration, therefore, may elucidate statistical descriptors that may forecast the collapse of an ecological network. In this particular scenario, it would point toward critical values of habitat destruction at which a mutualistic network breaks down. Even when species and interactions may still be regionally present, there is almost no overlap of species in enough habitat patches, and so mutualism may be disrupted. This will arguably become an area of fast development (Scheffer, Carpenter, et al., 2009; Scheffer, Carpenter, et al., 2012). Until now we have reviewed theoretical results. Interestingly enough, these theoretical consequences of network structure have been adduced to explain a puzzling observation in the field. Contrary to expectation, we would predict that specialist species would be much more susceptible to habitat loss than generalist species. This, however, was not observed in a real system

C O N S E Q U E N C E S O F N E T W O R K S T RU C T U R E

123

1.0

Proportion of Interactions

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Habitat Loss Figure 6.7. The disassembly of mutualistic networks following habitat destruction. The figure is based on a spatially explicit metacommunity model, where the cumulative fraction of plant-animal interactions is plotted as a function of habitat destruction. The figure summarizes the result for 30 metacommunities and represents the average (±SD) fraction of the original interactions for a given fraction of sites randomly destroyed across 10 replicates and all networks. Modified from Fortuna, Krishna, et al. (2013).

(Ashworth, Aguilar, et al., 2004). It has been argued that—because of the asymmetry in specialization—both specialist and generalist plant species seem to have similar reproductive susceptibility to habitat loss: although specialists depend on a single resource, this is highly abundant (Ashworth, Aguilar, et al.). Similarly, Aizen, Sabatino, et al. (2012) showed that interactions were lost nonrandomly and more rapidly than species as habitat area decreases. Furthermore, rare interactions or those with high average dependence were lost first, so specialist species could be saved only if they interacted with generalists. These studies are among the first examples of applicability of network theory to conservation issues. Some already clear messages are that due to asymmetrical specialization, rare plants depend largely on generalist pollinators, which, in turn, rely on common plants. Thus, protection of rare plants requires the management of the more common plant species (Gibson, Nelson, et al., 2006). The fact that mutualistic networks form well-defined

124

CHAPTER 6

1.0

Cummulative distribution

0.8 0.6 0.4 0.2 0.0 1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Fraction of interactions Figure 6.8. Spatial distribution of interactions across a pristine and a fragmented landscape. Each panel represents the dynamics of a spatially explicit metacommuntiy model based on a real mutualistic network. The cumulative distribution of the number of interactions per patch is represented when all habitat is pristine (darker dots) and at the point of destruction leading to the regional extinction of interactions (lighter dots). Modified from Fortuna, Krishna, et al. (2013).

and predictable patterns of interdependences provides a community-wide perspective for species conservation. Biological Invasions on Networks The invasion of exotic species is one of the leading factors of mutualism disruptions (Traveset and Richardson, 2006; Bronstein, Dieckmann, et al., 2004). Certainly, as with many other examples of global change, this needs a community-wide perspective, and recently the network approach has been used to understand both the likelihood and the consequences of biological invasions (Memmott and Waser, 2002; Olesen, Eskildsen, et al., 2002; Morales and Aizen, 2006; Bartomeus, Vilà, et al., 2008). All these studies have concluded that the structure of plant-animal mutualistic networks facilitates invasion: invasive species become well integrated into the existing pollination network. Memmott and Waser, for example, conclude that flowers of alien plants were visited by a lower number of pollinator species, but these insects were extreme generalists. This asymmetry in specialization is in agreement with the predictions of a nested community, as noted in previous chapters. Network architecture thus provides alien species with a more abundant and reliable resource.

C O N S E Q U E N C E S O F N E T W O R K S T RU C T U R E

125

This being said, however, there is some disagreement about the existence of the so-called invader complexes, namely, groups of plant and animal invaders that rely more on each other than on native species (Olesen, Eskildsen, et al., 2002). The possible existence of such invader complexes is important because they can highly increase invasion speed and establishment success. Whereas a paper by Morales and Aizen (2006) finds that alien flower visitors were more closely associated with alien than with native species, the previous paper by Olesen, Eskildsen et al. showed little evidence of invader complexes when noting that introduced pollinators and plants do not seem to interact as much as expected by chance. A possible reason for this apparent discrepancy is that, as shown later on by Aizen, Morales, et al. (2008), invader complexes seem to occur at later stages of the invasion process. There is some emerging consensus from this preliminary work on the structure of mutualistic networks facilitating the integration of native plant species that these species become supergeneralists, and on their consequences being buffered by the structure of mutualistic networks. There seems to be some variability on the consequences of these invasions for the visitation rates of native species. Aizen, Morales, et al. (2008), for example, have found that the invasion of species has not changed network connectivity but has transferred some links from native generalist species to supergeneralist alien species. Invasion, however, can modify some basic patterns of network structure such as interaction strength and the distribution of asymmetries (Aizen, Morales, et al.). On average, highly invaded networks have weaker dependence values between species than noninvaded ones, which is related to the invasive species being involved among the most asymmetric interactions of all. Regarding previous theoretical findings, this seems to suggest that invasion leads to more stable networks. This paper is also important in providing a network-based explanation for the dynamical process of alien invasion. Similarly, Bartomeus, Vilà, et al. (2008) experimentally compared invaded and noninvaded plots to explore how two alien plants with high flowers, Carpobrotus affine and Opuntia stricta, affected the structure of the pollination networks where they become integrated. As found in similar studies, both alien species received a higher number of insect visits compared to native species and acted as supergeneralists. The consequence of this for the pollinator visitation rates on the other plants, however, differed between these two plant species. Although Carpobrotus facilitated the pollination of the other plant species, Opuntia competed for pollinators with native species (Bartomeus, Vilà, et al., 2008). The preceding results, however, assess species invasion from static measures of network structure. They did not evaluate the dynamic implications

126

CHAPTER 6

of these invaders. The first step in addressing dynamic changes induced by alien plant species was performed through an extension of the metacommunity model by Fortuna and Bascompte (2006), in which some species were invaders (Valdovinos, Ramos-Jiliberto, et al., 2009). The removal of alien plants had the effect of decreasing the persistence of the remaining species in the network. This removal of alien plants was also associated with changes in the topology of the networks through time. This paper calls for caution on the management plans for the eradication of alien species. Invasion is one end of a spectrum of changing species number, with defaunation being the other end. This is an increasingly important problem in tropical ecosystems, with important consequences for biotic interactions (Dirzo and Miranda, 1990). Hunting preferentially targets large species of mammals and birds that play a paramount role in seed dispersal. These large-sized species are highly mobile and, therefore, contribute greatly to maintaining the connectivity in fragmented landscapes. The communitywide effects of the extinction of such large species depend on the structure of mutualistic networks and their ecological correlates (see previous chapters). For example, are large-bodied frugivores randomly scattered through the matrix of interactions, or do they tend to be the generalists forming the core? If the latter, the nested structure of mutualistic networks implies that losing these few species may induce a collapse of the whole network. In sum, basic work on the structure and robustness of mutualistic networks has been recently applied to restoration ecology (Forup, Henson, et al., 2008), impact of alien species (Olesen, Eskildsen, et al., 2002; Memmott and Waser, 2002; Morales and Aizen, 2006; Bartomeus, Vilà, et al., 2008; Aizen, Morales, et al., 2008), and the conservation of rare species (Gibson, Nelson, et al., 2006).

COEVOLUTIONARY IMPLICATIONS There is very little direct information on the coevolutionary implications of network structure. As a matter of fact, some evolutionary biologists would claim that these studies all refer to networks of ecological interactions but not necessarily coevolutionary networks. Although it is true that coevolution has not been proven in these networks, it is also true that mutualistic networks are the template over which such coevolutionary processes would take place. In some of the networks studied, there is ample information over the years that clearly prove that coevolution is at work (Jordano, García, et al., 2007). It is also clear that these complex networks embed the interactions shaping

C O N S E Q U E N C E S O F N E T W O R K S T RU C T U R E

127

trait evolution in these diversified mutualisms (Jordano, 2010; Kiers, Palmer, et al., 2010). Therefore, we can conclude that considering networks as invariant coevolved structures can potentially provide lots of insight into coevolution in species rich communities. Complex network theory represents an alternative approach to diffuse coevolution, where the main implied driver for evolutionary change was simply the diversity of multiple selective pressures. Network theory allows specific predictions of how a given pattern of multispecies interactions can result in multiple instances of coevolved changes and the cascading effects these can produce. The heterogeneous, nested networks built upon weak and asymmetric interactions confer a predictable pattern of links among species that can both be generated by and affect coevolution. The evolution and coevolution of pairwise interactions within networks can depend on and be constrained, to a large extent, to the structural properties we examined in Chapter 3, yet the structure of the network can, in turn, be reshaped by the evolutionary and coevolutionary processes. As already mentioned in Chapter 3, network modules may define coevolutionary units in the sense of reflecting small groups of plants and animals with strong and biologically meaningful interactions. These groups of strongly interacting species can be regarded as the basic building blocks of mutualistic networks (Olesen, Bascompte, et al., 2007). Note that this is a bottom-up approach that numerically identifies these modules, in contrast with a more top-down classification based on the observer’s more subjective classification. We could assume that coevolution is strong within these modules and is probably much weaker between modules. This, however, is a hypothesis that needs testing. Assuming that coevolution takes place on mutualistic networks, there are two coevolutionary forces that, in combination, can potentially generate a nested network: coevolutionary complementarity and coevolutionary convergence (Thompson, 2005, 2006). Pairwise interactions build up on traits that are complementary between a plant and an animal, such as the length of the pollinator’s tongue and the corolla tube’s length. This complementarity is key for the success of the pairwise interaction, and it is based on phenotypic traits that play a role in the fitness outcome of the interaction for the two partners. Once this pairwise interaction is defined, other species can become attached to the network through convergence of traits. One example is the syndromes or convergence in fruit shape and color among species that are dispersed by mammals, as opposed to by birds. Support for the role of coevolutionary complementarity comes from simulations indicating that highly nested networks can emerge from phenotypic complementarity, particularly

128

CHAPTER 6

when several traits are involved (Rezende, Jordano, et al., 2007; Santamaría and Rodríguez-Gironés, 2007). However, it is one thing to note that nested structure is compatible with coevolutionary forces, but another thing is to prove that such a force is the main one shaping these networks. Next, we turn to the first theoretical papers explicitly addressing coevolution in mutualistic networks. Guimarães, Jordano, et al. (2011) were the first to combine a model for trait evolution with data for 20 plant-animal assemblages to explore how short-term evolution and coevolution shape the spread of traits through mutualistic plant-animal networks. Their approach thus represents a first step to formalize models of coevolution in multispecific assemblages of mutualists by explicitly considering trait evolution within a complex network architecture. Recent work by Nuismer, Jordano, et al. (2013) further explored how these coevolutionary events might, in turn, reshape the structure of the network and contribute to its evolution. Guimarães, Jordano, et al. (2011) used simulations to model single-trait evolution in animals and plants linked to each other within a complex network of interactions. Trait evolution was modeled as discrete events of change caused by selective pressures imposed by mutualistic partners. In this approach, each node (species) has a state (trait value), and the state of each node is updated in each time step. These authors incorporated techniques from population genetics—the Breeder’s equation—to estimate the response to selection in mutualistic networks and to explore the role of heritability in the observed patterns. This approach allowed the authors to follow specific trait changes, to quantify coevolutionary and evolutionary events, and to measure the degree of trait complementarity and convergence in simulations with and without some network properties—for example, the presence of supergeneralist species (Fig. 6.9). Their results revealed three fundamental aspects of coevolution in species-rich mutualisms. First, coevolution shapes species traits throughout mutualistic networks by speeding up the overall rate of evolution. This generalizes to entire networks previous results for mutually specific pairwise models (Nuismer, Thompson, et al., 1999) and studies of experimental evolution (Forde et al., 2008). Second, coevolution results in higher trait complementarity in interacting partners and trait convergence in species in the same trophic level. Third, convergence is higher in the presence of supergeneralists, which are species that interact with multiple groups of species. The analysis by Guimarães, Jordano, et al. (2011) also shows the multiple types of distinct evolutionary events embedded in complex webs of species interactions (Figs. 6.9 and 6.10), including simple evolutionary

C O N S E Q U E N C E S O F N E T W O R K S T RU C T U R E

Cascades

Pairwise

a

b

c

d

f

Coevolution

Non-coevolution

129

Indirect coevolution

e

Figure 6.9. The forms of evolution and coevolution within multispecific networks. Squares and circles represent animals and plants, respectively. Dark symbols with thick contours represent species that show shifts in phenotype, and arrows indicate which species are showing directional phenotypic change. Some of these events (dark grey arrows) include (a) simple directional change related to a partner, which may lead to evolutionary cascades (b). Other evolutionary responses may lead to coevolutionary events (light grey arrows), in which a species responds to changes in other species that were directly or indirectly caused by the first species, such as in (c) pairwise coevolution or (d) direct and (e) indirect coevolutionary events within cascades. (f) Coevolutionary and evolutionary events may cascade through species-rich networks, affecting several species simultaneously. This network depicts interactions among plants and frugivorous animals in a local community, Nava de las Correhuelas, southeast Spain. Modified from Guimarães, Jordano, et al. (2011).

changes, noncoevolutionary cascading events, pairwise coevolutionary events, and indirect coevolutionary cascading events. As a consequence of the general and invariant properties of these complex networks, they predicted that worldwide shifts in the occurrence of supergeneralists will alter how coevolution shapes webs of interacting species. For example, introduced species such as

a

Number of changes

10000

1000

100

10

1

0.04

0.03

0.02

0.01

er (p gen la ce nt s)

nv Co

Co

ar en t Co

m

pl

em

–0.01

nv e (a rge ni n m ce al s)

0

ity

Relative effect of coevolution

b

Figure 6.10. Coevolution and the emergence of complementarity and convergence. (a) The frequency (mean SD) of the different types of evolutionary and coevolutionary events (Fig. 6.9) at the end of simulated runs. (b) The effects of coevolution were computed for simulation runs in which cascading effects were allowed (black bars) or not (white bars). The effects of coevolution were estimated as the ratio between the least squares means of complementarity and convergence values for simulations allowing or not coevolution. Positive ratios indicate that coevolution increases the values of the metric of interest, whereas negative values indicate decreases in the values. Modified from Guimarães, Jordano, et al. (2011).

C O N S E Q U E N C E S O F N E T W O R K S T RU C T U R E

131

honeybees will favor trait convergence in invaded communities, whereas the loss of large frugivores will lead to increased trait dissimilarity in tropical ecosystems. These coevolutionary models in complex networks identify two central points about how evolutionary change might occur in large networks. First, coevolution is likely to be a key process shaping trait evolution within megadiversified communities. It would act through direct and indirect influences on the rates and pathways of evolutionary change. Second, both coevolutionary and noncoevolutionary changes are intrinsically interwoven in species-rich networks, with coevolutionary events generating noncoevolutionary events through a complex set of cascading effects. Therefore, the importance of coevolution in shaping traits in large species-rich networks cannot be assessed simply by determining the relative proportion of current selection that involves reciprocal selection between pairs of species. By generating cascading effects and speeding up the overall rate of evolutionary change, coevolution generates additional noncoevolutionary events (Fig. 6.10). Hence, as complex networks evolve, coevolution may appear to be increasingly rare within species-rich mutualisms specifically because it fuels further nonreciprocal, evolutionary events. The preceding approach clearly showed that coevolution plays a major role in mutualistic networks by increasing the levels of trait convergence and complementarity. However, it assumed that the mutualistic network was fixed, whereas, in fact, coevolution will likely affect the structure of the mutualistic network. Nuismer, Jordano, et al. (2013) addressed the extent to which coevolution shapes the architecture of mutualistic networks. These authors extended theory on population genetics—previously developed for pairwise interactions—to address entire networks. These models essentially describe how the fitness of an individual of a plant (or animal) species depends on both the distance of its phenotype to the optimal one favored by the abiotic environment and to the phenotype of a certain partner with which it has a probability to interact. This probability of successfully interacting is given by two different models: phenotype matching and phenotype differences. Whereas the first model describes a system where interaction depends on the match between the trait of the plant and that of the animal (e.g., corolla and tongue lengths), the second model describes a scenario where the probability of interaction depends on the extent to which the animal’s trait is longer than the plant’s one. Nuismer, Jordano, et al. showed that coevolution tends to increase interaction efficiency and connectance under both interaction scenarios. On the other hand, convergence and complementarity are much more sensitive to the strength of coevolutionary selection

132

CHAPTER 6

in the scenario of phenotype matching than on the scenario of phenotypic differences. Regarding network structure, when mutualistic interactions are mediated by a mechanism of phenotype matching, networks tend to be more specialized and compartmentalized than when those interactions are mediated by a mechanism of phenotype differences (Nuismer, Jordano, et al., 2013). A nested architecture is compatible only with weak coevolution through phenotype differences, which is arguably the mechanism at work in species-rich communities of low specificity, such as generalized pollination and seed-dispersal interactions. Overall, these results show that coevolution drives changes in community function, distribution of traits, and pattern of interactions in mutualistic networks, even with infrequent pairwise coevolutionary events. Therefore, coevolution seems to be important in species-rich communities characterized by generalized interactions, something that is often dismissed. At the same time, however, the specific consequences of coevolution for network structure and function will depend on the particular mechanism of coevolution (matching vs. differences).

IMPLICATIONS FOR NONBIOLOGICAL SYSTEMS The analysis of mutualistic networks described in this book has also permeated other systems. For example, as noted by May, Levin, et al. (2008), thinking on ecological mutualistic networks was also one of the approaches discussed during a workshop in systemic risk and finance organized by the New York Reserve and the National Academy of Sciences in the United States. As the authors concluded, there is common ground in the study of ecological and financial networks (May, Levin, et al.). Specifically, the Fedwire interbank payment network is involved in US$1.2 trillion of daily transactions between banks. This represents a weighted network in which small, specialist banks tend to interchange with the largest, generalist banks, which interact with many others. This disassortativity, or asymmetric specialization, is combined with symmetric interactions among the most connected banks, as happened in the nested plant-animal mutualistic networks. The lesson is that there is little investment in systemic risk, that is, the risk of a global crisis, as opposed to conventional risk management in local firms, that is, a single bank or company. However, the potential consequences are so important that it is well justified to think at the level of the entire network. As the authors note, this is a general trend in many areas. Building on this analogy between ecology and finance, Haldane and May (2011) were able to come up with specific recommendations

C O N S E Q U E N C E S O F N E T W O R K S T RU C T U R E

133

1.0

0.0

a

b

0.8

0.2 0.4

0.6

Plants

Pollinators

0.4 0.2

0.0

0.0

1.0

c

0.0 0.2 0.4

d

0.8 0.6

Designers

Contractors

0.4 0.2 0.0 –4

–2

0

2

4

6

8

10

–4

–2

0

2

4

6

8

0.4 0.2 0.0 10

Individual nestedness contribution Figure 6.11. The survival probability of a node in a cooperative network decreases with its individual nestedness contribution. The histograms depict the probability density of individual nestedness contribution for nodes that survive, on the top at y = 1, and for those that do not survive, on the bottom at y = 0 (y-axis, right side). The curves represent the survival probability estimated via a logistic regression (y-axis, left side) and indicate a significant negative relationship in all cases. Here, (a) and (b) represent plants and pollinators in the ecological networks; (c) and (d) represent designers and contractors in the socioeconomic networks. Based on Saavedra, Stouffer, et al. (2011).

to shape the topology of the financial network in order to decrease systemic risk. Another social system that has benefited from research on mutualistic networks is that of the New York garment industry, a system that can also be represented as a bipartite cooperative network between several manufacturer companies and several contractor companies that use their product. Thinking networks can bring new insight into understanding how these social systems are organized and what the implications are for their robustness in front of perturbations. Saavedra, Reed-Tsochas, et al. applied the same network analysis we have reviewed through this book to measure the degree distribution, nestedness, and modularity of their humans made network. Interestingly enough, the architecture of this network is very similar to the one reported previously

Probability density

Survival probability

0.4 0.2

134

CHAPTER 6

for mutualistic communities. On top of these statistical similarities, a common, general model based on a description of the number of interactions per species, complementarity of traits, and a hierarchical organization describes equally well both types of bipartite networks. This speaks up of similar architectonic patterns in seemingly different systems. Similar types of interaction constraints seem to explain equivalent structures in different systems (Saavedra, ReedTsochas, et al., 2009). The mutalistic model of equation (6.2) can also describe these socioeconomic cooperative networks. Thus, the same analytical expressions describing how the size of the network is determined by its architecture can determine how many companies can coexist in this web of interdependence. At the level of the individual node, Saavedra, Stouffer, et al. (2011) extended their analysis on the relationship between a node’s individual contribution to nestedness and its probability of survival across the 15-year time series of the cooperative network in the New York City garment industry. Whereas for the ecological networks we had to rely on simulations because there were no temporal series available, here we can benefit from the observed temporal series. As with the ecological model, the observed survival probability of a firm decreases as its individual nestedness contribution increases (Saavedra, Stouffer, et al., 2011; Fig. 6.11). This conclusion was obtained both when looking at the companies going out of business by the end of the 15-year period and when calculating this on a yearly basis across the dataset. This paradox resembles the Tragedy of the Commons in heterogeneous networks of cooperation. The results obtained can provide a basis for identifying those companies that increase the common good and to develop policies that enhance the collective interest.

SUMMARY In this chapter we have analyzed the consequences of network structure for community robustness in front of perturbations such as species extinctions, habitat loss, and biological invasions. The nested, heterogeneous structure of mutualistic networks allows a higher number of coexisting species, a higher robustness to perturbations, and a higher rate of return to equilibrium. In spatially extended models, such network architecture increases spatial heterogeneity. When looking at species identities, however, there is a phylogenetic signal on network patterns. This implies that coextinction cascades following species extinctions tend to involve phylogenetically similar species, therefore leading to a higher loss of taxonomic diversity and a biased pruning of the evolutionary tree. Regarding the implications of network structure for

C O N S E Q U E N C E S O F N E T W O R K S T RU C T U R E

135

coevolution, models suggest that even very infrequent pairwise coevolutionary events might trigger evolutionary changes in other species, increasing the cohesiveness of the interaction network. Other systems beyond ecology have recently been described as similar cooperative networks, and this has brought interesting comparisons across systems and a better assessment of systemic risk. Both in ecology and finance, stronger contributors to network persistence are the most vulnerable to extinction.

CHAPTER SEVEN

Epilogue

In this book we have examined one of the most intriguing and central components of biodiversity: ecological interactions. A particular type of interaction, mutualisms, has been largely neglected in the ecological literature until very recently, yet mutualisms represent one of the main mechanisms for shaping animal’s and plant’s life histories. Understanding mutualistic interactions and their consequences in species-rich communities remains one of the most challenging tasks in ecology. We are all fascinated by the enormous variety of details in the natural history of biotic interactions. Despite these daunting details, we should not be trapped in a reductionistic scenario precluding the analysis of global patterns of complexity. We need an interdisciplinary approach to deal with this challenge. In this book, we have reviewed some basic tools and illustrated their application to species-rich mutualistic interactions. Our analysis has revealed some patterns in the structure of mutualistic networks that pervade their organization despite the variable environmental settings where they occur. Network theory thus describes an intermediate scenario to the extremes of highly specific one-on-one coevolution and largely intractable diffuse coevolution. Network theory also provides useful tools aimed at the identification of the key elements supporting these complex networks. For instance, these tools can help to unequivocally identify species roles in local communities, helping to set conservation priorities. Similarly, understanding the modular structure of the interaction networks is central to identifying the basic coevolutionary units. The extinction of interactions can occur well before the actual extinction of the interacting species. The ecological services and functions provided by these interactions (i.e., pollination, dispersal of seeds) may collapse well before the mutualistic agents are gone. One challenge for the next few years will be to quantify the services provided by mutualistic networks and to predict how these services will be eroded as the networks collapse. This can benefit from recent advances in the identification of early-warning indicators of ecosystem shift in combination with their application to the identification of architectural features that may cause the existence of tipping points in mutualistic networks.

EPILOGUE

137

The Tree of Life provides an extremely useful paradigm for understanding species diversification. However, deeply embedded in this tree is the Web of Life, describing how interactions among species on Earth have driven this diversification. Recent evidences of lateral gene transfer in prokaryotes, for instance, have unveiled how the branches of the Tree of Life are de facto connected not only by the relations of shared ancestry, but also by horizontal connections among them. More generally, codiversification is a large-scale pattern in the Tree of Life, mediated through coevolutionary interactions among mega-diversified lineages of hosts and parasites or plants and mutualistic seed dispersers and pollinators. The approach outlined in this book will become central to understanding these gene transfers and uncovering the dynamics of prokaryote chromosome evolution. Thus, we envision that we will be moving from considering the Tree of Life to considering the Web of Life. This book started with a quotation from Alexander Calder. Because each of his mobiles has his very particular style defined by the way the parts are interconnected, we have to embrace the fact that biodiversity is more than a list of species. These species form webs of interdependence that also show a distinct style. Understanding how biodiversity is organized and how it will respond to perturbation depends largely on thinking in terms of these networks. In ecology, as well as other fields, shifting the emphasis from competition to mutualism will surely bring new insight to lasting challenges in a time of unprecedented change.

APPENDIX A

Indices Used in Mutualistic Network Analyses

Table A.1 summarizes the main set of parameters, indices, and functions used in the analysis of bipartite networks. Note that some of these have essentially the same meaning or analogous meanings for unipartite projections and for unipartite webs (e.g., food webs); however, this is by no means the rule. For instance, expressions for centrality, clustering, modularity, and so on, are slightly different when bipartite and unipartite graphs are analyzed. Recent work summarizing several of these network indices include Bascompte and Jordano (2007); Ulrich, Almeida-Neto, et al. (2009); Dormann, Fründ, et al. (2009), and Vázquez, Chacoff, et al. (2009). Also see Rayfield, Fortin, et al. (2011) for an overview of network parameters specific to spatial networks. Table A.1. Parameters and indices frequently used in ecological network analyses. A P S I

Number of animal species. Number of plant species. Total number of species. Number of pairwise interactions.

m

Number of interaction records.

ai j

Number of pairwise interactions between plant species i and animal species j. Connectance.

C kj
km 

Degree of animal species j. Average degree of the animals.

Species richness of the higher trophic level. Species richness of the lower trophic level. S = A + P. Number of links present in the network, meaning species i and j interact, i.e., nonzero elements in the interaction matrix. Number of feeding records, visits, or similar currencies for interaction frequency in a quantitative (weighted) web. Cell value for the pairwise interaction recorded between partners i and j (e.g., feeding records, visits) in quantitative (weighted) networks. Number of interactions relative to the I . maximum potential, C = AP Number of interactions between animal species j and plant species. Average number of interactions for animal species.

140

APPENDIX A

Table A.1. (cont.) ki

Degree of plant species i.


kn 

Average degree of the plants.


k W

Average number of interactions per species. Web asymmetry.

T

Matrix temperature.

N

Nestedness (matrix temperature based).

NODF Nestedness of AlmeidaNeto et al.

η

Nestedness of Bastolla et al.

Ai , A j Total number of interaction records for plant species i or animal species j.

Number of interactions between plant species i and animal species. Average number of interactions for plant species. I .
k = A+ P Balance between species richness of animals P−A and plants, W = ; positive numbers A+ P indicate more plant (lower-trophic level) species; negative, more animal (higher-trophic level) species; rescaled to [−1, 1]. Departure from a perfectly nested interaction matrix. T = 0◦ is defined for maximum nestedness 100 − T ; values ranging from 0 to 1 N= 100 (maximum nestedness).
A
P i< j Mi j + i< j Mi j    , where NODF =  A(A − 1) P(P − 1) + 2 2 ni j Mi j = 0 if ki = k j and Mi j = min(ki , k j ) otherwise. Metric based on overlap and decreasing fill. A simpler version has also been used without the rule of decreasing fill, namely, Mi j = n i j / min(ki , k j ) even if ki = k j (Saavedra, Stouffer, et al., 2011).
(P) i< j n i j η(P) =
, here defined for (P) (P) i< j min(n i , n j ) the plant set P, similar to NODF but without the rule of decreasing fill and sum in both numerator and denominator. This definition is linked to a dynamic model. Column or row
sums in weighted
interaction P ai j . matrices. Ai = Aj=1 ai j ; A j = i=1

Hi , Hj

Diversity (Shannon) of interactions per plant or animal species.

Hi = −

ES

Evenness (Shannon) of interactions across the matrix.

ES =

A   ai j j=1

−

Ai



i

j

· ln

 ai j . Ai

pi j ln pi j

ln(A P)

.

I N D I C E S U S E D I N M U T UA L I S T I C N E T W O R K ANALYSES

141

Table A.1. (cont.) G qw

Generality, or effective mean number of links per animal mutualist.

Vulnerability, or effective mean number of links per plant species.

G qw =

A  A j Hj 2 . This is analogous to the m j=1

“effective” weighted mean number of prey per predator in Bersier, Banašek-Richter, et al. (2002) Replace j by i and A by P in the equation for G qw .   A P  1  ak• a•k L Dq = m A,k + m P,k , 2 k=1 a•• a k=1 •• and the quantitative, weighted connectance L Dq is calculated as , where A P is the total AP number of potential interactions in the web. Mean number of links per species weighted by the number of interactions. It is the mean Lq =  of generality and vulnerability.  A P   A j Hj Ai Hi  2 + 2 0.5  . m m j=1 i=1

L Dq

Linkage density.

Lq

Weighted interaction density.

H2

Interaction diversity.

H2

Network-level measure of diversity of j i   ai j  ai j · ln . interactions. H2 = − m m i=1 j=1

Niche-oriented, networklevel measure of specialization. Mutual dependence asymmetry.

A standardized H2 to account for the total number of records each species has (Ai ,A j ).

AS

AS 

Scaled mutual dependence asymmetry.

Difference between the interaction strengths | (bi j − b ji ) | of partner species. ASi j = ; max(bi j , b ji ) ai j ai j and b ji = and b values where bi j = Ai Aj are the proportion of interactions between two partners, ai j , relative to the totals of each partner (Ai ), (A j ). Average difference in the (scaled) interaction strengths of partner species. ASi j =
 (bi j − bji ) , where ki is the number of ki interactions for species i, and bi j and bji are scaled interaction-strength values. AS  values close to 1 indicate strong effects on the interaction partners, with no strong reciprocal effects. AS  values close to −1 indicate strong effects from the interaction partners that do not exert a strong reciprocal effect.

142

APPENDIX A

Table A.1. (cont.) M

Modularity.

ci

Among-module connectivity.

A measure of the degree to which the network is organized into clearly delimited modules.     n   dP dA 1 ei − i i , M ∈ 0; 1 − . M= L L L n i=1 A measure of how connected species i is to all  NM   kis 2 modules. ci = 1 − , c ∈ [0, 1]. ki S=1

APPENDIX B

Fitting Degree Distributions

The frequency distributions of the number of links per node, k, have been of interest because of their potential to infer topological properties of the networks (Dorogovtsev and Mendes, 2002). The most-used approach has been to examine the cumulative degree distributions and look for the best model function fitting to an empirical pattern. This distribution depicts the probability P(≥ k) of finding nodes with k or more links. For bipartite plant-animal interaction networks, we obtain two degree sequences that include the k values for the interacting species. One sequence corresponds to the animal species, the other to the plant species (Guillaume and Latapy, 2004). Formally, the bipartite graph, G = (, ⊥, E), has two distinct sets of nodes, animals () and plants (⊥), with interactions between sets represented by E. The unipartite projections of each set, for example, G  = (, E  ) and G  = (⊥, E  ), can easily be obtained (Guillaume and Latapy 2006; Fig. 3.4). The two degree sequences include information on the number of interactions per species and the number of repeats there are in each set of the bipartite graph. The degree distributions for plants and animals can then be plotted using frequency-degree and/or rank-degree plots on a log-log scale (Fig. B.1). The data have a large number of nodes (species) with low degree and a smaller collection of species with a large number of interactions, which may suggest a fat-tailed distribution. The most frequent approach is to fit models to the frequency-degree plots by using different theoretical distribution functions (Table B.1). However, many authors use the tail of the degree distribution—that is; for k sufficiently large—to assess the model fit. These authors, for example, consider a model to be scale free if it takes an asymptotic power-law form (Stumpf and Wiuf, 2005). Three frequently used functions include the exponential, the truncated power-law, and the power-law functions. Stretched exponential and log-normal distributions typically show intermediate decays between those characteristic of the exponential and the linear decay of the power law.

a

b 1e+00

Cumulative distribution P(k)

Number of species

20

15

10

5

1e−01

1e−02

1e−03

1e−04

0 0

10

20

30

40

50

1

Degree (k, Animals)

2

5

10

20

50

100

Number of links (k, Animals)

1e+00

Cumulative distribution P(k)

50

Number of species

40

30

20

10

1e−01

1e−02

1e−03

1e−04

0 0

5

10

15

20

25

Degree (k, Plants)

30

35

1

2

5

10

20

50

Number of links (k, Plants)

Figure B.1. (a) Frequency distributions of degree values k for an empirical plantanimal interaction network (Silva, de Marco, et al., 2002). Top, animals; bottom, plants. The cumulative frequency distributions for P(k) (b) are also frequently used for this type of graph. Lines show fits to exponential (light grey), truncated power law (dark grey) and power law (black) functions.

200

FITTING DEGREE DISTRIBUTIONS

145

Cumulative frequency

1e-01

1e-03

1 0.9 0.8 0.7 0.6 1e-05 1

2

5

10

20

50

100

200

500

Degree Figure B.2. Cumulative frequency distributions of degree values k for simulated networks with different connectivity patterns derived from variations of a nonlinear preferential attachment. We plot how the (cumulative) degree distribution of a graph changes if we change the exponent of the preferential attachment process (values in the inset).

Given a probability model (Table B.1) we proceed in two steps: First, we determine the parameters that describe the degree distribution best; second, we want to compare the different models to find out which one provides the best fit to the empirical data. Thus, our first step is to estimate the likelihoods for these standard degree distribution models (Stumpf, Ingram, et al., 2005). Maximum likelihood estimation is used to estimate the parameter values so that the same rank and frequency plots can be overlaid with the best-fit lines from the different models (see, e.g., Fig. B.1b). The likelihoods and parameter values are used to find the Akaike weights to determine the best model to

146

APPENDIX B

Table B.1. Main types of distribution functions fitted to empirical degree distributions when studying plant-animal interaction networks. Function

Expression −γ

Notes

Power law

P(k) ∝ k

Truncated power law

P(k) ∝ k −γ e−k/kc .

kc is the threshold k value. Beyond it, the degree distribution progressively deviates from a power law.

Exponential

P(k) ∝ e−αk/k k −γ .

Exponential decays are also found in truncated power-law distributions. The exponential decay is less marked than in Poisson.

.

There is very high variance in k values.

¯

represent the data. These weights represent a transformation of the Akaike information criterium (AIC) to take into account the maximum and minimum AIC values of competing models and the number of parameters used to fit them. The shape of the connectivity correlation is related to the process of network formation. As an example, Figure B.2 shows how the degree distribution changes with the exponent of the preferential attachment process.

APPENDIX C

Measures of Nestedness

Nestedness analysis originated in island biogeography. Here we will not review this research systematically; the interested reader can find more details in Ulrich and Gotelli (2007) and Ulrich, Almeida-Neto, et al. (2009), where eight different indices of nestedness are analyzed and compared. Instead, here we will review the most frequently used measures in the context of mutualistic networks. One of the most popular measures of nestedness was the nestedness temperature calculator (NTC; Atmar and Patterson, 1993), which was the software employed by Bascompte, Jordano, et al. (2003) in their application of nestedness to interaction networks. This software starts by rearranging rows and columns from the most generalist to the most specilist species, so that nestedness is maximized. For each matrix, the method proceeds by drawing an isocline of perfect nestedness. This isocline separates the interaction matrix in two submatrices. In a perfectly nested matrix, all 1s (interactions) would be placed in the left-hand-side matrix, whereas the right-hand-side matrix would be empty. For each unexpected absence (holes in the left-hand-side submatrix) and presence (interactions in the righthand submatrix), the software calculates a normalized measure of distance to the isocline, and these values are averaged. Using an analogy with thermal disorder, this measure was called temperature, T , and ranged from 0◦ to 100◦ . Nestedness can then be measured as in Bascompte, Jordano, et al.: N=

100 − T , 100

(C.1)

with values between 0 and 1. One problem with this measure is that it changes with matrix size, shape, and degree of filling. Thus, absolute values of nestedness tell us little and cannot be compared across networks differing in the preceding quantities. To sort this out, Bascompte, Jordano, et al. (2003) advocated the use of a

148

APPENDIX C

relative value of nestedness N ∗ defined as N∗ =

N − N¯ R , N¯ R

(C.2)

where N¯ R is the average nestedness of a population of randomizations (see Appendix F). Other authors have used the equivalent expression, dividing by the standard deviation of the randomizations, that is, a z-score. Both measures are equivalent for the purpose of normalizing nestedness. The NTC has played a very important role in the studies of nestedness and spurred lots of subsequent work. For example, a weighted extension has been recently developed, which is equivalent to the preceding but uses information on the frequency of interactions at each cell (Galeano, Pastor, et al., 2009). Two potential drawbacks of the NTC measure are that it relies on a black box and that it has some methodological issues. For example, Rodríguez-Gironés and Santamaría (2006) made the point that the isocline of perfect nestedness is not uniquely defined and developed a new software (BINMATNEST) that solves this problem by deriving a uniquely defined isocline and by improving the rearrangement of the matrix that maximizes nestedness. Despite the improvement in the calculation of nestedness, however, BINMATNEST still works as a black box, and it is difficult for the user to understand its mechanism. A different type of measure has more recently been proposed that does not rely on a complex software but is instead based on a simple calculation. Thus, Almeida-Neto, Guimarães, et al. (2008) developed a measure of nestedness based on what the authors distill as the two main properties of a nested matrix: node overlap (NO) and decrease filling (DF), which are the basis for its name, NODF. The overall nestedness of the matrix can be defined as
A Mi j + i< j Mi j    , P(P−1) A(A−1) + 2 2


P N= 

i< j

(C.3)

where the first sum is across all pairs of plant species, the second sum is across all pairs of animal species, and P and A are the total number of plant species and animal species, respectively. For every pair of nodes i and j, Mi j = 0 if ki = k j and Mi j = n i j / min(ki , k j ) otherwise. Here, k is a node’s number of interactions; n i j is the number of common interactions between nodes i and j; and min(ki , k j ) refers to the minimum of the two values ki and k j . This nestedness metric takes values in the interval N ∈ [0, 1], where 1 would correspond to a perfectly nested network. Almeida-Neto, Guimarães,

MEASURES OF NESTEDNESS

149

et al. (2008) study the statistical properties of this measure of nestedness and conclude that it reduces potential bias introduced by network size and shape compared with alternative measures. Note that the preceding measure can also be defined for the plant or the animal set, in which case we would have only one sum (across plants or animals) both in the numerator and denominator of the previous equation. A version of NODF without the rule of decrease filling has been used by Saavedra, Stouffer, et al. The difference depends on the precise definition of the measure. This measures the tendency of a network to be nested irrespective of the degree of each row or column. Also, we could claim that the rule in NODF of adding a zero if the degree of the two rows or colums is the same is somehow arbitrary. At any rate, however, the two versions are very much correlated, as noted in the supplementary material by Saavedra, Stouffer, et al. (2011). Simultaneously, Bastolla, Fortuna, et al. (2009) derived an equivalent measure to NODF that has a dynamical basis. For the plant set, this measure reads as follows:
η

(P)

=
i< j

i< j

n i(P) j

min(ki(P) , k (P) j )

.

(C.4)

Note that this is slightly different than the previous equation in (1) not having the rule of decrease fill and (2) having the sum signs in both numerator and denominator. This measure can be related to a dynamic model of mutualistic networks, which allows relating nestedness to ecological processes.

APPENDIX D

Measures of Modularity

In this appendix we provide a brief historical and methodological overview of the methods used to measure modularity. It is just for illustration purposes, and by no means is it an exhaustive account. Interested readers can refer to the thoughtful reviews by Danon, Diaz-Guilera, et al. (2006) and Fortunato (2010). Here we will distinguish between previous measures of compartmentalization made in the ecological literature, which can be considered as surrogates of modularity, from the most recent measures, mainly in the physics and sociological literature, that are born from the analysis of complex networks. In ecology, Pimm and Lawton (1980) were among the first addressing the issue of food-web compartmentalization. Their approach consisted of using a statistic based on presence-absence data that measure the tendency of the food web to be compartmentalized. This statistic was compared to an appropriate null distribution. Cluster analysis was then used to identify species within modules. In a similar context, both Raffaelli and Hall (1992) and Dicks, Corbet, et al. (2002) used the Jaccard’s similarity index to analyze pairwise trophic similarity. The method is also based on presence-absence data, and species within modules are identified using cluster analysis. Although this procedure provides a measure of how compartmentalized the network is, it cannot be tested statistically because the observed value is not compared to a population of randomizations. Variations of this approach have been used by Fonseca and Ganade (1996). In this case, the log-likelihood-ratio test statistic is tested by Monte Carlo simulations (random matrices in which row and column totals are fixed). As noted in Chapter 3, a significantly negative connectivity correlation is indicative of a modular organization (highly connected nodes are linked to nodes with few connections; Melián and Bascompte 2002a). The advantage of this approach is that it is based on a simple correlation. Among the drawbacks are that connectivity correlation appears to be a measure of network transitivity rather than modularity (Girvan and Newman, 2002) and that this does not allow the identification of the different modules. The preceding approaches have been very useful in different stages of research in food webs and other ecological networks, but they have strong

MEASURES OF MODULARITY

151

limitations. In the last few years, modularity studies have benefited from work at the core of complex network research. The results presented in Chapter 3, based mainly on Olesen, Bascompte, et al. (2007) and further extensions, are based on the module-finding algorithm by Newman and Girvan (2004), implemented through Simulated Annealing by Guimerà and Amaral (2005). First, let us define modularity following Newman and Girvan (2004). The following development benefits from the technical insight by Rudolf Rohr. Modularity is defined as follows:
 fraction of edges   expected fraction of edges  M= − . within module i within module i all modules i (D.1) Here we will focus on the exact expression for the expected number of links, which will be slightly different for unipartite and bipartite networks. Thus, for one-mode, undirected networks, the previous expression should be written as (Newman and Girvan):     e
km kn   i  M=  −  L  2L 2L all modules i   all pairs of


species in i






=

all modules i

di di ei − L 2L 2L

 ,

(D.2)

where ei is the number of edges within module i, km is the degree of node m, and di is the sum of the degrees of all nodes in module i. For bipartite networks such as the mutualistic networks described in this book, the modularity function can be written as (Barber, 2007; Guimerà, SalesPardo, et al., 2007; Sawardecker, Amundsen, et al., 2009):    
  ei  − M= L all modules i   

=


all modules i




all pairs of species in i but with m in set P and n in set A

ei diP diA − L L L

   km kn  = L L   

 ,

(D.3)

152

APPENDIX D

where P and A indicate plant and animal sets, respectively, and diP and diA is the sum of the degrees of nodes in plant and animal sets, respectively. The last equation is provided in compact notation in Chapter 2 as equation (2.3). So far, we have derived the expression for modularity, which has to be optimized. To do so, we need to rely on an algorithmic approach. Guimerà and Amaral (2005) base their approach on simulated annealing. This algorithm is time consuming but can handle moderately large datasets for the ecological standards. In other research areas, such as the network of cell-phone calls within a country, the network is just too large, and this method is not practical. The greedy algorithm was recently created to solve this problem (Schuetz and Caflisch, 2008). This approach allows merging several modules at one time step, which prevents the premature condensation into a few large modules. This finds partitions with higher modularity values than in previous approaches. It has the advantages of being quite easy to understand and very fast, even for very large networks. It may not be as precise, though, and thus for ecological networks the approach by Guimerà and Amaral seems the most reasonable (Danon, Diaz-Guilera, et al., 2006). More recently, spectral algorithms based on eigenvectors and eigenvalues have been used (Newman, 2006). Olesen, Bascompte, et al. (2007) used the algorithm available at the time for one-mode networks. Right after that, Guimerà, Sales-Pardo, et al. (2007) developed an extension for two-mode, presence-absence networks. Olesen, Bascompte, et al. discussed, in their supplementary material, the pros and cons of using one or another version, building on arguments provided by R. Guimerà (pers. com.). Essentially, the choice largely depends on the type of question addressed. Thus, the algorithm for one-mode networks attempts to identify the most dense modules in the network, involving both plants and animals, whereas the algorithm for two-mode networks identifies groups of plants (animals) that have a large fraction of shared interactions with animals (plants). As emphasized by Olesen, Bascompte, et al., they were interested in identifying coevolutionary groups of plants and animals. This was a way to find a bottom-up classification of pollination syndromes, which contrasts with the usual a priori definition of these modules by the scientist. In contrast, the algorithm for two-mode networks makes groups of plants somehow independent of the grouping of animals. That is, an animal group does not necessarily contain the species interacting with a particular group of plants, just animals with a similar pattern of interactions. Specifically, if we focus on one of the sets, plants or animals, the resulting partition in modules would be quite similar for both algorithms. The preceding being said, there were two limitations in the study by Olesen, Bascompte, et al. (2007), which do not qualitatively change

MEASURES OF MODULARITY

153

their results but should be mentioned. Both are related to the bipartivity of the data. The first limitation relates with the measure of modularity employed by Olesen, Bascompte, et al. These authors used equation (D.2) for unipartite networks, as opposed to using the version in equation (D.3) that explicitly acknowledges the bipartivity of the data. Thus, equation (D.2) assumes that all nodes within a module can potentially interact, irrespective of whether they belong to the animal set or plant set, something addressed in equation (D.3). Second, as acknowledged in the supplementary material by Olesen, Bascompte, et al. (2007), these authors used a null model for one-mode network, which randomizes the interactions across all nodes. Because the null model is blind to whether a node is a plant or an animal, randomizations may draw interactions between two plants or two animals. This would artificially increase the difference between the observed network structure and that of the population of randomizations (see Appendix F on null models). This point was already addressed by Fortuna, Stouffer, et al. (2010), who used a randomization in the test of modularity that preserved the two-mode nature of the network. That is, only interactions between a plant and an animal were allowed.

APPENDIX E

Phylogenetic Methods and Network Analysis

Our framework for assessing the role of evolutionary history and shared ancestry in shaping interaction patterns is based on an explicit consideration of the phylogenetic affinity of plant and animal species interacting in a complex network (Figs. 4.1 and E.1). Phylogenetic relatedness among participating species can influence interaction patterns in two ways. First, species-specific traits (ecological characteristics) can show phylogenetic load and influence the position of the species within the network. These influences can be viewed as correlates between traits of the animals (X A1 , ..., X Ai , e.g., body mass, gape width, local abundance, etc.) or the plants (X P1 , ..., X Pi , e.g., fruit size, fruit production, phenology, etc., Fig. E.1) and their positions as defined by degree (k) and eigenvalue centrality (eigc). Second, the whole interaction matrix itself can be affected by the phylogenetic history of both groups if related species tend to show similar interaction patterns. Phylogenetic patterns are frequently summarized as supertrees (BinindaEmonds, Gittleman, et al., 2002; Webb, Ackerly, et al., 2002) obtained from separate phylogenies of the animals and the plants. Therefore, comparative analyses within networks extensively use phylogenetic data of the interacting species sets and rely on well-supported phylogenies. Statistical analyses can be performed with the R package (R Development Core Team, 2010) with functions available in the ade4, apes, and picante libraries (Thioulouse, Chessel, et al., 1997; Paradis, Claude, et al., 2004; Kembel, Cowan, et al., 2010). Phylogenetically independent contrasts (Felsenstein, 1985; Garland, Harvey, et al., 1992) can be used to examine the correlations between topological characteristics of the species (k and eigc) and their ecological traits. Due to the fact that many ecological traits can show a significant phylogenetic signal, these types of phylogenetic effects need to be accounted for with the use of phylogenetically-independent contrasts. The library ape (Paradis, Claude, et al., 2004) in the R package can be used to compute the independent contrast correlations. Then, generalized least squares (GLS

P H Y L O G E N E T I C M E T H O D S A N D N E T W O R K A NA LY S I S

155

Paradis [2006]) can be used for correlation tests between the independent contrasts. These tests often assume specific evolutionary models, such as Brownian or Ornstein-Uhlenbeck models of character evolution (Martins and Hansen, 1997) to test the correlations. Given that ecological traits and k and eigc can show significant phylogenetic effects (Rezende, Lavabre, et al., 2007) specific tests can be designed to assess the association between phylogenetic distance and similarity of interaction patterns across taxa. The procedure analyzes the sets of plant and animal species separately. Two distance matrices are compared; the first matrix is based on the phylogenetic distance among taxa, as the pairwise distances between the pairs of tips from a phylogenetic tree using its branch lengths. The second matrix is obtained from the interaction matrix and can be estimated with any distance (or similarity) index, such as the S4 index of Gower and Legendre (1986) on the binary interaction matrix: S4 = (a + d)/(a + b + c + d), where a, b, c, and d are the entries of the 2 × 2 contingency table of the binary interactions for two species. If a similarity index such as S4 is used, the √ distance can then be estimated as d = (1 − S4). Then the correlation between the two matrices can be estimated with a Mantel test (see Rezende, Lavabre, et al. for a similar approach and additional details). A frequently used procedure for estimating the phylogenetic signal of a given trait across a species assemblage uses a generalized least-squares (GLS) approach that calculates a K statistic, which is the ratio between the observed signal and that expected under a Brownian evolution model (Blomberg, Garland, et al., 2003). The null model represents the lack of phylogenetic signal (K = 0). Significant signals may take values of either K < 1, indicating that closely related species resemble each other less than expected under Brownian motion evolution, or K > 1, indicating that related species are more similar than expected under Brownian motion. The statistical significance of the phylogenetic signal value is calculated from a null model constructed by randomization of the data values across the tips of the tree. To account for the identity of the interacting partners, the phylogenetic signal of the mutual dependence can be obtained following the estimated generalized least-squares (EGLS) procedure of Ives and Godfray (2006). This procedure is similar to the method by Blomberg, Garland, et al. (2003) described previously to calculate phylogenetic signal but considering the matrix of interactions as the target trait. The goal is to assess the relative contributions of the two phylogenies (Fig. E.1) to the overall interaction pattern. Ives and Godfray’s (2006) procedure calculates the strength of the phylogenetic signal in the mutual dependence interactions acting through both the animal’s (d A ) and plant’s (d P ) phylogenies (see Fig. 4.8 for a graphical

T

Plant phylogeny

Traits, plants

Animal phylogeny

1 1 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 0 1 0 1 0 1 1 1 1 1 1

1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 0

1 1 1 1 1 1 1 0 1 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

1 1 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0

1 1 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

yP1

...

...

...

Fruit size

yP2

...

...

...

Abundance

yP3

...

...

...

Phenology

...

1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Geographic area

0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Traits, animals 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Degree Centrality

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

xA1

xA2

xA3...

... ...

... ...

... ...

Body mass Gape width Degree of frugivory Abundance Phenology Geographic area Degree Centrality

Figure E.1. A framework to assess the phylogenetic effects of animal and plant evolutionary histories on the interaction matrix. The goal is to explain variation in the interaction matrix in terms of species-specific traits that influence interaction patterns for each species. These traits are pivotal in determining the outcomes of the interactions (e.g., beak or tongue size, body mass, fruit size, etc.). They can be correlated (phylogenetically independent contrast correlations) with variables such as degree and centrality, illustrating species positions in the network. In addition, the phylogenetic histories of both plants and animals can influence the overall interaction pattern to variable extents.

158

APPENDIX E

explanation). This is the key difference from the previous analyses based on the K statistic, where we were analyzing the phylogenetic signal on either the plant or the animal position in the network of interactions. The interaction rate of animal species i on plant species k ( Aik ) estimates the strength of mutual dependence between animals and plants, following Ives and Godfray.
 Fik , Aik = − log 1 − Hi

(E.1)

where Hi is the total number of recorded visits to fruiting plants or flowers (or fruits consumed) by species i and Fik is the number of fruits or flowers species i obtains from plant species k. The measure of strength of association (Aik ) makes sense in the context of mutualism because Aik depends, as stated by Ives and Godfray, on both the selectivity and abundance of species, which are two crucial variables explaining the number of interactions occurring in mutualistic networks (Bascompte and Jordano, 2007). Similar expressions have been used to describe energy or mass flows between trophic levels in food webs (Bersier, Banašek-Richter, et al., 2002) or heterogeneity of interactions (Tylianakis, Tscharntke, et al., 2007). Thus, different impacts of consumers on their hosts are evaluated and increase with their overall frequency of interaction, such as in communities where links represent pollination, seed dispersal, predation, or herbivory events (Vázquez, Morris, et al., 2005; Sahli and Conner, 2006). As mentioned in main text, recent analyses of complex networks from a purely niche-based perspective (Blüthgen, Menzel, et al., 2006; Blüthgen, Fründ, et al., 2008; Blüthgen, 2010) miss this important point, because they attempt to control for differential abundances of the participating species and thus ignore an extremely central variable in the analysis of interaction frequencies and their impacts. Altered interaction frequencies in food webs after environmental perturbations could be driven by relative shifts in the abundance of different species. As noted by Tylianakis (2009), metrics of food web structure that are advocated for their insensitivity to differing relative abundances of mutualist species may not detect these ecologically important changes. The procedure first considers the interaction matrix with elements Aik (i = 1,...,n; k = 1,...,m and N = nm total number of pairwise interactions) depicting the intensity of interaction, as described before. If both animals and plants have traits that influence these interaction strengths, closely related species will show more similar trait values. Phylogenetic relatedness gives rise to covariance matrices V and U, whose elements vi j and u kl describe the covariance in trait values for the plant and animal interacting species. The amount similarity of the interaction strengths between i − k and j − l

P H Y L O G E N E T I C M E T H O D S A N D N E T W O R K A NA LY S I S

159

mutualists is indicated by the product vi j u kl . Given a plant species i, the phylogenetic correlation between its animal partners k and l is given by the animal phylogeny u kl . The same would be true for the plant phylogeny vi j influencing the correlation between plants i and j used by an animal partner k. A positive covariance between the interaction strength of two plant-animal species pairs, that is, i − k and j − l, will exist only if both the plant and animal species are phylogenetically related. To model the strength of interaction between n plants and m frugivores in the web, Ives and Godfray (2006) first vectorize the nm matrix of Aik values and estimate: A = b0 +
,

(E.2)

where b0 is the phylogenetically corrected mean of interaction strength and
is a zero-mean vector of random variables with covariance matrix given by E[

 ] = W. Matrix W incorporates the two phylogenies of animals and  plants as the Kroenecker (outer) product W = U V. The magnitude of phyogenetic signal in A depends on the covariances in U and V, with larger values of u kl and vi j indicating stronger signals due to the plant and animal phylogenies, respectively. To measure the strength of phylogenetic signal based on covariances in U and V, the procedure models the evolution of trait values of both plants and fruits based on an Ornstein-Uhlenbeck (OU) process (Hansen and Martins, 1996; Blomberg, Garland, et al., 2003). The OU model of evolution with stabilizing selection detects the presence of a phylogenetic signal through the parameter d. This parameter determines the strength of phylogenetic signal, with d = 1 indicating absence of selection on plant and animal traits under a Brownian model of evolution in the two phylogenies; 0 < d < 1 is indicative of stabilizing selection; d = 0 indicates absence of phylogenetic effects (i.e., a star phylogeny in both plant and animal groups); and d > 1 indicates disruptive selection. The goodness of fit of competing models can be estimated by comparing the mean-squared error calculated for (1) the full model (the actual phylogenies), (2) a star phylogeny, and (3) a Brownian evolution model. Current applications of phylogenetic methods to mutualistic networks therefore combine phylogenetically independent contrasts (Felsenstein, 1985) to examine to what extent the evolutionary history of plant and animal species influences their specific positions in the network and phylogenetic analysis of trophic associations (Ives and Godfray, 2006) to test for the joint influence of both phylogenies on the pattern of mutual dependence depicted by the whole interaction matrix.

APPENDIX F

Null Models for Assessing Network Structure

Looking for network patterns requires having a benchmark to asses the significance of a meaningful variable such as nestedness or modularity. The degree of absolute nestedness, for example, depends on a long list of properties, such as network size, number of interactions, connectivity distribution, and so on. In this regard, research on network structure depends very much on appropriate null models. A null model is a randomization that deliberately excludes all biological mechanisms to test whether the observed level of structure (e.g., nestedness or modularity) can be explained just out of chance using a derived probability of cell occupancy (Gotelli, 2001; Gotelli and McCabe, 2002). This has been the subject of a broad debate in island biogeography, from which research on ecological networks has benefited. For example, Diamond (1975) published pioneering work on the patterns of distribution of birds across the Bismarck Archipelago, arguing that interspecific competition was a driving force of the observed patterns of co-occurrences between species. However, with a finite and noisy sample, a pattern may be produced by chance, and inferring competition (or any other biological mechanism) may be misleading. How do we decide? The trick is to calculate how many times we get the observed pattern by chance. Only if we never obtain this pattern by chance or it happens in less than 5% of the randomizations, we can conclude that there is some meaningful mechanism underlying our data. Needless to say, null models can never point out this mechanism. However, null models make some explicit assumptions about what they preserve and have distinct statistical behaviors; inferences on the real data may largely depend on the properties of the null model used. Following this rationale, Diamond’s finding of assembly rules was questioned by Connor and Simberloff (1979). These authors claimed that many of Diamond’s previous results attributed to competition could arise by chance. This debate—kept alive for about two decades—was settled more recently by

N U L L M O D E L S F O R A S S E S S I N G N E T W O R K STRUCTURE

161

Gotelli and McCabe, who approached the problem by using appropriate null models. Their findings showed that in the majority of communities studied, there was a significantly lower number of species co-occurrence than expected by chance, therefore supporting Diamond’s assembly-rule mechanism (Gotelli and McCabe, 2002). Similar examples of this debate about whether the significance of a result depends on the type of null model arose in the context of the measures of nestedness in island biogeography. As noted in Appendix C, the nestedness temperature calculator (NTC) is not insensitive to matrix size, shape, and filling, so we need to put it in perspective. The NTC uses a null model in which each cell of the interacting matrix has the same probability of having a 1. This probability is estimated by the connectivity of the matrix (fraction of all cells occupied). The problem is that this null model has a large type I error, that is, it mistakenly rejects a true null hypothesis. Consider that this null model tends to distribute on average the same number of interactions per species. We already know that the distribution of interactions per species in mutualistic networks is quite heterogeneous (some species interact only with one or a few other species, whereas a few species are supergeneralists). Thus, it would be easy to find statistical differences between the observed matrix and the randomizations, but it would probably be wrong to adduce nestedness for this difference. There are several potential explanations. If we want to focus on nestedness, then we have to control for these differences in the number of interactions per species. This problem has been widely discussed in the context of island biogeography, and several alternative null models have been introduced (Roberts and Stone, 1990; Sanderson, Moulton, et al., 1998; Cook and Quinn, 1998; Fischer and Lindenmayer, 2002; Rodríguez-Gironés and Santamaría, 2006; Ulrich and Gotelli, 2007; Miklós and Podani, 2004; Gotelli and Entsminger, 2001). For example, Fischer and Lindenmayer (2002) developed a null model where cells were occupied on the basis of the actual probability of encountering a given species in the real data. A species observed in 60% of the sites has a probability of 0.6 of occupying any site in the randomization. Thus, occupancy probabilities varied across species but were kept constant across sites for a given species (Fischer and Lindenmayer). A similar strategy for maintaining the observed interspecies variability but in a deterministic way (i.e., exactly fixing the number of interactions per species rather than using this information to calculate the probability) was used by Prado and Lewinsohn (2004). These authors were testing whether interactions between plants and their insect herbivores were organized in compartments. Their null model reassigned at random the host plants of each insect species, thus preserving

162

APPENDIX F

the host breadth of insects but randomizing the identity of their interactors. This null model was originally proposed by Sale (1974). However, not only does the number of islands occupied by a species change across species, the number of species per island also changes across islands. Thus, why not fix rows instead of columns? Or even better, why not fix rows and columns simultaneously? If, in the latter case, nestedness is significant, we will be sure this definitely speaks about nestedness because all species have the same number of interactions as the real matrix. This sounds great and seems like it should be the end of our quest for the ideal null model. Unfortunately, this is not so easy. There are two main difficulties. First, there is not only a single way to implement a null model fixing both rows and columns simultaneously, and the different implementations lead to considerably different results. Second, fixing both rows and columns imposes huge constraints, so that the randomization becomes very conservative. At the extreme of a perfectly nested matrix, for example, no other matrix rearrangements are allowed (Ulrich and Gotelli). In general, it is the type II error (i.e., incorrectly accepting a false null hypothesis) that becomes very high this time (Ulrich and Gotelli). Thus, the probability of detecting a pattern when it is actually present is very low. We will explain these two problems next. Fixing exactly the number of interactions of rows and columns has been implemented through two major algorithmic families. One is called the random-fill procedure. This approach starts with a matrix containing only 0s and proceeds by sequentially choosing a 0 and replacing it by a 1 as long as the marginal row and column sums do not exceed the observed marginals in the original matrix (Sanderson, Moulton, et al., 1998). One difficulty associated with this method is that the process may get trapped so that no additional 1s can be replaced without failing to preserve the observed marginals. In this case, the algorithm would go backward and try to find a different path to keep filling 1s. The second major strategy for randomizing a matrix by fixing the number of 1s in both rows and columns is the swap algorithm; it starts with the actual matrix and then proceeds by sequentially reshuffling 2 × 2 submatrices with the same row and column total, as follows:

 1 0 0 1 −→ . 0 1 1 0 Some discussion has arisen on whether swap or filling algorithms perform better on the basis of their statistical properties (Sanderson, Moulton, et al., 1998; Gotelli and Entsminger, 2001), with Gotelli and Entsminger concluding that swapping methods are more efficient and reliable.

N U L L M O D E L S F O R A S S E S S I N G N E T W O R K STRUCTURE

163

The situation is more complicated because even if we chose the swap algorithm, there are several implementations. Gotelli and Entsminger (2001) considered two versions of the swap algorithm: sequential swap and independent swap. The difference is that the sequential swap uses a very large number of initial transpositions of the original matrix—let’s say 30,000—to ensure that every submatrix has enough chances to be swapped. Next, each subsequent transposition is kept as a different null matrix. In the independent swap, on the other hand, each null matrix is created through a series of independent swaps of the original matrix (Gotelli and Entsminger). Again, results may vary, depending on the method used to implement the swapping procedure. At any rate, both methods have a methodological problem: they do not sample the potential presence-absence matrices uniformly (Miklós and Podani, 2004; Rodríguez-Gironés and Santamaría, 2006). As a consequence, the distribution of any statistic measured over this population of randomizations will be biased because of the dependence of these randomizations on the initial configuration. To try to remedy this problem, Miklós and Podani proposed a third swapping procedure, the trial swap (for details see Miklós and Podani, 2004). These authors claim that the procedure leads to a uniform distribution, but its convergence is slow for large matrices. Miklós and Podani finally, suggested a combined approach to overcome this slowness, where a fast initialization is first used, followed by the trial swap. Some concerns, however, have been raised about how efficiently this combination achieves the goal (RodríguezGironés and Santamaría). The second problem with the null models fixing both the number of 1s simultaneously in both rows and columns has to do with the type I versus type II error. Null models fixing both the observed row and column totals have very low type I errors. This means that when they reject the null hypothesis, we can be strongly convinced that there is, indeed, a significant pattern. The problem is that this low type I error comes at a price: fixed-fixed null models have a high type II error. This means that it is very difficult to detect a pattern such as nestedness even when this pattern is present (Cook and Quinn, 1998; Ulrich and Gotelli, 2007). Specifically, Ulrich and Gotelli made a compelling study of the type I versus type II statistical errors of several randomization schemes. These authors created perfectly nested matrices and then progressively increased randomness. In this scenario, a good null model should detect nestedness, but the most constrained null models only did so in less than 25% of the cases (Ulrich and Gotelli). On top of the preceding constraints of the fixed-fixed null model, other authors have presented more conceptual concerns. These authors would argue that an observation would be just one realization of a random process if several

164

APPENDIX F

replicates were allowed, and, therefore, we should allow marginal totals to vary to some extent (Cook and Quinn, 1998). The equivalent to fixing both rows and columns is to probabilistically do so. This is the null model 2 in Bascompte, Jordano, et al. (2003). In this case, the probability ρi j of drawing an interaction in cell wi j is proportional to the generalization level of plant i and animal j. Specifically, this probability can be described as the arithmetic mean of the fraction of occupied cells on row (plant) i ( pi ) and on column (animal) j (q j ). This can be written as ρi j =

pi + q j . 2

(F.1)

It can be proven that this null model gives an unbiased estimate of the overall connectivity. The following demonstration is due to Joel E. Cohen (pers. com.) during a visit to our lab. Consider a matrix with P plant species in rows and A animal species in columns. Each realized element of this matrix will be wi j =1 if plant i and P A animal j interact and 0 otherwise. L = i=1 j=1 wi j is the total number of interactions (1s) in the matrix. Now, let xi j be the predicted element of row i and column j under null model (F.1). The interaction probabilities of plant i ( pi ) and of animal j (q j ) are estimated by A j=1

pi =

A

wi j

P ,

qj =

i=1

wi j

P

.

 P pi = L/A and Aj=1 q j = L/P. Then, by definition we have that i=1 Thus, the expected number of links if xi j ∼ B[( pi + q j )/2] is 



E

i, j



1  1 xi j  = ( pi + q j ) = 2 i=1 j=1 2 P

A




L L A +P A P



1 = 2L = L . (F.2) 2

Therefore, the expected number of links under null model (F.1) is the observed number of links. In a comparative analysis of several probabilistic null models used in interaction networks, Rodríguez-Gironés and Santamaría (2006) concluded that null model (F.1) is the one that performs better in the sense of simultaneously having smaller type I and type II errors. Their clever statistical analysis was based on building a “data matrix” with each one of the three probabilistic null models and, for each, calculating the p-value with each of the three null models. This provided 3 × 3 comparisons of the null model used to generate

N U L L M O D E L S F O R A S S E S S I N G N E T W O R K STRUCTURE

165

the matrix and the one used to estimate the p-value. Because the original data were already random, the p-values are expected to be uniformly distributed. Deviations from this expectation speak of the tendency of the null model to be more or less conservative, allowing the quantification of both type I and type II errors (Rodríguez-Gironés and Santamaría, 2006). Perhaps we have to conclude that there is no magic null model, all have pros and cons, and the best strategy as advocated by Melián and Bascompte (2004) is to use a suite of null models. The most robust results are those obtained consistently across a gradient of null models. For example, the original result on the nested structure (Bascompte, Jordano, et al., 2003) and asymmetric specialization (D. P. Vázquez and Aizen, 2004) in mutualistic networks was confirmed with the most conservative null model maintaining both row and column marginals (Joppa, Bascompte, et al. [2009] and Joppa, Montoya, et al. [2010]). So far, we have focused on the traditional debate of presence-absence null models derived from island biogeography. Within the context of mutualistic networks, other sources of null models have been generated that consider the frequency of interactions. For example, a recent null model by Blüthgen, Fründ, et al. (2008) treats the interaction matrix as a contingency table, in an effort to generate expected frequencies of interaction. These authors use Patefield’s algorithm to estimate the expected frequencies of an R × C table (treating the interaction matrix as a table of R rows and C columns). The problem is that Patefield’s algorithm merely keeps the marginals of the table fixed. So, for any row species with 25 interactions recorded, we can allocate them either in a single cell (with k = 1 and Ai j = 25) or among five cells (with k = 5 and Ai j = 5 for each of the five cells). The algorithm maintains neither the degree distribution nor the overall connectivity of the table. Obviously, this results in inflated estimated interaction frequencies for a large number of cells, especially those in the lower-right corner of the interaction matrix, once the interaction matrix has been reordered by row-column sorting in decreasing order of both the row and column totals. Thus, Blüthgen, Fründ, et al. (2008) use a KullbackKleiber distance to obtain a residual interaction value that is simply a weighted frequency: the observed interaction frequency minus the expected from the Patefield’s algorithm. They find that even for rare species, there are more interactions than expected by the null model and so specialization is more frequent than previously acknowledged. This is not surprising. Patefield’s procedure is adequate for statistical analysis of contingency tables that do not necessarily require preserving a given frequency of filled cells (i.e., do not necessarily preserve a k value, as is badly needed when dealing with

166

APPENDIX F

an interaction matrix). The null matrices obtained invariably have larger connectance values than real matrices because the null model does not impose any constraint on the k-values (the frequency of nonempty cells, where it allocates a given row-column combination of interactions). The basic rationale for generating appropriate null models is that abundance is commonly considered as a nuisance factor that needs to be controlled for. Thus, another problem with more elaborate null models attempting to control for species abundance is the circular reasoning underlying them: species abundance is derived from the interaction frequency data to estimate the expected interaction frequencies. This leads to an impossibility to reject the null hypothesis and to the unavoidable conclusion that abundance is all that matters to explain network patterns (Vázquez, 2005; Blüthgen, Fründ, et al., 2008). Species abundance has become an important property of community structure due to the influence of Steve Hubbell’s neutral theory of biodiversity (Hubbell, 2001). Abundance is certainly an important component of network patterns, as stressed in early work by Jordano (1987), but we claim that it has to be considered as another ecological correlate or attribute of the species in the interaction network and taken into account simultaneously with other trait variables controlling their effects (see Chapter 4). Subtracting abundance because it is the basis of neutrality is as wrong as subtracting phylogenetic effects in comparative analysis as a way to “control” the undesirable effects of phylogeny. Blüthgen, Fründ, et al. confound the abundance effects with the bias due to sampling, which might itself be the consequence of an incorrectly designed sampling protocol that is influenced by abundance variation among species. The solution when we need to factor out abundance variation across species is to have species-abundance data derived independently of the interaction data itself. This can be done by carrying out separate censuses specifically designed for estimating abundance and separate routine focal observations at plants to record visits. A most useful approach would be to have robust estimates of sampling bias and to incorporate them in the analysis together with independently derived abundance estimates. Vázquez, Poulin, et al. (2005) used path analysis for the Cold Lake host-parasite interaction dataset and concluded that even when the sampling effect is considered separately from host abundance, the indirect effect of host abundance on parasite richness is substantially high. Their results adequately account for sampling effects and are consistent with the hypothesis that observed patterns of interaction in host–parasite interaction networks result partly from abundance variation among species.

APPENDIX G

An Analytical Theory of Mutualistic Networks

FIXED POINTS AND LOCAL STABILITY

We consider here the general model of a mutualistic community described in Chapter 6 (model 6.2) and analyzed by Bastolla, Fortuna, et al. (2009). What follows is a modified and updated version of the supplementary material and methods in Bastolla, Fortuna, et al. (2009). Let us start by calculating the fixed points of the dynamical system, defined by the equations dNi(A,P) /dt = 0, and analyze their stability. In order to be able to obtain analytic solutions, we will take advantage of the fact that the handling time h is small compared with the typical intrinsic time of growth, 1/α. There are two different types of solutions. The first one is characterized by small equilibrium biomasses, N
1/ hγ . In this limit, one can expand the functional response in a Taylor series, whose dominant term yields a linear system of fixed point equations. Bastolla, Fortuna, et al. (2009) named this regime weak mutualism. A second type of fixed points correspond to equilibrium biomasses N of order 1/ hγ . The linear system is not a valid approximation anymore, but it is possible to obtain analytic insight by neglecting the terms hα with respect to hγ N . This regime was refered to as strong mutualism. Furthermore, in order to simplify the analytic expressions, we will consider (P) mainly direct competition matrices βi(P) j of mean-field type, with βi j =
 β0(P) ρ (P) + (1 − ρ (P) )δi j (Lassig, Bastolla, et al., 2001), where δi j is Kronecker’s delta (one if i = j and zero otherwise). The dimensionless parameters ρ (P) < 1 measure the extent of interspecific competition between different species of the same group. For a purely competitive system, i.e., when γik ≡ 0, the fixed point densities {Ni } satisfy the system of equations:  j

(P) (P) βi(P) j N j = αi .

(G.1)

168

APPENDIX G

The analytical expressions are symmetrical for the case of the animals. The necessary and sufficient conditions for dynamic stability are that (i) all equilibrium biomasses must be positive; and (ii) the direct competition matrix β must be positive definite. We now integrate mutualistic interactions into the competitive community. If the equilibrium densities are small, N
1/ hγ , which represents a valid approximation within the weak mutualism regime. Under these assumptions, the fixed point equations for the plant communities at the dominant order in h can be written in the form of a linear system: 

(P) (P) Ci(P) j N j = pi .

(G.2)

j

These equations are mathematically equivalent to the fixed points of a purely competitive system (equation G.1). The vector pi and matrix Ci j were termed the effective productivity and the effective competition, respectively (Bastolla, Fortuna, et al., 2009). In analogy with the purely competitive system, the equilibrium fixed point is stable if and only if all the equilibrium densities are positive and the effective competition matrix is positive (i.e., all the eigenvalues are positive). At zero order in h, the effective productivity and the effective competition are given by the expressions:  (P)
(−1) γik β (A) kl αl(A) , (G.3) pi(P) = αi(P) + k,l

(P) Ci(P) j = βi j −




(−1) γik(P) β (A) kl γl(A) j .

(G.4)

k,l

First order corrections in h, although straightforward to compute, will be omitted in what follows. They do not change the qualitative picture. Let us first consider mean field mutualist interactions, with all plant and animal species interacting between each other with equal per capita mutualistic effect, γik(A,P) = γ0(A,P) . Later on, we will relax this assumption. With this assumption, the effective competition matrix turns out to be of mean field type:     (P) (P) (P) (P) (G.5) Ci(P) j = β0 (1 − a ) δi j 1 − ρmut + ρmut ,

a (P) =

γ0(P) γ0(A) β0(A) β0(P)




S (A) . S (A) ρ (A) + (1 − ρ (A) )

(G.6)

A N A NA LY T I C A L T H E O RY O F M U T UA L I S T I C N E T W O R K S

169

The effective interspecific competition is given by (P) ρmut =

ρ (P) − a (P) < ρ (P) . 1 − a (P)

(G.7)

As can be seen from the above expression, for the mean-field system and for values of the parameter a (P) ∈ [0, ρ (P) ], the effective interspecies competition (P) is smaller than the bare competition ρ (P) . Thus, one can conclude that ρmut mutualistic interactions of mean-field type reduce the effective interspecific competition. Equation (G.7) is valid for a (P) < ρ (P) + (1 − ρ (P) )/S (P) . At this point, the main eigenvalue λ1 of the effective competition matrix becomes negative and the community enters the strong mutualism regime. Stability of the weak mutualism fixed point requires that the effective competition matrix is positive. The eigenvalues can be written  (P) (P) as follows: λ1 = (1 − a (P) ) S (P) ρmut + (1 − ρmut ) = S (P) (ρ (P) − a (P) ) + (1 − (P) ρ (P) ) and λk = (1 − a (P) )(1 − ρmut ) = (1 − ρ (P) ) (k > 1). Positivity of the com(P) (P) petition matrix requires that S (ρ − a (P) ) + (1 − ρ (P) ) > 0, which in turn yields the condition:

  1 − ρ (A) 1 − ρ (P) (P) γ0(P) γ0(A) < β0(P) β0(A) ρ (A) + + ρ , S (A) S (P)

(G.8)

which generalizes the result by Bascompte, Jordano, et al. (2006a) for nonzero interspecific competition values. Note that when ρ (P) and ρ (A) are different from zero, the maximum value of mutualistic interactions in the weak mutualism regime does not vanish for large ecosystems (large S (P) and S (A) ), but it is limited as γ0(P) γ0(A) < β0(P) β0(A) ρ (A) ρ (P) .

EFFECTS OF NETWORK STRUCTURE ON COMPETITION AND BIODIVERSITY

Local or dynamic stability is an important requirement to assess how meaningful are the feasible solutions of model ecosystems. This stability condition has been frequently addressed in theoretical ecology. Structural stability, on the other hand, is a different type of stability, defined as the degree to which a type of solution (e.g., a steady state) is robust to changes in the parameters. Here we refer to structural stability as the volume in parameter space compatible with positive densities at the fixed point. The basis of our following development is that structural stability is negatively correlated with the number of species in

170

APPENDIX G

competitive systems. Here we show that it is possible to extend this analytic insight to communities in which competition and mutualism coexist. Effective competition and structural stability

Let us consider a community in which the fixed point equations can be written

in the form j Ci j N j = pi . We refer to Ci j and pi as the effective competition matrix and the effective productivity vector, respectively. Next, let us normalize the effective competition matrix, so that it can be written as: Ci j Bi j =  , Cii C j j

(G.9)

in such a way that Bii = 1. From the main eigenvalue of this matrix, we can derive the following quantity: ρ˜ ≡

λ1 (B) − 1 . S−1

(G.10)

If the effective competition matrix is a direct competition matrix of mean-field type, Bi j = ρ + (1 − ρ)δi j , it holds that λ1 (B) = Sρ + (1 − ρ) and, consequently, ρ˜ = ρ. Thus, the quantity ρ˜ measures the effective interspecific competition, generalizing the mean field parameter ρ. To have all equilibrium densities to be positive imposes more stringent conditions on the productivity parameters { pi } as either the number of species or the interspecific competition ρ˜ is increased (Bastolla, Lassig, et al., 2005; Bastolla, Fortuna, et al., 2009). This result generalizes the mean-field result in Lassig, Bastolla, et al. (2001) and in Chesson (1994, 2000). In other words, the larger ρ˜ is, the less structurally stable the system is, in the sense that the productivity vectors must be fine tuned to get positive equilibrium densities. Assuming that the fluctuations of the productivity vector are limited from below by the environmental variability , one can obtain the following limit to the maximum biodiversity S (Bastolla, Lassig, et al., 2005): 

1 − ρ˜ S ≤ 1+ ρ˜



˜ − λ2 (B)/(1 − ρ) 

.

(G.11)

For mean-field competition matrices, ρ˜ = ρ and λ2 (B) = 1 − ρ, whence S ≤ S(1 − )/. Therefore, the maximum biodiversity parameter can be

A N A NA LY T I C A L T H E O RY O F M U T UA L I S T I C N E T W O R K S

171

defined as: 1 − ρ˜ , ρ˜

S≡

(G.12)

which sets the scale for the maximum biodiversity that a competitive community can host. To distinguish the different scenarios (purely competitive versus competitive and mutualistic), we will use the following notation: S = (1 − ρ)/ρ indicates the maximum biodiversity for the purely competitive system, ρmut refers to the effective interspecific competition in the presence of mutualistic interactions, and S mut = (1 − ρmut )/ρmut depicts the maximum biodiveristy in the presence of mutualistic interactions. Weak mutualism: mean field

The above calculations remain valid in the weak mutualist regime. As noted above, if the mutualistic interactions and the direct competition matrix are of mean-field type, the effective competition matrix is also mean field, and the effective interspecific competition parameter can be analytically computed as in equation (G.7). It can be derived from this equation that mean-field mutualism reduces interspecific competition, increasing as a consequence the number of species that can stably coexist, which is now given by: (P)

S mut ≡

(P) 1 − ρmut (P) ρmut

(P)

=

S . 1 − a (P) /ρ (P)

(G.13)

Weak mutualism beyond the mean field: nestedness

Next, the mean-field assumption that plant and animal species interact mutualistically with all species in the other group will be relaxed, while for the sake of mathematical tractability we will maintain the assumption that the strength of all existing mutualistic interactions are equal. This model will be referred to as the soft mean field. A binary matrix gik can be defined; its elements are 1 if the interactions is present and 0 otherwise, such that γik = γ0 gik . It holds that

(P) (P) (A) = gki . We further denote the number of links of plant i as n i(P) = k gik gik and the number of common links of plants i and j as n i(P) j ≡

 k

which we refer to as the overlap matrix.

(P) (P) gik g jk ,

(G.14)

172

APPENDIX G


The elements of the overlap matrix fulfill the inequality n i j ≤ min n i , n j . The interaction matrix g is maximally nested if all the elements of 
the overlap matrix take their maximum possible value, i.e., if n i j = min n i , n j (see equation C.3). Weak mutualism beyond the mean field: soft mean field

In order to obtain a simple analytical expression depicting the effect of network structure on species richness, one can introduce the soft mean-field model. In this case, all parameters are equal but the mutualistic network is not fully connected as in the mean-field case. In this model, the direct competition matrix βi j is of mean-field type and all non-zero mutualistic interactions are (P) (P) (P) equal, γi(P) j = γ0 gi j , where the binary matrix gi j is the adjacency matrix of (P) the mutualistic network and gi(A) j is the transpose of gi j . The effective productivity vector can be written down from equation (G.3) as follows: pi(P) = αi(P) +

γ0(P) β0(A) (1 − ρ (A) )

  (P) (A)  gi j α j − j

 S (A) S (A)

+S

(A)

α (A) , (G.15)

where α (A) is the average growth rate (or death rate, if α is negative) of animal species. This will make explicit the intrinsic growth rates αi . As we have already seen, a necessary condition for species coexistence under competition is that the effective productivity vector has a narrow distribution. Therefore, Bastolla, Fortuna, et al. (2009) assumed that the assembly process leads to a narrow distribution of effective productivities, and that its dispersion , which appears in equation (G.11), is the smallest one compatible with the unavoidable environmental variability. Furthermore, this is assumed not to change in the presence of mutualism. This assumption, which has to be justified through an explicit model of network assembly, implies that αi must be chosen negatively correlated to the number of mutualistic links, leaving the focus to the effective competition matrix. This depends on the network architecture but it does not depend on αi , which will not play any role in the following analytic computation. Alternatively, one could formulate the soft mean field model in such a way that the higher the number of mutualistic interactions a species has, the weaker these links are, defining the mutualistic parameters as γi j = γ0 / f (n i )gi j , where f (n i ) is a growing function of the number of links. This equation assumes that specialist species are more efficient than generalist species in dealing with their mutualistic partner. In this formulation,

A N A NA LY T I C A L T H E O RY O F M U T UA L I S T I C N E T W O R K S

173

the function f (n) should be chosen such that the effective productivity vector given by equation (G.3) is uncorrelated with the number of mutualistic interactions n i . This formulation of the soft mean field model would lead to different expressions for the effective competition matrix from the one introduced below. In the weak mutualism regime, the normalized competition matrix Bi(P) j is defined through Ci(P) j β0(P) (1 − ρ (P) )

= δi j +

Bi(P) j



1 S

(P)

+R

n i(P) n (P) j S (A) + S

− n i(P) j (A)

Ci(P) j

= , Cii(P) C (P) jj

,

(G.16)

(G.17)

where R=

γ0(P) γ0(A) β0(P) β0(A) (1 − ρ (P) )(1 − ρ (A) )

.

(G.18) (P)

Note that the matrix B (P) depends only on three numerical parameters: R, S (A) and S . To obtain further analytic insight on how mutualism influences biodiversity, Bastolla, Fortuna, et al. (2009) computed the derivative of
the main eigenvalue of the normalized effective competition matrix, λ1 B (P) , with respect to the mutualistm-to-competition ratio R at the point R = 0 (absence of mutualism). This proves that the effective interspecific competition decreases with increasing nestedness for a given distribution of number of links {n i(P) } (P) and fixed parameters. The maximum predicted biodiversity S mut increases with decreasing effective competition. Hence, the model predicts that, for perfectly nested mutualist networks, the effective competition is weakest and the maximum biodiversity is largest. Nested mutualistic interactions, thus, increase biodiversity.
 Specifically, by calculating the derivative of λ1 B (P)  , one can  easily obtain (P)

(P) (P) , where /ρmut the derivative of the maximum biodiversity S mut = 1 − ρmut

  (P) (P) (P) ρmut = λ1 B − 1 /(S − 1) is the effective interspecific competition parameter, with respect to the mutualism-to-competition ratio. This measures the relative increment of the maximum biodiversity due to mutualism, and

174

APPENDIX G

is equal to 1 (P)

S mut

 (P) ∂ S mut   ∂R 

 = 1+ R=0

+

1 S

(P)







n (P)  S

(n (P) )2  − n (P) 2 n (P) (S (A) + S

(A)

)

(P)



ηˆ (P) −

n (P)  S (A) + S

 (P)

S (P) + S S (P) − 1

,

(A)

− (1 − ηˆ (P) )

(G.19)


2

 2 where n (P)  = i n i(P) /S (P) and  n (P)  = i n i(P) /S (P) are the mean and mean square number of mutualistic interactions per plant species. The para

(P)  (P) meter ηˆ (P) = i= j n i(P) / (S − 1) n appears to be strongly correlated j i i with nestedness as defined in equation (C.3) (for real networks, the correlation coefficient is 0.97). The derivative in the previous equation is not necessarily positive. Actually, it is typically negative if there are few shared interactions (small η) ˆ together (A) with strong direct competition (small S ), so that the term ηˆ − n (P) /(S (A) + (A) S ) is negative. As a consequence, mutualism can also increase the effective competition and reduce biodiversity. Although it looks counter-intuitive, this result can be easily understood by considering that, if plant species i and j do not share any animal species (n i(P) j = 0), the direct competition between the animals interacting with them has the net effect to increase the effective competition that i and j experience. This illustrates how the direct competition for resources explicitly described by the βi j terms in the mutualistic model (equation 6.2) is now mediated by the use of a common set of mutualismstic partners. A second more stringent condition for having a mutualism-induced increase of biodiversity is that the reduction in interspecific effective competition Ci= j must be larger than the reduction in intraspecific effective competition Cii . This (P) requires that the parameter S must be large enough so that the denominator of equation (G.19) is positive. Notice that, for networks in which ηˆ attains the maximum possible value ηˆ = 1, as in the fully connected mean-field network, the increment of biodiversity (equation G.19) is always positive, independent (A) (P) of the parameters S and S .

APPENDIX H

Software for the Analysis of Complex Networks

PAJEK http://vlado.fmf.uni-lj.si/pub/networks/pajek/ Batagelj, V., and A. Mrvar. 2003. Pajek-Analysis and Visualization of Large Networks. In Graph Drawing Software, eds., M. Jnger and P. Mutzel. pp. 77–103. Berlin: Springer. PAJEK is a widely used analysis and visualization program for complex networks. Its development started in 1996, oriented to data mining. Implements many graph operators, graph exporting in bitmap, vector-based and 3D formats, and macro support. Can be connected to other packages (e.g., R). igraph http://necsi.org/events/iccs6/viewabstract.php?id=88. Csa´rdi, G., and T. Nepusz. 2006. The igraph software package for complex network research. InterJournal Complex Systems. Visualization package for complex networks, implemented as an R library (with GUI visualization) and also in Python, with low-level routines implemented in C. A stand-alone (desktop) version is also available. NetworkX http://networkx.lanl.gov/ Libraries in Python for network analysis and visualization. Has specific generators and functions implemented for analysis of bipartite graphs. Can handle extremely large networks (i.e., ≥ 106 nodes). Relies on other software package for visualization.

176

APPENDIX H

Gephi http://gephi.org/ Bastian M., Heymann S., et al., 2009. Gephi: an open source software for exploring and manipulating networks. International AAAI Conference on Weblogs and Social Media. Visualization and analysis of large networks. Its development started in 2008, based in Java, and implementing plugins, multiple formats and a very broad range of graph-editing options. Some functions are still limited (e.g., bipartite graphs, community detection, etc.). FoodWeb3D http://www.foodwebs.org/index_page/wow2.html Written by R. J. Williams and provided by the Pacific Ecoinformatics and Computational Ecology Lab (http://www.foodwebs.org). FoodWeb3D was specifically designed to visualize and explore food webs. Can also be used to represent bipartite networks, such as plant-animal interaction webs. Cytoscape http://www.cytoscape.org Shannon, P., Markiel, A., Ozier, O., Baliga, N. S., Wang, J. T., Ramage, D., Amin, N., Schwikowski, B., and Ideker, T. 2003. Cytoscape: A software environment for integrated models of biomolecular interaction networks. Genome Research, 13: 2498–2504. This is a versatile, open source, visualization package for large complex networks. Libraries, e.g., sna, network, networksis, bipartite, netZ, degreenet, tnet, and nettheory in the R package http://www.r-project.org/ Most of the R libraries (R Development Core Team, 2010) related to network analysis are included in the statnet bundle:

S O F T WA R E F O R T H E A NA LY S I S O F C O M P L E X NETWORKS

177

Handcock, M. S., Hunter, D. R., Butts, C. T., Goodreau, S. M. and Morris, M. 2003. Statnet: Software Tools for the Statistical Modeling of Network Data. Statnet Project, Seattle, WA. Version 2. http://www.statnetproject.org In particular, the bipartite library (Dormann, Fründ, et al. 2008) in the R package (R Development Core Team, 2010) is specifically designed to analyze bipartite graphs depicting species interactions, and implements most statistics summarized in Appendix A. The R-package bipartite also includes example datasets, including most of the datasets available in the interaction web database: http://www.nceas.ucsb.edu/interactionweb SONIA http://www.stanford.edu/group/sonia/index.html Bender-deMoll, S., and McFarland, D. A. 2006. The Art and Science of Dynamic Network Visualization. Journal of Social Structure. Volume 7(2). SONIA is a package based in Java for the visualization and analysis of temporally-dynamic networks. It implements visualizations of Kamada-Kawai and Fruchterman-Reingold ordinations of time-sliced interaction graphs. Nestedness Temperature Calculator, NTC http://www.aics-research.com/nestedness/tempcalc.html The Nestedness Temperature Calculator measures the extent of the order present in nested presence–absence matrices. It was the package first used in the analysis of nestedness in mutualistic networks, later on replaced by more analytic measures such as NODF. Atmar, W. and B. D. Patterson. 1993. The measure of order and disorder in the distribution of species in fragmented habitat. Oecologia 96: 373–382. ANINHADO http://www.guimaraes.bio.br/soft.html Guimarães Jr., P. R., and P. Guimarães. 2006. Improving the analyses of nestedness for large sets of matrices. Environmental Modelling and Software 21: 1512–1513.

178

APPENDIX H

A software package for estimation of nestedness statistics, implementing several different null models and nestedness indexes. NetCarto Guimerà, R., and Amaral, L.A.N. 2005. Functional cartography of complex metabolic networks. Nature, 433: 895–900. A software package for estimation of modularity. Determines the presence and identity of modules and classifies species (nodes) according to their roles (e.g., network hubs, module hubs, connectors, peripherals). Library vegan in the R Package Implements several null models for nestedness analysis in R (R Development Core Team, 2010). It finds the mean IAC and its standard deviation from random permutations of the data, or subsampling without replacement (Gotelli and Colwell, 2001).

Bibliography

Aizen, M. A., Morales, C. L., and Morales, J. M. 2008. Invasive mutualists erode native pollination webs. Plos Biology. 6: e31. Aizen, M. A., Sabatino, M., and Tylianakis, J. M. 2012. Specialization and rarity predict nonrandom loss of interactions from mutualist networks. Science. 335: 1486–1489. Alarcón, R., Waser, N. M., and Ollerton, J. 2008. Year-to-year variation in the topology of a plant-pollinator interaction network. Oikos. 117: 1796–1807. Albert, R., and Barabási, A. L. 2002. Statistical mechanics of complex networks. Reviews of Modern Physics. 74: 47–97. Albert, R., Jeong, H., and Barabási, A. L. 2000. Error and attack tolerance of complex networks. Nature. 406: 378–382. Allesina, S., Alonso, D., and Pascual, M. 2008. A general model for food web structure. Science. 320: 658–661. Almeida-Neto, M., Guimarães, P., Guimarães P. R., Jr., Loyola, R. D., and Ulrich, W. 2008. A consistent metric for nestedness analysis in ecological systems: reconciling concept and measurement. Oikos. 117: 1227–1239. Amaral, L. A., and Ottino, J. M. 2004. Complex networks. Augmenting the framework for the study of complex systems. European Physical Journal B. 38: 147–162. Amaral, L. A., Scala, A., Barthelemy, M., and Stanley, H. E. 2000. Classes of small-world networks. Proceedings of the National Academy of Sciences. 97: 11149–11152. Anderson, R. M., and May, R. M. 1992. Infectious Diseases of Humans. Dynamics and Control. Oxford University Press, Oxford. Arim, M., and Marquet, P. 2004. Intraguild predation: A widespread interaction related to species biology. Ecology Letters. 7: 557–564. Ashworth, L., Aguilar, R., Galetto, L., and Aizen, M. A. 2004. Why do pollination generalist and specialist plant species show similar reproductive susceptibility to habitat fragmentation? Journal of Ecology. 92: 717–719. Atmar, W., and Patterson, B. D. 1993. The measure of order and disorder in the distribution of species in fragmented habitat. Oecologia. 96: 373–382. Baldock, K.C.R., Memmott, J., Ruiz-Guajardo, J. C., Roze, D., and Stone, G. N. 2011. Daily temporal structure in African Savanna flower visitation networks and consequences for network sampling. Ecology. 92: 687–698. Banašek-Richter, C., Bersier, L.-F., Cattin, M.-F., Baltensperger, R., Gabriel, J.-P., Merz, Y., Ulanowicz, R. E., Tavares, A. F., et al. 2009. Complexity in quantitative food webs. Ecology. 90: 1470–1477.

180

BIBLIOGRAPHY

Barabási, A. L., and Albert, R. 1999. Emergence of scaling in random networks. Science. 286: 509–512. Barber, M. J. 2007. Modularity and community detection in bipartite networks. Physical Review E. 76: 066102. Barrat, A., Barthelemy, M., Pastor-Satorras, R., and Vespignani, A. 2004. The architecture of complex weighted networks. Proceedings of the National Academy of Sciences. 101: 3747–3752. Bartomeus, I., Vilà, M., and Santamaría, L. 2008. Contrasting effects of invasive plants in plant-pollinator networks. Oecologia. 155: 761–770. Bascompte, J. 2007. Networks in ecology. Basic and Applied Ecology. 8: 485–490. ———. 2009. Disentangling the web of life. Science. 325: 416–419. Bascompte, J., and Jordano, P. 2007. Plant-animal mutualistic networks: The architecture of biodiversity. Annual Review of Ecology, Evolution, and Systematics. 38: 567–593. Bascompte, J., Jordano, P., Melián, C. J., and Olesen, J. M. 2003. The nested assembly of plant-animal mutualistic networks. Proceedings of the National Academy of Sciences. 100: 9383–9387. Bascompte, J., Jordano, P., and Olesen, J. M. 2006a. Asymmetric coevolutionary networks facilitate biodiversity maintenance. Science. 312: 431–433. ———. 2006b. Response to comment on “Asymmetric coevolutionary networks facilitate biodiversity maintenance.” Science. 313: 1887. Bascompte, J., and Melián, C. J. 2005. Simple trophic modules for complex food webs. Ecology. 86: 2868–2873. Bascompte, J., Melián, C. J., and Sala, E. 2005. Interaction strength combinations and the overfishing of a marine food web. Proceedings of the National Academy of Sciences. 102: 5443–5447. Bascompte, J., and Solé, R. V. 1996. Habitat fragmentation and extinction thresholds in spatially explicit models. Journal of Animal Ecology. 65: 465–473. ———. 1998. Modeling Spatiotemporal Dynamics in Ecology. Springer-Verlag, Berlin. Basilio, A. M., Medan, D., Torretta, J. P., and Bartoloni, N. J. 2006. A year-long plantpollinator network. Austral Ecology. 31: 975–983. Bastolla, U., Fortuna, M. A., Pascual-García, A., Ferrera, A., Luque, B., and Bascompte, J. 2009. The architecture of mutualistic networks minimizes competition and increases biodiversity. Nature. 458: 1018–1020. Bastolla, U., Lassig, M., Manrubia, S. C., and Valleriani, A. 2005. Biodiversity in model ecosystems, i: Coexistence conditions for competing species. Journal of Theoretical Biology. 235: 521–530. Beal, W. J. 1898. Seed dispersal. Ginn and Co., Boston. Belgrano, A., Scharler, U. M., Dunne, J., and Ulanowicz, R. E., eds. 2005. Aquatic Food Webs: An Ecosystem Approach. Oxford University Press, Oxford. Berlow, E. L., Dunne, J. A., Martinez, N. D., Williams, R. J., and Brose, U. 2009. Simple prediction of interaction strengths in complex food webs. Proceedings of the National Academy of Sciences. 106: 187–191. Bersier, L. F. 2007. A history of the study of ecological networks. In A. Képès, ed., Biological networks, 365–416. World Scientific Publishing Co., Singapore. Bersier, L. F., Banašek-Richter, C., and Cattin, M. 2002. Quantitative descriptors of food-web matrices. Ecology. 83: 2394–2407.

BIBLIOGRAPHY

181

Bersier, L. F., Dixon, P., and Sugihara, G. 1999. Scale-invariant or scale-dependent behavior of the link density property in food webs: a matter of sampling effort? American Naturalist. 153: 676–682. Bersier, L. F., and Kehrli, P. 2008. The signature of phylogenetic constraints on foodweb structure. Ecological Complexity. 5: 132–139. Bininda-Emonds, O.R.P., Gittleman, J. L., and Steel, M. A. 2002. The (super)tree of life: procedures, problems, and prospects. Annual Review of Ecology, Evolution, and Systematics. 33: 265–289. Bloch, E., and Ritter, M. 1977. Nonsynchronism and the obligation to its dialectics. New German Critique. 11: 22–38. Blomberg, S. P., Garland, T. J., and Ives, A. R. 2003. Testing for phylogenetic signal in comparative data: behavioral traits are more labile. Evolution. 57: 717–745. Blüthgen, N. 2010. Why network analysis is often disconnected from community ecology: A critique and an ecologist’s guide. Basic and Applied Ecology. 11: 185–195. Blüthgen, N., Fründ, J., Vázquez, D. P., and Menzel, F. 2008. What do interaction network metrics tell us about specialization and biological traits? Ecology. 89: 3387–3399. Blüthgen, Nico, Menzel, F., and Blüthgen, Nils. 2006. Measuring specialization in species interaction networks. BMC Ecology. 6: 9. Borrell, B. J. 2005. Long tongues and loose niches: Evolution of euglossine bees and their nectar flowers. Biotropica. 37: 664–669. Boucher, D. H. 1985a. The idea of mutualism, past and future. In D. Boucher, ed., The Biology of Mutualism, 1–28. Croom Helm, London. ———. 1985b. Mutualism in agriculture. In D. Boucher, ed., The Biology of Mutualism, 375–386. Croom Helm, London. Bronstein, J. L. 1994. Conditional outcomes in mutualistic interactions. Trends in Ecology and Evolution. 9: 214–217. ———. 1995. The plant-pollinator landscape. In L. Hansson, L. Fahrig, and G. Merriam, eds., Mosaic Landscapes and Ecological Processes, 257–288. Chapman and Hall, New York. Bronstein, J. L., Dieckmann, U., and Ferrière, R. 2004. Coevolutionary dynamics and the conservation of mutualisms. In R. Ferrière, U. Dieckmann, and D. Couvet, eds., Evolutionary Conservation Biology, 305–326. Cambridge University Press, Cambridge. Bronstein, J. L., Wilson, W. G., and Morris, W. E. 2003. Ecological dynamics of mutualist/antagonist communities. American Naturalist. 162: S24–S39. Buldyrev, S. V., Parshani, R., Paul, G., Stanley, H. E., and Havlin, S. 2010. Catastrophic cascade of failures in interdependent networks. Nature. 464: 1025–1028. Burgos, E., Ceva, H., Perazzo, R.P.J., Devoto, M., Medan, D., Zimmermann, M., and Delbue, A. M. 2007. Why nestedness in mutualistic networks? Journal of Theoretical Biology. 249: 307–313. Burns, K. C., and Zotz, G. 2010. A hierarchical framework for investigating epiphyte assemblages: Networks, meta-communities, and scale. Ecology. 91: 377–385. Camacho, J., Guimerà, R., and Amaral, L.A.N. 2002. Robust patterns in food web structure. Physical Review Letters. 88: 228102.

182

BIBLIOGRAPHY

Carnicer, J., Abrams, P. A., and Jordano, P. 2008. Switching behavior, coexistence and diversification: Comparing empirical community-wide evidence with theoretical predictions. Ecology Letters. 11: 802–808. Carnicer, J., Jordano, P., and Melián, C. J. 2009. The temporal dynamics of resource use by frugivorous birds: A network approach. Ecology. 90: 1958–1970. Cattin, M. F., Bersier, L. F., Banašek-Richter, C., Baltensperger, R., and Gabriel, J. P. 2004. Phylogenetic constraints and adaptation explain food-web structure. Nature. 427: 835–839. Chacoff, N. P., Vázquez, D. P., Lomáscolo, S. B., Stevani, E. L., Dorado, J., and Padrón, B. 2012. Evaluating sampling completeness in a desert plant-pollinator network. Journal of Animal Ecology. 81: 190–200. Chamberlain, S. A., and Holland, J. N. 2009. Quantitative synthesis of context dependency in ant-plant protection mutualisms. Ecology. 90: 2384–2392. Chesson, P. 1994. Multispecies competition in variable environments. Theoretical Population Biology. 45: 227–276. ———. 2000. Mechanisms of maintenance of species diversity. Annual Review of Ecology, Evolution, and Systematics. 31: 343–366. Clauset, A., Moore, C., and Newman, M.E.J. 2008. Hierarchical structure and the prediction of missing links in networks. Nature. 453: 98–101. Clements, R. E., and Long, F. L. 1923. Experimental Pollination: An Outline of the Ecology of Flowers and Insects. Carnegie Institute of Washington, Washington, DC. Cocucci, A. A., Moré, M., and Sérsic, A. N. 2009. Restricciones mecánicas en las interacciones planta-polinizador: estudio de casos en plantas polinizadas por esfíngidos. In R. Medel, R. Zamora, M. Aizen, and R. Dirzo, eds., Interacciones Planta—Animal y la Conservación de la Biodiversidad, 43–59. CYTED, Madrid. Coddington, J. A., Agnarsson, I., Miller, J. A., Kuntner, M., and Hormiga, G. 2009. Undersampling bias: The null hypothesis for singleton species in tropical arthropod surveys. Journal of Animal Ecology. 78: 573–584. Cohen, J. E. 1978. Food Webs and Niche Space. Princeton University Press. Princeton. ———. 1994. Lorenzo Camerano’s contribution to early food web theory. In S. Levin, ed., Frontiers in Mathematical Biology. Springer-Verlag, Berlin. ———. 2004. Mathematics is biology’s next microscope, only better; biology is mathematics’ next physics, only better. PLoS Biology. 2: 2017–2023. Cohen, J. E., Briand, F., and Newman, C. M. 1990. Community Food Webs: Data and Theory. Springer-Verlag, Berlin. Connor, E. F., and Simberloff, D. 1979. The assembly of species communities: Chance or competition? Ecology. 60: 1132–1140. Cook, J. M., and Rasplus, J. Y. 2003. Mutualisms with attitude: Coevolving fig wasps and figs. Trends in Ecology and Evolution. 18: 241–248. Cook, R. R., and Quinn, J. F. 1998. An evaluation of randomization models for nested species subset analysis. Oecologia. 113: 584–592. Cramp, S. 1988. The Birds of the Western Palaearctic, vol. 5. Oxford University Press, Oxford. Dale, M., and Fortin, M. J. 2010. From graphs to spatial graphs. Annual Review of Ecology, Evolution, and Systematics. 41: 21–38.

BIBLIOGRAPHY

183

Danon, L., Diaz-Guilera, A., and Arenas, A. 2006. The effect of size heterogeneity on community identification in complex networks. Journal of Statistical Mechanics—Theory and Experiment P11010. Darwin, C. 1859. On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life. John Murray, London. ———. 1862. On the Various Contrivances by which British and Foreign Orchids are Fertilised by Insects, and on the Good Effects of Intercrossing. John Murray, London. Davies, K. F., Margules, C. R., and Lawrence, J. F. 2004. A synergistic effect puts rare, specialized species at greater risk of extinction. Ecology. 85: 265–271. de Bary, H. A. 1879. Die Erscheinung der Symbiose. de Bary, Strasbourg, France. Devoto, M., and Montaldo, N. H. 2005. Patterns of interaction between plants and pollinators along an environmental gradient. Oikos. 109: 461–472. Diamond, J. M. 1975. Assembly of species communities. In M. Cody, and J. Diamond, eds., Ecology and Evolution of Communities, 342–444. Harvard University Press, Cambridge. Dicks, L. V., Corbet, S. A., and Pywell, R. F. 2002. Compartmentalization in plantinsect flower visitor webs. Journal of Animal Ecology. 71: 32–43. Dirzo, R., and Miranda, A. 1990. Contemporary neotropical defaunation and forest structure, function, and diversity. Conservation Biology. 4: 444–447. Dodd, M. E., Silvertown, J., and Chase, M. W. 1999. Phylogenetic analysis of trait evolution and species diversity variation among angiosperm families. Evolution. 53: 732–744. Donatti, C. I., Grüber, P. R., Jr., Galetti, M., Pizo, M. A., Marquitti, F.M.D., and Dirzo, R. 2011. Analysis of a hyper-diverse seed dispersal network: Modularity and underlying mechanisms. Ecology Letters. 14: 773–781. Dormann, C. F., Fründ, J., Blüthgen, N., and Gruber, B. 2009. Indices, graphs and null models: Analyzing bipartite ecological networks. Open Ecology Journal. 2: 7–24. Dormann, C. F., Gruber, B., and Fründ, J. 2008. Introducing the bipartite package: Analyzing ecological networks. R News. 8: 8–11. Dorogovtsev, S. N., and Mendes, J.F.F. 2002. Evolution of networks. Advances in Physics. 51: 1079–1187. Dunne, J. A., Williams, R. J., and Martinez, N. D. 2002a. Food-web structure and network theory: The role of connectance and size. Proceedings of the National Academy of Sciences. 99: 12917–12922. ———. 2002b. Network structure and biodiversity loss in food webs: Robustness increases with connectance. Ecology Letters. 5: 558–567. Dyer, R. 2007. The evolution of genetic topologies. Theoretical Population Biology. 71: 71–79. Dyer, R. J., and Nason, J. 2004. Population graphs: The graph theoretical shape of genetic structure. Molecular Ecology. 13: 1713–1727. Egerton, F. N. 2007. Understanding food chains and food webs, 1700–1970. Bulletin of the Ecological Society of America. 88: 50–69. Ehrlich, P. R., and Raven, P. H. 1964. Butterflies and plants: A study in coevolution. Evolution. 18: 586–608. Elton, C. 1927. Animal Ecology. Sidgwick and Jackson. London.

184

BIBLIOGRAPHY

Encinas-Viso, F., Revilla, T. A., and Etienne, R. S. 2012. Phenology drives mutualistic network structure and diversity. Ecology Letters. 15: 198–208. Erdös, P., and Rényi, A. 1959. On random graphs. Publicationes Mathematicae. 6: 290–297. Euler, L. 1736. Solutio problematis ad geometriam situs pertinentis. Commentarii Academiae Scientiarum Imperialis Petropolitanae. 8: 128–140. Fabina, N. S., Abbott, K. C., and Gilman, R. T. 2010. Sensitivity of plant-pollinatorherbivore communities to changes in phenology. Ecological Modelling. 221: 453–458. Fagan, W. F. 2002. Connectivity, fragmentation, and extinction risk in dendritic metapopulations. Ecology. 83: 3243–3249. Fagan, W. F., and Hurd, L. E. 1994. Hatch density variation of a generalist arthropod predator: Population consequences and community impact. Ecology. 75: 2022– 2032. Farrell, B. D. 1998. “Inordinate fondness” explained: Why are there so many beetles? Science. 281: 555–559. Feinsinger, P. 1978. Ecological interactions between plants and hummingbirds in a successional tropical community. Ecological Monographs. 48: 269–287. ———. 1983. Coevolution and pollination. In D. J. Futuyma, and M. Slatkin, eds., Coevolution, 282–310. Sinauer Associates, Inc., Sunderland, MA. Felsenstein, J. 1985. Phylogenies and the comparative method. American Naturalist. 125: 1–15. Filotas, E., Grant, M., Parrott, L., and Rikvold, P. A. 2010. Positive interactions and the emergence of community structure in metacommunities. Journal of Theoretical Biology. 266: 419–429. Fischer, J., and Lindenmayer, D. B. 2002. Treating the nestedness temperature calculator as a “black box” can lead to false conclusions. Oikos. 99: 193–199. Fonseca, C. R., and Ganade, G. 1996. Asymmetries, compartments and null interactions in an Amazonian ant-plant community. Journal of Animal Ecology. 65: 339–347. Fontaine, C., Guimarães, P. R., Jr., Kéfi, S., Loeuille, N., Memmott, J., van der Putten, W. H., van Veen, F.J.F., and Thébault, E. 2011. The ecological and evolutionary implications of merging different types of networks. Ecology Letters. 14: 1170–1181. Forde, S. E., Thompson, J. N., Holt, R. D., and Bohannan, B.J.M. 2008. Coevolution drives temporal changes in fitness and diversity across environments in a bacteria-bacteriophage interaction. Evolution. 62: 1830–1839. Fortin, M. J., and Dale, M. 2005. Spatial Analysis. A Guide for Ecologists. Cambridge University Press, Cambridge. Fortuna, M. A., Albaladejo, R. G., Fernández, L., Aparicio, A., and Bascompte, J. 2009. Networks of spatial genetic vatiation across species. Proceedings of the National Academy of Sciences. 106: 19044–19049. Fortuna, M. A., and Bascompte, J. 2006. Habitat loss and the structure of plant-animal mutualistic networks. Ecology Letters. 9: 281–286. Fortuna, M. A., García, C., Guimarães, P. R., Jr, and Bascompte, J. 2008. Spatial mating networks in insect-pollinated plants. Ecology Letters. 11: 490–498.

BIBLIOGRAPHY

185

Fortuna, M. A., Gómez-Rodríguez, C., and Bascompte, J. 2006. Spatial network structure and amphibian persistence in stochastic environments. Proceedings of the Royal Society B. 273: 1429–1434. Fortuna, M. A., Krishna, A., and Bascompte, J. 2013. Habitat loss and the disassembly of mutualitstic networks. Oikos. 122: 938–942. Fortuna, M. A., Popa-Lisseanu, G., Ibáñez, C., and Bascompte, J. 2009. The roosting spatial network of a bird-predator bat. Ecology. 90: 934–944. Fortuna, M. A., Stouffer, D. B., Olesen, J. M., Jordano, P., Mouillot, D., Krasnov, B. R., Poulin, R., and Bascompte, J. 2010. Nestedness versus modularity in ecological networks: Two sides of the same coin? Journal of Animal Ecology. 79: 811–817. Fortunato, S. 2010. Community detection in graphs. Physics Reports. 486: 75–174. Forup, M. L., Henson, K.S.E., Craze, P. G., and Memmott, J. 2008. The restoration of ecological interactions: Plant-pollination networks on ancient and restored heathlands. Journal of Applied Ecology. 45: 742–752. Fox, L. R. 1981. Defense and dynamics in plant-herbivore systems. American Zoologist. 21: 853–864. ———. 1988. Diffuse coevolution within complex communities. Ecology. 69: 906–907. Freckleton, R. P., Harvey, P. H., and Pagel, M. 2002. Phylogenetic analysis and comparative data: A test and review of evidence. American Naturalist. 160: 712–726. Galeano, J., Pastor, J. M., and Iriondo, J. M. 2009. Weighted-Interaction Nestedness Estimator (WINE): A new estimator to calculate over frequency matrices. Environmental Modelling & Software. 24: 1342–1346. Galil, L. 1977. Fig biology. Endeavour. 1: 52–56. Garland, T., Jr, Bennett, A. F., and Rezende, E. L. 2005. Phylogenetic approaches in comparative physiology. Journal of Experimental Biology. 208: 3015–35. Garland, T., Jr., Harvey, P. H., and Ives, A. R. 1992. Procedures for the analysis of comparative data using phylogenetically independent contrasts. Systematic Biology. 41: 18–32. Gause, G. F., and Witt, A. A. 1935. Behavior of mixed populations and the problem of natural selection. American Naturalist. 69: 596–609. Gibson, R. H., Nelson, I. L., Hopkins, G. W., Hamlett, B. J., and Memmott, J. 2006. Pollination webs, plant communities, and the conservation of rare plants: Arable weeds as a case study. Journal of Applied Ecology. 43: 246–257. Gilarranz, L. J., and Bascompte, J. 2012. Spatial network structure and metapopulation persistence. Journal of Theoretical Biology. 297: 11–16. Gilbert, G. S., and Webb, C. O. 2007. Phylogenetic signal in plant pathogen-host range. Proceedings of the National Academy of Sciences. 104: 4979–4983. Gilbert, L. E., and Raven, P. H. 1975. Coevolution of Animals and Plants. University of Texas Press, Austin. Girvan, M., and Newman, M.E.J. 2002. Community structure in social and biological networks. Proceedings of the National Academy of Sciences. 99: 7821–7826. Gómez, J. M., Jordano, P., and Perfectti, F. 2011. The functional consequences of mutualistic network architecture. PLoS ONE. 6: e16143. Gómez, J. M., Verdú, M., and Perfectti, F. 2010. Ecological interactions are evolutionarily conserved across the entire tree of life. Nature. 465: 918–921.

186

BIBLIOGRAPHY

Gomulkiewicz, R., Drown, D. M., Dybdahl, M. F., Godsoe, W., Nuismer, S. L., Pepin, K. M., Ridenhour, B. J., Smith, C. I., and Yoder, J. B. 2007. Dos and don’ts of testing the geographic mosaic theory of coevolution. Heredity. 98: 249–258. Gotelli, N. J. 2001. Research frontiers in null models. Global Ecology and Biogeography. 10: 337–343. Gotelli, N. J., and Colwell, R. K. 2001. Quantifying biodiversity: Procedures and pitfalls in the measurement and comparison of species richness. Ecology Letters. 4: 379–391. Gotelli, N. J., and Entsminger, G. L. 2001. Swap and fill algorithms in null model analysis: Rethinking the knight’s tour. Oecologia. 129: 281–291. Gotelli, N. J., and McCabe, D. J. 2002. Species co-ocurrence: a meta-analysis of J. M. diamond’s assembly rules model. Ecology. 83: 2091–2096. Gower, J. C., and Legendre, P. 1986. Metric and Euclidean properties of dissimilarity coefficients. Journal of Classification. 3: 5–48. Gravel, D., Canard, E., Guichard, F., and Mouquet, N. 2011. Persistence increases with diversity and connectance in trophic metacommunities. PLoS ONE. 6: e19374. Guillaume, J. L., and Latapy, M. 2004. Bipartite structure of all complex networks. Information Processing Letters. 90: 215–221. Guillaume, J.-L., and Latapy, M. 2006. Bipartite graphs as models of complex networks. Physica A. 371: 795–813. Guimarães P. R., Jr., Argollo de Menezes, M., Baird, R. W., Lusseau, D., Guimarães, P., and dos Reis, S. F. 2007. Vulnerability of a killer whale social network to disease outbreaks. Physical Review E. 76: 042901. Guimarães, P. R., Jr., de Aguiar, M. A., Bascompte, J., Jordano, P., and dos Reis, S. F. 2005. Random initial condition in small Barabási-Albert networks and deviations from the scale-free behavior. Physical Review E. 71: 037101. Guimarães, P. R., Jr., Jordano, P., and Thompson, J. N. 2011. Evolution and coevolution in mutualistic networks. Ecology Letters. 14: 877–885. Guimarães, P. R., Jr., Machado, G., de Aguiar, M. A. M., Jordano, P., Bascompte, J., Pinheiro, A., and Dos Reis, S. F. 2007. Build-up mechanisms determining the topology of mutualistic networks. Journal of Theoretical Biology. 249: 181–189. Guimarães, P. R., Jr., Rico-Gray, V., Furtado dos Reis, S., and Thompson, J. N. 2006. Asymmetries in specialization in ant-plant mutualistic networks. Proceedings of the Royal Society B. 273: 2041–2047. Guimarães, P. R., Jr., Rico-Gray, V., Oliveira, P. S., Izzo, T. J., dos Reis, S. F., and Thompson, J. N. 2007. Interaction intimacy affects structure and coevolutionary dynamics in mutualistic networks. Current Biology. 17: 1797–1803. Guimarães P. R., Jr., Sazima, C., Furtado dos Reis, S., and Sazima, I. 2007. The nested structure of marine cleaning symbiosis: Is it like flowers and bees? Biology Letters. 3: 51–54. Guimerà, R., and Amaral, L. N. 2005. Functional cartography of complex metabolic networks. Nature. 433: 895–900. Guimerà, R., Mossa, S., Turtschi, A., and Amaral, L.A.N. 2005. The worldwide air transportation network: Anomaluous centrality, community structure, and cities’ global roles. Proceedings of the National Academy of Sciences. 102: 7794–7799.

BIBLIOGRAPHY

187

Guimerà, R., Sales-Pardo, M., and Amaral, L.A.N. 2007. Module identification in bipartite and directed networks. Physical Review. 76: 036102. Guimerà, R., Stouffer, D. B., Sales-Pardo, M., Leicht, E. A., Newman, M.E.J., and Amaral, L.A.N. 2010. Origin of compartmentalization in food webs. Ecology. 91: 2941–2951. Guttal, V., and Jayaprakash, C. 2008. Changing skewness: An early warning signal of regime shifts in ecosystems. Ecology Letters. 11: 450–460. Haldane, A. G., and May, R. M. 2011. Systemic risk in banking ecosystems. Nature. 469: 351–355. Hall, S. J., and Raffaelli, D. 1991. Food-web patterns: Lessons from a species-rich web. Journal of Animal Ecology. 60: 823–842. Hansen, T. F., and Martins, E. P. 1996. Translating between microevolutionary process and macroevolutionary patterns: The correlation structure of interspecific data. Evolution. 50: 1404–1417. Hanski, I. 1999. Metapopulation Ecology. Oxford University Press, New York. Hanski, I., and Gilpin, M. E. 1997. Metapopulation Biology: Ecology, Genetics, and Evolution. Academic Press, New York. Harper, J. L. 1977. Population Biology of Plants. Academic Press, London. Hartwell, L. H., Hopfield, J. J., Leibler, S., and Murray, A. W. 1999. From molecular to modular cell biology. Nature. 402: C47–C52. Harvey, P. H., and Pagel, M. D. 1991. The Comparative Method in Evolutionary Biology. Oxford University Press, Oxford. Hassell, M. P. 1978. The Dynamics of Arthropod Predator Prey Systems. Princeton University Press, Princeton. Heithaus, E. R. 1974. The role of plant-pollinator interactions in determining community structure. Annals of the Missouri Botanical Garden. 61: 675–691. Herrera, C. M. 1982. Seasonal variation in the quality of fruits and diffuse coevolution between plants and avian dispersers. Ecology. 63: 773–785. Hill, E. J. 1883. Means of plant dispersion. American Naturalist. 17: 811–1028. Hoffman, P. 1998. The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth. Hyperion Books, New York. Holland, J. N., Ness, J. H., Boyle, A., and Bronstein, J. L. 2005. Mutualisms as consumer-resource interactions. In Evolution and Coadaptation in Ant Communities, 17–35. Oxford University Press, Oxford. Holland, J. N., Okuyama, T., and DeAngelis, D. L. 2006. Comment on “Asymmetric coevolutionary networks facilitate biodiversity maintenance.” Science. 313: 1887b. Holland, M. D., and Hastings, A. 2008. Strong effect of dispersal network structure on ecological dynamics. Nature. 456: 792–794. Holt, R. D. 2002. Food webs in space: on the interplay of dynamic instability and spatial processes. Ecological Research. 17: 261–273. Hubbell, S. P. 2001. The Unified Neutral Theory of Biodiversity and Biogeography. Princeton Monographs in Population Biology. Princeton University Press, Princeton. Inouye, B., and Stinchcombe, J. R. 2001. Relationships between ecological interaction modifications and diffuse coevolution: Similarities, differences, and causal links. Oikos. 95: 353–360.

188

BIBLIOGRAPHY

Ives, A. R., and Godfray, H. C. 2006. Phylogenetic analysis of trophic associations. American Naturalist. 168: E1–E14. Iwao, K., and Rausher, M. D. 1997. Evolution of plant resistance to multiple herbivores: Quantifying diffuse coevolution. American Naturalist. 149: 316–335. Jabot, F., and Bascompte, J. 2012. Bi-trophic interactions shape biodiversity in space. Proceedings of the National Academy of Sciences. 109: 4521–4526. Jalas, J., and Suominen, J. E. 1985. Atlas Florae Europaeae, vol. 1. Cambridge University Press, Cambridge. Janzen, D. H. 1974. The deflowering of Central America. Natural History, NY. 83: 48–53. ———. 1980. When is it coevolution? Evolution. 34: 611–612. ———. 1983. Dispersal of seeds by vertebrate guts. In D. J. Futuyma, and M. Slatkin, eds., Coevolution, 232–262. Sinauer Associates, Inc., Sunderland, MA. ———. 1985. The natural history of mutualisms. In D. H. Boucher, ed., The Biology of Mutualism, 40–99. Croom Helm, London. Jeong, H., Neda, Z., and Barabási, A. L. 2003. Measuring preferential attachment in evolving networks. Europhysics Letters. 61: 567–572. Johnson, S. D., and Steiner, K. E. 1997. Long-tongued fly pollination and evolution of floral spur length in the Disa draconis complex (orchidaceae). Evolution. 51: 45–53. Joppa, L. N., Bascompte, J., Montoya, J. M., Solé, R. V., Sanderson, J., and Pimm, S. L. 2009. Reciprocal specialization in ecological networks. Ecology Letters. 12: 961–969. Joppa, L. N., Montoya, J. M., Solé, R., Sanderson, J., and Pimm, S. 2010. On nestedness in ecological networks. Evolutionary Ecology Research. 12: 35–46. Jordano, P. 1987. Patterns of mutualistic interactions in pollination and seed dispersal: Connectance, dependence asymmetries, and coevolution. American Naturalist. 129: 657–677. ———. 1988. Diet, fruit choice and variation in body condition of frugivorous warblers in mediterranean scrubland. Ardea. 76: 193–209. ———. 2000. Fruits and frugivory. In M. Fenner, ed., Seeds: the Ecology of Regeneration in Natural Plant Communities, 125–166. Commonwealth Agricultural Bureau International, Wallingford. ———. 2010. Coevolution in multispecific interactions among free-living species. Evolution, Education, and Outreach. 3: 40–46. Jordano, P., and Bascompte, J. In preparation. Ecological correlates of interaction in complex plant-animal mutualistic networks. Jordano, P., Bascompte, J., and Olesen, J. M. 2003. Invariant properties in coevolutionary networks of plant-animal interactions. Ecology Letters. 6: 69–81. ———. 2006. The ecological consequences of complex topology and nested structure in pollination webs. In N. M. Waser, and J. Ollerton, eds., Plant-Pollinator Interactions. From Specialization to Generalization, 173–199. University of Chicago Press, Chicago. Jordano, P., García, C., Godoy, J. A., and García-Castaño, J. L. 2007. Differential contribution of frugivores to complex seed dispersal patterns. Proceedings of the National Academy of Sciences. 104: 3278–3282.

BIBLIOGRAPHY

189

Jordano, P., Vázquez, D., and Bascompte, J. 2009. Redes complejas de interacciones planta–animal. In R. Medel, R. Zamora, M. Aizen, and R. Dirzo, eds., Interacciones Planta–Animal y la Conservación de la Biodiversidad, 17–41. CYTED, Madrid. Junker, B. H., and Schreiber, F. 2008. Analysis of Biological Networks. WileyInterscience, Hoboken, NJ. Kaiser-Bunbury, C. N., Muff, S., Memmott, J., Müller, C. B., and Caflisch, A. 2010. The robustness of pollination networks to the loss of species and interactions: a quantitative approach incorporating pollinator behaviour. Ecology Letters. 13: 442–452. Kaiser-Bunbury, C. N., Valentin, T., Mougal, J., Matatiken, D., and Ghazoul, J. 2010. The tolerance of island plant-pollinator networks to alien plants. Journal of Ecology. 99: 202–213. Kashtan, N., and Alon, U. 2005. Spontaneous evolution of modularity and network motifs. Proceedings of the National Academy of Sciences. 102: 13773–13778. Kearns, C. A., Inouye, D. W., and Waser, N. M. 1998. Endangered mutualisms: The conservation of plant-pollinator interactions. Annual Review of Ecology, Evolution, and Systematics. 29: 83–112. Keitt, T. H., and Stanley, H. E. 1998. Dynamics of North American breeding bird populations. Nature. 393: 257–260. Kelly, D. 1994. The evolutionary ecology of mast seeding. Trends in Ecology and Evolution. 9: 465–470. Kelly, D., and Sork, V. L. 2002. Mast seeding in perennial plants: Why, how, where? Annual Review of Ecology, Evolution, and Systematics. 33: 427–447. Kembel, S. W., Cowan, P. D., Helmus, M. R., Cornwell, W. K., Morlon, H., Ackerly, D. D., Blomberg, S., and Webb, C. 2010. Picante: R tools for integrating phylogenies and ecology. Bioinformatics. 26: 1463–1464. Kiers, E. T., Palmer, T. M., Ives, A. R., Bruno, J. F., and Bronstein, J. L. 2010. Mutualisms in a changing world: An evolutionary perspective. Ecology Letters. 13: 1459–1474. Knuth, P. 1898. Handbuch der Blütenbiologie. I. Band: Einleitung und Literatur. Verlag Wilhelm Engelmann, Leipzig. Kokkoris, G. D., Troumbis, A. Y., and Lawton, J. H. 1999. Patterns of species interaction strength in assembled theoretical competition communities. Ecology Letters. 2: 70–74. Köllreuter, J. G. 1761. Vorläufige Nachricht von einigen das Geschlecht der Pflanzen betreffenden Versuchen und Beobachtungen. Gleditschischen Handlung, Leipzig. Konagurthu, A. S., and Lesk, A. M. 2008. On the origin of distribution patterns of motifs in biological networks. BMC Systems Biology. 2: 730. Kondoh, M. 2003. Foraging adaptation and the relationship between food-web complexity and stability. Science. 299: 1388–1391. Kondoh, M., and Kato, S. 2010. Food webs are built up with nested subwebs. Ecology. 91: 3123–3130. Krapivsky, P. L., Redner, S., and Leyvraz, F. 2000. Connectivity of growing random networks. Physical Review Letters. 85: 4629–4632.

190

BIBLIOGRAPHY

Krause, A. E., Frank, K. A., Mason, D. M., Ulanowicz, R. E., and Taylor, W. W. 2003. Compartments revealed in food-web structure. Nature. 426: 282–285. Krishna, A., Guimarães P. R., Jr., Jordano, P., and Bascompte, J. 2008. A neutral-niche theory of nestedness in mutualistic networks. Oikos. 117: 1609–1618. Labandeira, C. C. 2002. The history of associations between plants and animals. In C. M. Herrera and O. Pellmyr, eds., Plant-Animal Interactions. An Evolutionary Approach, 26–74. Blackwell, Oxford. Labandeira, C. C., Johnson, K. R., and Wilf, P. 2002. Impact of the terminal cretaceous event on plant-insect associations. Proceedings of the National Academy of Sciences. 99: 2061–2066. Lafferty, K. D., Dobson, A. P., and Kurtis, A. M. 2006. Parasites dominate food web links. Proceedings of the National Academy of Sciences. 103: 11211–1216. Laliberté, E., and Tylianakis, J. M. 2010. Deforestation homogenizes tropical parasitoid-host networks. Ecology. 91: 1740–1747. Lande, R. 1987. Extinction thresholds in demographic models of territorial populations. American Naturalist. 130: 624–635. Lassig, M., Bastolla, U., Manrubia, S. C., and Valleriani, A. 2001. The shape of ecological networks. Physical Review Letters. 86: 4418–4421. Leibhold, A., and Bascompte, J. 2004. The Allee effect, stochastic dynamics and the eradication of alien species. Ecology Letters. 6: 133–140. Leibold, M., Holyoak, M., Mouquet, N., Amarasekare, P., Chase, J. M., Hoopes, M. F., Holt, R. D., et al. 2004. The metacommunity concept: a framework for multiscale community ecology. Ecology Letters. 7: 601–613. Levins, R. 1969. Some demographic and generic consequences of environmental heterogeneity for biological control. Bulletin of the Entomological Society of America. 15: 237–240. Lewinsohn, T. M., Novotny, V., and Basset, Y. 2005. Insects on plants: diversity of herbivore assemblages revisited. Annual Review of Ecology, Evolution, and Systematics. 36: 597–620. Lewinsohn, T. M., Prado, P. I., Jordano, P., Bascompte, J., and Olesen, J. M. 2006. Structure in plant-animal interaction assemblages. Oikos. 113: 174–184. Liben-Nowell, D., and Kleinberg, J. M. 2007. The link-prediction problem for social networks. Journal of the American Society for Information Science and Technology. 58: 1019–1031. Lieberman, E., Hauert, C., and Nowak, M. 2005. Evolutionary dynamics on graphs. Nature. 433: 312–316. Liljeros, F., Edling, C. R., Amaral, L. A., Stanley, H. E., and Aberg, Y. 2001. The web of human sexual contacts. Nature. 411: 907–908. Lindeman, R. L. 1942. The trophic-dynamic aspect of ecology. Ecology. 23: 399–418. Lloyd, A. L., and May, R. M. 2001. Epidemiology: How viruses spread among computers and people. Science. 292: 1316–1317. Lundgren, R., and Olesen, J. M. 2005. The dense and highly connected world of greenland’s plants and their pollinators. Arctic, Antarctic, and Alpine Research. 37: 514–520. Luscombe, N. M., Babu, M. M., Yu, H., Snyder, M., Teichmann, S. A., and Gerstein, M. 2004. Genomic analysis of regulatory network dynamics reveals large topological changes. Nature. 431: 308–12.

BIBLIOGRAPHY

191

MacArthur, R. H. 1972. Geographical Ecology: Patterns in the Distribution of Species. Princeton University Press, Princeton. Martinez, N. D., Hawkins, B. A., Dawah, H. A., and Feifarek, B. P. 1999. Effects of sampling effort on characterization of food-web structure. Ecology. 80: 1044–1055. Martins, E. P., and Hansen, T. F. 1997. Phylogenies and the comparative method: a general approach to incorporating phylogenetic information into the analysis of interspecific data. American Naturalist. 149: 646–667. May, R. M. 1972. Will a large complex system be stable? Nature. 238: 413–414. ———. 1973. Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton. ———. 1976. Models of two interacting populations. In R. M. May, ed., Theoretical Ecology: Principles and Applications, 49–70. W. B. Saunders, Philadelphia. ———. 1982. Mutualistic interactions among species. Nature. 290: 401–402. ———. 2006. Network structure and the biology of populations. Trends in Ecology and Evolution. 21: 394–9. May, R. M., Levin, S. A., and Sugihara, G. 2008. Ecology for bankers. Nature. 451: 893–895. McCann, K, A., Hastings, A., and Huxel, G. R. 1998. Weak trophic interactions and the balance of nature. Nature. 395: 794–798. McCann, K. S., Rasmussen, J. B., and Umbanhowar, J. 2005. The dynamics of spatially coupled food webs. Ecology Letters. 8: 513–523. McKenna, D. D., Sequeira, A. S., Marvaldi, A. E., and Farrell, B. D. 2009. Temporal lags and overlap in the diversification of weevils and flowering plants. Proceedings of the National Academy of Sciences. 106: 7083–7088. Medan, D., Basilio, A. M., Devoto, M., Bartoloni, N. J., Torretta, J. P., and Petanidou, T. 2006. Measuring generalization and connectance in temperate, year-long active systems. In N. M. Waser, and J. Ollerton, eds., Plant–Pollinator Interactions: From Specialization to Generalization, 245–259. University of Chicago Press, Chicago. Melián, C. J., and Bascompte, J. 2002a. Complex networks: Two ways to be robust? Ecology Letters. 5: 705–708. ———. 2002b. Food web structure and habitat loss. Ecology Letters. 5: 37–46. ———. 2004. Food web cohesion. Ecology. 85: 352–358. Melián, C. J., Bascompte, J., Jordano, P., and Kˇrivan, V. 2009. Diversity in a complex ecological network with two interaction types. Oikos. 118: 122–130. Memmott, J. 1999. The structure of a plant-pollination food web. Ecology Letters. 2: 276–280. Memmott, J., Craze, P. G., Waser, N. M., and Price, M. V. 2007. Global warming and the disruption of plant-pollinator interactions. Ecology Letters. 10: 710–717. Memmott, J., Godfray, H. C. J., and Gauld, I. D. 1994. The structure of a tropical host parasitoid community. Journl of Animal Ecology. 63: 521–540. Memmott, J., and Waser, N. M. 2002. Integration of alien plants into a native flowerpollinator visitation web. Proceedings in Biological Sciences. 269: 2395–2399. Memmott, J., Waser, N. M., and Price, M. V. 2004. Tolerance of pollination networks to species extinctions. Proceedings in Biological Sciences. 271: 2605–2611.

192

BIBLIOGRAPHY

Miklós, I., and Podani, J. 2004. Randomization of presence-absence matrices: comments and new algorithms. Ecology. 85: 86–92. Milo, R., Itzkovitz, S., Kashtan, N., Levitt, R., Shen-Orr, S., Ayzenshtat, I., Sheffer, M., and Alon, U. 2004. Superfamilies of evolved and designed networks. Science. 303: 1538–1542. Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., and Alon, U. 2002. Network motifs: Simple building blocks of complex networks. Science. 298: 824–827. Montoya, J. M., Pimm, S. L., and Solé, R. V. 2006. Ecological networks and their fragility. Nature. 442: 259–264. Montoya, J. M., and Solé, R. V. 2002. Small-world patterns in food webs. Journal of Theoretical Biology. 214: 405–412. Morales, C. L., and Aizen, M. A. 2006. Invasive mutualisms and the structure of plant-pollinator interactions in the temperate forests of North-West Patagonia, Argentina. Journal of Ecology. 94: 171–180. Morales, J. M., and Vázquez, D. P. 2008. The effect of space in plant-animal mutualistic networks: Insights from a simulation study. Oikos. 117: 1362–1370. Morris, W. F. 2003. Which mutualisms are most essential? Buffering of plant reproduction against the extinction of different kinds of pollinator. In P. Kareiva, and S. Levin, eds., The Importance of Species, 260–280. Princeton University Press, Princeton. Mossa, S., Barthélémy, M., Stanley, H. E., and Amaral, L.A.N. 2002. Truncation of power law behavior in “scale-free” network models due to information filtering. Physical Review Letters. 88: 138701. Müller, C. B., Adriaanse, I.C.T., Belshaw, R., and Godfray, H.C.J. 1999. The structure of an aphid-parasitoid community. Journal of Animal Ecology. 68: 346–370. Müller, H., Thompson, D. W., and Darwin, C. 1883. The Fertilisation of Flowers trans. and ed. D’Arcy Thompson. Macmillan and Co., London. Nee, S. 1994. How populations persist? Nature. 367: 123–124. Neutel, A. M., Heesterbeek, J.A.P., and de Ruiter, P. C. 2002. Stability in real food webs: Weak links in long loops. Science. 296: 1120–1123. Neutel, A. M., Heesterbeek, J.A.P., van de Koppel, J., Hoenderboom, G., Vos, A., Kaldeway, C., Berendse, F., and de Ruiter, P. C. 2007. Reconciling complexity with stability in naturally assembling food webs. Nature. 449: 599–602. Newman, M.E.J. 2003. The structure and function of complex networks. SIAM Review. 45: 167–256. ———. 2004. Analysis of weighted networks. Physical Review E. 70: 056131. ———. 2006. Finding community structure in networks using the eigenvectors of matrices. Physical Review E. 74: 036104. Newman, M.E.J., Barabási, A., and Watts, D. J. 2006. The Structure and Dynamics of Networks. Princeton University Press, Princeton. Newman, M.E.J., and Girvan, M. 2004. Finding and evaluating community structure in networks. Physical Review E. 69: 026113. Nielsen, A., and Bascompte, J. 2007. Ecological networks, nestedness and sampling effort. Journal of Ecology. 95: 1134–1141. Nilsson, L. A. 1988. The evolution of flowers with deep corolla tubes. Nature. 334: 147–149.

BIBLIOGRAPHY

193

Nilsson, L. A. Jonsson, L., Ralison, L., and Randrianjohany, E. 1987. Angraecoid orchids and hawkmoths in Central Madagascar: Specialized pollination systems and general foragers. Biotropica. 19: 310–318. Novotny, V., Basset, Y., Miller, S. E., Weiblen, G. D., Bremer, B., Cizek, L., and Drozd, P. 2002. Low host specificity of herbivorous insects in a tropical forest. Nature. 416: 841–844. Nuismer, S. L., Jordano, P., and Bascompte, J. 2013. Coevolution and the architecture of mutualistic networks. Evolution. 67: 338–354. Nuismer, S. L., Thompson, J. N., and Gomulkiewicz, R. 1999. Gene flow and geographically structured coevolution. Proceedings of the Royal Society B. 266: 605–609. Odum, H. T. 1956. Primary production in flowing waters. Limnology and Oceanography. 1: 102–117. Okuyama, T., and Holland, J. N. 2008. Network structural properties mediate the stability of mutualistic communities. Ecology Letters. 11: 208–216. Olesen, J. M., Bascompte, J., Dupont, Y. L., Elberling, H., and Jordano, P. 2010. Missing and forbidden links in mutualistic networks. Proceedings of the Royal Society B. 278: 725–732. Olesen, J. M., Bascompte, J., Dupont, Y. L., and Jordano, P. 2006. The smallest of all worlds: pollination networks. Journal of Theoretical Biology. 240: 270–276. ———. 2007. The modularity of pollination networks. Proceedings of the National Academy of Sciences. 104: 19891–19896. Olesen, J. M., Bascompte, J., Elberling, H., and Jordano, P. 2008. Temporal dynamics in a pollination network. Ecology. 89: 1573–1582. Olesen, J. M., Eskildsen, L. I., and Venkatasamy, S. 2002. Invasion of pollination networks on oceanic islands: Importance of invader complexes and epidemic super generalists. Diversity and Distributions. 8: 181–192. Olesen, J. M., and Jordano, P. 2002. Geographic patterns in plant-pollinator mutualistic networks. Ecology. 83: 2416–2424. Olesen, J. M., Stefanescu, C., and Traveset, A. 2011. Strong long-term temporal dynamics of an ecological network. Plos One. 6: e26455. Ollerton, J. 1996. Reconciling ecological processes with phylogenetic patterns: The apparent paradox of plant-pollinator systems. Journal of Ecology. 84: 767–769. Ollerton, J., Steven, J., Johnson, D., Cranmer, L., and Kellie, S. 2003. The pollination ecology of an assemblage of grassland asclepiads in south africa. Annals of Botany. 92: 807–834. Otto, S. B., Rall, B. C., and Brose, U. 2007. Allometric degree distributions facilitate food-web stability. Nature. 450: 1226–U7. Paine, R. T. 1969. A note on trophic complexity and community stability. American Naturalist. 103: 91–93. ———. 1980. Food webs: Linkage, interaction strength and community infrastructure the third Tansley lecture. Journal of Animal Ecology. 49: 667–685. ———. 1992. Food-web analysis through field measurement of per capita interaction strength. Nature. 355: 73–75. Paradis, E. 2006. Analysis of Phylogenetics and Evolution with R. Springer-Verlag, New York.

194

BIBLIOGRAPHY

Paradis, E., Claude, J., and Strimmer, K. 2004. APE: Analyses of phylogenetics and evolution in R language. Bioinformatics. 20: 289–290. Parchman, T. L., and Benkman, C. W. 2002. Diversifying coevolution between crossbills and black spruce on Newfoundland. Evolution. 56: 1663–1672. Pascual, M., and Dunne, J. A. 2006. Ecological Networks. Linking Structure to Dynamics in Food Webs. Oxford University Press. Oxford. Pastor-Satorras, R., and Vespignani, A. 2001a. Epidemic dynamics and endemic states in complex networks. Physical Review E. 63: 066117. ———. 2001b. Epidemic spreading in scale-free networks. Physical Review Letters. 86: 3200–3203. Patterson, B. D., and Atmar, W. 1986. Nested subsets and the structure of insular mammalian faunas and archpielagos. Biological Journal of the Linnean Society. 28: 65–82. Pellmyr, O. 1992. Evolution of insect pollination and angiosperm diversification. Trends in Ecology and Evolution. 7: 46–49. ———. 2003. Yuccas, yucca moths, and coevolution: A review. Annals of the Missouri Botanical Garden. 90: 35–55. Petanidou, T., and Ellis, W. E. 1993. Pollinating fauna of a phryganic ecosystem: composition and diversity. Biodiversity Letters. 1: 9–22. Petanidou, T., and Ellis, W. N. 1996. Interdependence of native bee faunas and floras in changing Mediterranean communities. In A. Matheson, S. Buchmann, C. O’Toole, P. Westrich, and I. Williams, eds., The Conservation of Bees, 201–226. Academic Press, London. Petanidou, T., Kallimanis, A. S., Tzanopoulos, J., Sgardelis, S. P., and Pantis, J. D. 2008. Long-term observation of a pollination network: Fluctuation in species and interactions, relative invariance of network structure and implications for estimates of specialization. Ecology Letters. 11: 564–575. Petanidou, T., and Potts, S. 2006. Mutual use of resources in Mediterranean plant– pollinator communities: How specialized are pollination webs? In N. M. Waser, and J. Ollerton, eds., Plant–Pollinator Interactions: From Specialization to Generalization, 220–244. University of Chicago Press, Chicago. Petchey, O. L., Eklof, A., Borrvall, C., and Ebenman, B. 2008. Thophically unique species are vulnerable to cascading extinction. American Naturalist. 171: 568–579. Pierce, W. D., Cushman, R. A., and Hood, C. E. 1912. The insect enemies of the cotton boll weevil. U.S. Department of Agriculture, Bureau of Entomology Bulletin. 100: 1– 99. Pimm, S. L. 1979. The structure of food webs. Theoretical Population Biology. 16: 144–158. ———. 1982. Food Webs. Chapman and Hall, London. Pimm, S. L., and Lawton, J. H. 1980. Are food webs divided into compartments? Journal of Animal Ecology. 49: 879–898. Pocock, M.J.O., Evans, D. M., and Memmott, J. 2012. The robustness and restoration of a network of ecological networks. Science. 335: 973–977. Poulin, R., and Valtonen, E. T. 2001. Nested assemblages resulting from host size variation: The case of endoparasite communities in fish hosts. International Journal for Parasitology. 31: 1194–1204.

BIBLIOGRAPHY

195

Pound, R. 1893. Symbiosis and mutualism. American Naturalist. 27: 509–520. Prado, P. I., and Lewinsohn, T. M. 2004. Compartments in insect–plant associations and their consequences for community structure. Journal of Animal Ecology. 73: 1168–1178. Price, D. J. de S. 1965. Networks of scientific papers. Science. 149: 510–515. Prill, R., Iglesias, P., and Levchenko, A. 2005. Dynamic properties of network motifs contribute to biological network organization. PLoS Biology. 3: e343. Proulx, S. R., Promislow, D. E., and Phillips, P. C. 2005. Network thinking in ecology and evolution. Trends in Ecology and Evolution. 20: 345–353. R Development Core Team 2010. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. http://www. R-project.org. Raffaelli, D., and Hall, S. 1992. Compartments and predation in an estuarine food web. Journal of Animal Ecology. 61: 551–560. Raffaelli, D. G., and Hall, S. J. 1995. Assessing the importance of trophic links in food webs. In G. Polis, and K. Winemiller, eds., Food Webs: Integration of Pattern and Dynamics, 185–191. Chapman & Hall, New York. Ramírez, S. R., Eltz, T., Fujiwara, M. K., Goldman-Huertas, B., Gerlach, G., Tsutsui, N. D., and Pierce, N. E. 2011. Asynchronous diversification in a specialized plant-pollinator mutualism. Science. 333: 1742–1746. Rayfield, B., Fortin, M.-J., and Fall, A. 2011. Connectivity for conservation: A framework to classify network measures. Ecology. 92: 847–858. Rezende, E. L., Albert, E. M., Fortuna, M. A., and Bascompte, J. 2009. Compartments in a marine food web associated with phylogeny, body mass, and habitat structure. Ecology Letters. 12: 779–788. Rezende, E. L., Jordano, P., and Bascompte, J. 2007. Effects of phenotypic complementarity and phylogeny on the nested structure of mutualistic networks. Oikos. 116: 1919–1929. Rezende, E.L., Lavabre, J., Guimarães, P. R., Jr., Jordano, P., and Bascompte, J. 2007. Non-random coextinctions in phylogenetically structured mutualistic networks. Nature. 448: 925–928. Rico-Gray, V., and Oliveira, P. S. 2007. The Ecology and Evolution of Ant-Plant Interactions. The University of Chicago Press, Chicago. Roberts, A., and Stone, L. 1990. Island-sharing by archipelago species. Oecologia. 83: 560–567. Robertson, C. 1929. Flowers and Insects: Lists of Visitors to Four Hundred and Fiftythree Flowers. Carnegie Institute of Washington, Carlinville, IL. Rodríguez-Gironés, M. A., and Santamaría, L. 2006. A new algorithm to calculate the nestedness temperature of presence–absence matrices. Journal of Biogeography. 33: 924–935. Rooney, N., McCann, K. S., and Moore, J. C. 2008. A landscape theory for food web architecture. Ecology Letters. 11: 867–881. Rozenfeld, A. F., Arnaud-Haond, S., Hernández-García, E., Eguíluz, V. M., Serrão, E. A., and Duarte, C. M. 2008. Network analysis identifies weak and strong links in a metapopulation system. Proceedings of the National Academy of Sciences. 105: 18824–18829.

196

BIBLIOGRAPHY

Ryder, T. B., McDonald, D. B., Blake, J. G., Parker, P. G., and Loiselle, B. A. 2008. Social networks in the lek-mating wire-tailed manakin (Pipra filicauda). Proceedings of the Royal Society B. 275: 1367–1374. Saavedra, S., Reed-Tsochas, F., and Uzzi, B. 2009. A simple model of bipartite cooperation for ecological and organizational networks. Nature. 457: 463–466. Saavedra, S., Stouffer, D. B., Uzzi, B., and Bascompte, J. 2011. Strong contributors to network persistence are the most vulnerable to extinction. Nature. 478: 233–235. Sahli, H. F., and Conner, J. K. 2006. Characterizing ecological generalization in plantpollination systems. Oecologia. 148: 365–372. Sale, P. 1974. Overlap in resource use and interspecific competition. Oecologia. 17: 245–256. Sales-Pardo, M., Guimerà, R., Moreira, A. A., and Amaral, L.A.N. 2007. Extracting the hierarchical organization of complex systems. Proceedings of the National Academy of Sciences. 104: 15224–15229. Sanderson, J. G., Moulton, M. P., and Selfidge, R. 1998. Null matrices and the analysis of species co-occurrences. Oecologia. 116: 275–283. Santamaría, L., and Rodríguez-Gironés, M. A. 2007. Linkage rules for plant-pollinator networks: Trait complementarity or exploitation barriers? PLoS Biology. 5: e31. Santos, F. C., Rodrigues, J. F., and Pacheco, J. M. 2006. Graph topology plays a determinant role in the evolution of cooperation. Proceedings of the Royal Society B. 273: 51–55. Sapp, J. 1994. Evolution by Association: A History of Symbiosis. Oxford University Press, Oxford. Sawardecker, E. N., Amundsen, C. A., Sales-Pardo, M., and Amaral, L.A.N. 2009. Comparison of methods for the detection of node group membership in bipartite networks. The European Physical Journal B. 72: 671–677. Sazima, C., Jordano, P., Guimarães P. R., Jr., dos Reis, S. F., and Sazima, I. 2012. Cleaning associations between birds and herbivorous mammals in Brazil: Structure and complexity. Auk. 129: 36–43. Scheffer, M., Bascompte, J., Brock, W. A., Brovkin, V., Carpenter, S. R., Dakos, V., Held, H., Van Nes, E. H., Rietkerk, M., and Sugihara, G. 2009. Early-warning signals for critical transitions. Nature. 461: 53–59. Scheffer, M., Carpenter, S., Foley, J. A., Folke, C., and Walker, B. 2001. Catastrophic shifts in ecosystems. Nature. 413: 591–596. Scheffer, M., Carpenter, S. R., Lenton, T. M., Bascompte, J., Brock, W., Dakos, V., van de Koppel, J., et al. 2012. Anticipating critical transitions. Science. 338: 344–348. Schemske, D. W. 1983. Limits to specialization and coevolution in plant-animal mutualisms. In M. Nitecki, ed., Coevolution, 67–110. Chicago University Press, Chicago. Schneider, A. 1897. The phenomena of symbiosis. Minnesota Botanical Studies. 1: 923–948. Schroeder, M. 1991. Fractals, Chaos, and Power Laws. W. H. Freeman and Co., New York. Schuetz, P., and Caflisch, A. 2008. Efficient modularity organization by multistep greedy algorithm and vertex mover refinement. Physical Review E. 77: 046112.

BIBLIOGRAPHY

197

Schupp, E. W. 1995. Seed seedling conflicts, habitat choice, and patterns of plant recruitment. American Journal of Botany. 82: 399–409. Schupp, E. W., Jordano, P., and Gómez, J. M. 2010. Seed dispersal effectiveness revisited: A conceptual review. New Phytologist. 188: 333–353. Schwöbbermeyer, H. 2008. Network motifs. In B. H. Junker, and F. Schreiber, eds., Analysis of Biological Networks. Wiley Interscience, Hoboken, NJ. Selva, N., and Fortuna, M. A. 2007. The nested structure of a scavenger community. Proceedings of the Royal Society B. 274: 1101–1108. Sernander, R. 1906. Entwurf einer Monographie der europäischen Myrmekochoren. Almquist and Wiksells Boktrycker, Uppsala and Stockholm. Silva, W. R., de Marco, P., Jr, Hasui, E., and Gomes, V.S.M. 2002. Patterns of fruitfrugivore interactions in two atlantic forest bird communities of South-eastern Brazil: Implications for conservation. In D. Levey, W. Silva, and M. Galetti, eds., Seed Dispersal and Frugivory: Ecology, Evolution, and Conservation, 423–435. CAB International, Wallingford. Simon, H. A. 1955. On a class of skewed distribution functions. Biometrika. 42: 425–440. Snow, B. K., and Snow, D. W. 1972. Feeding niches of hummingbirds in a Trinidad valley. Journal of Animal Ecology. 41: 471–485. Solé, R. V., and Bascompte, J. 2006. Self-Organization in Complex Ecosystems. Princeton University Press, Princeton. Solé, R. V., and Montoya, J. M. 2001. Complexity and fragility in ecological networks. Proc. R. Soc. B. 268: 2039–2045. Solé, R. V., and Valverde, S. 2006. Are network motifs the spandrels of cellular complexity? Trends in Ecology and Evolution. 21: 419–422. Spirin, V., and Mirny, L. A. 2003. Protein complexes and functional modules in molecular networks. Proceedings of the National Academy of Sciences. 100: 12123–12128. Sporns, O., and Kotter, R. 2004. Motifs in brain networks. PLoS Biology. 2: e369. Sprengel, C. K. 1793. Das entdeckte Geheimniss der Natur im Bau und in der Befruchtung der Blumen, I. Vieweg sen., Berlin. Srinivasan, U. T., Dunne, J. A., Harte, J., and Martinez, N. D. 2007. Response of complex food webs to realistic extinction sequences. Ecology. 88: 671–682. Stang, M., Klinkhamer, P. G. L., and van der Meijden, E. 2006. Size constraints and flower abundance determine the number of interactions in a plant-flower visitor web. Oikos. 112: 111–121. ———. 2007. Asymmetric specialization and extinction risk in plant-flower visitor webs: A matter of morphology or abundance? Oecologia. 151: 442–453. Stebbins, G. L. 1970. Adaptive radiation of reproductive characteristics in angiosperms, I: Pollination mechanisms. Annual Review of Ecology, Evolution, and Systematics. 1: 307–326. Steuer, R., and Zamora-López, G. 2008. Global network properties. In B. Junker, and F. Schreiber, eds., Analysis of Biological Networks. Wiley-Interscience, Hoboken, NJ. Stouffer, D. B., and Bascompte, J. 2010. Understanding food-web persistence from local to global scales. Ecology Letters. 13: 154–161.

198

BIBLIOGRAPHY

———. 2011. Compartmentalization increases food-web persistence. Proceedings of the National Academy of Sciences. 108: 3648–3652. Stouffer, D. B., Camacho, J., and Amaral, L.A.N. 2006. A robust measure of food web intervality. Proceedings of the National Academy of Sciences. 103: 19015– 19020. Stouffer, D. B., Camacho, J., Guimerà, R., R, Ng, C. A., and Amaral, L.A.N. 2005. Quantitative patterns in the structure of model and empirical food webs. Ecology. 86: 1301–1311. Stouffer, D. B., Camacho, J., Jiang, W., and Amaral, L.A.N. 2007. Evidence for the existence of a robust pattern of prey selection in food webs. Proceedings of the Royal Society B. 274: 1931–1940. Strauss, S. Y., Sahli, H., and Conner, J. K. 2005. Toward a more trait-centered approach to diffuse (co)evolution. New Phytologist. 165: 81–90. Strogatz, S. H. 2001. Exploring complex networks. Nature. 410: 268–76. Stumpf, M.P.H., Ingram, P. J., Nouvel, I., and Wiuf, C. 2005. Statistical model selection methods applied to biological networks. In C. Priami, ed., Transactions in Computational Systems Biology, 65–77. Springer, New York. Stumpf, M.P.H., and Wiuf, C. 2005. Sampling properties of random graphs: the degree distribution. Physical Review E. 72: 036118. Sugihara, G. 1983. Holes in niche space: A derived assembly rule and its relation to intervality. In D. DeAngelis, W. Post, and G. Sugihara, eds., Current Trends in Food Web Theory. Report 5983, 25–35. Oak Ridge National Laboratory. Oak Ridge, TN. Sugihara, G., Schoenly, K., and Trombla, A. 1989. Scale invariance in food web properties. Science. 245: 48–52. Thébault, E., and Fontaine, C. 2008. Does asymmetric specialization differ between mutualistic and trophic networks? Oikos. 117: 555–563. ———. 2010. Stability of ecological communities and the architecture of mutualistic and trophic networks. Science. 329: 853–856. Thioulouse, J., Chessel, D., Dolédec, S., and Olivier, J. 1997. ADE-4: A multivariate analysis and graphical display software. Statistical Computing. 7: 75–83. Thompson, J. N. 1982. Interaction and Coevolution. Wiley, New York. ———. 1994. The Coevolutionary Process. The University of Chicago Press, Chicago. ———. 1999a. The evolution of species interactions. Science. 284: 2116–2118. ———. 1999b. Specific hypotheses on the geographic mosaic of coevolution. American Naturalist. 153: S1–S14. ———. 2005. The Geographic Mosaic of Coevolution. University of Chicago Press, Chicago. ———. 2006. Mutualistic webs of species. Science. 312: 372–373. Thompson, J. N., and Cunningham, B. M. 2002. Geographic structure and dynamics of coevolutionary selection. Nature. 417: 735–738. Thompson, J. N., and Pellmyr, O. 1992. Mutualism with pollinating seed parasites amid co-pollinators: Constraints on specialization. Ecology. 73: 1780–1791. Tiffney, B. 2004. Vertebrate dispersal of seed plants through time. Annual Review of Ecology, Evolution, and Systematics. 35: 1–29. Tilman, D., and Kareiva, P. 1997. Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions. Princeton University Press, Princeton.

BIBLIOGRAPHY

199

Traveset, A., and Richardson, D. M. 2006. Biological invasions as disruptors of plant reproductive mutualisms. Trends in Ecology and Evolution. 21: 208–16. Tylianakis, J. M. 2009. Warming up food webs. Science. 323: 1300–1301. Tylianakis, J. M., Didham, R. K., Bascompte, J., and Wardle, D. A. 2008. Global change and species interactions in terrestrial ecosystems. Ecology Letters. 11: 1351– 1363. Tylianakis, J.M., Tscharntke, T., and Lewis, O. 2007. Habitat modification alters the structure of tropical host–parasitoid food webs. Nature. 445: 202–205. Ulanowicz, R. E., and Wolff, W. F. 1991. Ecosystem flow networks: Loaded dice? Mathematical Biosciences. 103: 45–68. Ulrich, W., Almeida-Neto, M., and Gotelli, N. J. 2009. A consumer’s guide to nestedness analysis. Oikos. 118: 3–17. Ulrich, W., and Gotelli, N. J. 2007. Null model analysis of species nestedness patterns. Ecology. 88: 1824–1831. Urban, D., and Keitt, T. 2001. Landscape connectivity: A graph-theorerical perspective. Ecology. 82: 1205–1218. Vacher, C., Piou, D., and Desprez-Loustau, M.-L. 2008. Architecture of an antagonistic tree/fungus network: The asymmetric influence of past evolutionary history. PLoS ONE. 3: e1740. Valdovinos, F. S., Ramos-Jiliberto, R., Flores, J. D., Espinoza, C., and López, G. 2009. Structure and dynamics of pollination networks: The role of allien plants. Oikos. 118: 1190–1200. van Veen, F. J., Morris, R. J., and Godfray, H. C. 2006. Apparent competition, quantitative food webs, and the structure of phytophagous insect communities. Annual Review of Entomology. 51: 187–208. Vázquez, A., Dobrin, R., Sergi, D., Eckmann, J. P., Oltvai, Z. N., and Barabási, A. L. 2004. The topological relationship between the large-scale attributes and local interaction patterns of complex networks. Proceedings of the National Academy of Sciences. 101: 17940–17945. Vázquez, D. P. 2005. Degree distribution in plant-animal mutualistic networks: Forbidden links or random interactions? Oikos. 108: 421–426. Vázquez, D. P., and Aizen, M. A. 2004. Asymmetric specialization: A pervasive feature of plant-pollinator interactions. Ecology. 85: 1251–1257. Vázquez, D. P., Chacoff, N. P., and Cagnolo, L. 2009. Evaluating multiple determinants of the structure of plant-animal mutualistic networks. Ecology. 90: 2039–2046. Vázquez, D. P., Morris, W. F., and Jordano, P. 2005. Interaction frequency as a surrogate for the total effect of animal mutualists on plants. Ecology Letters. 8: 1088–1094. Vázquez, D. P., Poulin, R., Krasnov, B. R., and Shenbrot, G. I. 2005. Species abundance and the distribution of specialization in host-parasite interaction networks. Journal of Animal Ecology. 74: 946–955. Verdú, M., and Valiente-Banuet, A. 2008. The nested assembly of plant facilitation networks prevents species extinctions. American Naturalist. 172: 751–760. Volterra, M. V., and D’Ancona, M. U. 1935. Les Associations Biologiques au Point de Vue Mathématique. Hermann et Cie, Paris. Wallace, A. R. 1889. Darwinism. An Exposition of the Theory of Natural Selection with Some of Its Applications. Macmillan and Co., London. Wang, B. C., and Smith, T. B. 2002. Closing the seed dispersal loop. Trends in Ecology and Evolution. 17: 379–385.

200

BIBLIOGRAPHY

Waser, N. M. 2006. Specialization and generalization in plant-pollinator interactions: A historical perspective. In N. M. Waser, and J. Ollerton, eds., Plant-Pollinator Interactions. From Specialization to Generalization, 3–22. University of Chicago Press, Chicago. Waser, N. M., Chittka, L., Price, M. V., Williams, N. M., and Ollerton, J. 1996. Generalization in pollination systems, and why it matters. Ecology. 77: 1043– 1060. Waser, N. M., and Ollerton, J. 2006. Plant-Pollinator Interactions. From Specialization to Generalization. University of Chicago Press, Chicago. Watts, Duncan, J. 2003. Small Worlds: The Dynamics of Networks Between Order and Randomness. Princeton University Press, Princeton. Watts, Duncan, J., and Strogatz, S. H. 1998. Collective dynamics of “small-world” networks. Nature. 393: 440–442. Webb, C. O., Ackerly, D. D., McPeek, M. A., and Donoghue, M. J. 2002. Phylogenies and community ecology. Annual Review of Ecology, Evolution, and Systematics. 33: 475–505. Weiblen, G. D., Webb, C. O., Novotny, V., Basset, Y., and Miller, S. E. 2006. Phylogenetic dispersion of host use in a tropical insect herbivore community. Ecology. 87: S62–S75. Wheelwright, N. T., and Orians, G. H. 1982. Seed dispersal by animals: Contrasts with pollen dispersal, problems of terminology, and constraints on coevolution. American Naturalist. 119: 402–413. White, E. P., Enquist, B. J., and Green, J. L. 2008. On estimating the exponent of powerlaw frequency distributions. Ecology. 89: 905–912. Wiebes, J. T. 1979. Co-evolution of figs and their insect pollinators. Annual Review of Ecology, Evolution, and Systematics. 10: 1–12. Williams, R. J., Berlow, E. L., Dunne, J. A., Barabási, A. L., and Martinez, N. D. 2002. Two degrees of separation in complex food webs. Proceedings of the National Academy of Sciences. 99: 12913–12916. Williams, R. J., and Martinez, N. D. 2000. Simple rules yield complex food webs. Nature. 404: 180–183. Winemiller, K. O. 1990. Spatial and temporal variation in tropical fish trophic networks. Ecological Monographs. 60: 331–367. Wolf, J.B.W., and Trillmich, F. 2008. Kin in space: Social viscosity in a spatially and genetically substructured network. Proceedings of the Royal Society B. 275: 2063–2069. Wootton, J. T. 1997. Estimates and tests of per capita interaction strength: Diet, abundance, and impact of intertidally foraging birds. Ecological Monographs. 67: 45–64. Wright, D. H. 1989. A simple, stable model of mutualism incorporating handling time. American Naturalist. 134: 664–667. Wright, S. J. 2003. The myriad consequences of hunting for vertebrates and plants in tropical forests. Perspectives in Plant Ecology Evolution and Systematics. 6: 73–86. Yook, S. H., Jeong, H., Barabási, A. L., and Tu, Y. 2001. Weighted evolving networks. Physical Review Letters. 86: 5835–5838.

Index

abundance. See species abundance air traffic networks, 24, 28 allospecific interactions, 6 AMOVA, 33–34 analytic techniques, xi–xii; of bipartite networks, 139–42, 153; of phylogenetic signals, xi, 154–59; software for, 175–78; of stability and persistence, 167–74 ANINHADO, 177–78 antagonistic interactions, 3, 7 ant-plant mutualisms, 6 architecture of networks, xi–xii, 17–21; buildup models of, 31–32, 37, 41, 65; connectivity distributions in, 21–23; consequences of, 107–35; in ecological systems, 32–41; graph representations of, 17–20; hubs of, 23, 27; modularity function in, 25–29, 35, 150–53; motifs as building blocks of, 29–31, 52, 127; nodes of interactions in, 15, 16, 18, 19, 83–84; one- and two-mode types of, 20–21; preferential attachment mechanisms in, 31–32, 64–65; quantifiable components of, 15; small-world property in, 24–25, 26; strength distribution in, 24; three classes of, 23; weighting of, 21. See also structure of mutualistic plant-animal networks average distance, 97 Beal, W. J., 1–2 behavioral switching, 94–96 between-year turnover, 97 biodiversity, 4–5, 12, 102; architecture of, 14; coevolutionary events and, 131; coextinction cascades and, 111–12; complex networks of, 13–14; dynamic stability and, 115–20; global change and, 120–26; neutral theory of, 76–77, 166; soft mean field model of, 171–74

biological invasions, 107, 124–26 bipartite graphs, 20–21 bipartite library, 177 bipartite networks, 60–61; analysis of, 139–42, 153; spatial and temporal dynamics of, 89 Boucher, D. H., 4 boundary specification problems, 67–69 Breeder’s equation, 128 broad-scale distributions, 23, 90 cascade models, 37, 129–31 central species, 83–84, 96, 97 Chinese boxes, 47 Clements, R. E., 3 codiversification, 137 coevolution, xi–xii, 12–14, 107, 126–32; cascading effects of, 129–31; complementarity and convergence in, 127–32; complex network theory of, xi, 13–14, 17–21, 127; ecological and evolutionary mechanisms of, 64–86; geographic mosaic theory of, 12–13, 87, 98, 99; historical overview of, 1–6; remixing of traits in, 13; two-mode representation of, 20–21 coevolutionary hotspots, 13 coextinction cascades, 108–13; foraging adaptation and, 111–12; nested structures and, 111; size factors in, 112–13 cohesion, 54 community contexts, 5–6; assembly process in, 12, 88–96; consequences of network structure for, 107; of global change, 120–26; nodes of interactions in, 15, 16, 83–84; theory of complex networks in, xi, 13–14, 167–74. See also metacommunity models; networks compartmentalization, 34, 150 complementarity, 127–32

202

INDEX

complex network theory, xi, 13–14, 17–21, 127, 136; analysis of fixed points and stability in, 167–69; on competition and biodiversity, 169–74. See also networks connectance, 34, 97 connectivity distributions, 21–23, 90; in food webs, 60–61; forbidden links and, 65, 67–71; information filtering and, 65; network robustness and, 22–23, 108; power-law distributions and, 23, 42–44; truncated power-law distributions and, 43–44, 64–65. See also degree distributions consequences of network structure, 107–35; coevolution as, 126–32; coextinction cascades and, 108–13; dynamic stability and, 107, 113–20; global change and, 107, 120–26; for nonbiological systems, 132–34; species richness as, 115–20 convergence, 127–32 cooperative networks, 133–34 cumulative probability distributions, 21–22, 42–44; in daily interaction matrices, 92–93; scaling functions in, 44, 45 Cytoscape, 176

ecological and evolutionary mechanisms, 64–86; forbidden links in, 65, 67–71; information filtering in, 65; phylogenetic signals in, 66, 77–85; preferential attachment mechanisms in, 64–65; species abundance and, 71–77; truncated power-law connectivity distributions in, 64–65; types of links in, 67 ecological networks, 32–41; comparisons among, 60–62; epidemiological networks as, 38–39; food web networks as, 34–37; host-parasitoid networks as, 37–38; individual behavior in, 41; social networks as, 39–41; spatial models of, 32–34, 35; stability and persistence in, 36–37, 61–62 effective competition, 168 effective productivity, 168 Ehrlich, P. R., 4 epidemiological networks, 38–39 estimated generalized least-squares (EGLS) procedure, 155–59 exponential distribution, 43, 108, 143 extinction of interactions, xi. See also coextinction cascades

daily interactions, 88–96 D’Ancona, M. U., 3 Darwin, Charles, 1–3, 5, 34 de Bary, H. A., 3 defaunation, 126 degree distributions, 21–23, 97; fitting models in, 143–46; in plant-animal mutualistic networks, 42–44; scaling functions in, 44, 45; weighted equivalents of, 58–59. See also connectivity distributions demographic stability, 107, 113. See also stability and persistence diffuse coevolution, 12 dispersal stage, 8–11 diversification. See biodiversity dynamics. See spatial and temporal dynamics dynamic stability, 107, 113–20; analysis of, 167–69; asymmetry in, 113–15; species richness in, 115–20

fig wasps, 5 financial networks, 132–33 fitness effects, 7, 8–10 FoodWeb3D, 176 food webs, 18, 20, 21, 32, 34–37, 139; analysis of, 139–42; buildup models of, 37; coextinction cascades in, 113; community organization in, 37; compartmentalization in, 150; connectivity distributions in, 60–61; interaction strength and, 61–62, 158; modularity function in, 28–29; nestedness analysis of, 61–62; network motifs in, 29–31; power-law distributions in, 23; small-world property in, 25, 62; species-deletion experiments in, 110–11; stability and persistence in, 36–37, 61–62; as static descriptions, 87 foraging adaptation, 111–12 forbidden links, 65, 67–71; size mismatching in, 69, 71, 72–73, 74; temporal uncoupling in, 69, 70, 72–73, 74, 120 fraction of top predators, 34 fragility. See robustness to perturbation

early recruitment stage, 9 early-warning indicators of ecosystem shift, 136

INDEX

frequency distributions. See degree distributions Fst metrics, 33 garment industry networks, 133–34 Gause, G. F., 3 Gaussian distributions, 121 generalist species, 46, 47–48, 59; extinction rates of, 110–11; habitat loss of, 121–24 geographic mosaics of coevolution, 12–13, 87, 98–102 Gephi, 176 global change, 107, 120–26; biological invasions and, 124–26; habitat loss and, 121–24; temporal uncoupling and, 120 global structure of interactions, 84–85 graph theory, 19–21, 33 greedy algorithm, 152 habitat loss, 5, 107, 121–24; disassembly of mutualistic networks and, 122, 123, 124; rare plants and, 123–24 harvest mutualisms, 6, 8 heterogeneity, 39–44, 62 highly diversified plant-animal mutualisms, 107 host-parasitoid networks, 37–38 hubs, 23, 27 human-based domestication mutualisms, 6 hunting, 126 igraph, 175 individual-based networks, 41 information filtering, 65 interaction accumulation curves (IAC), 102–6 interaction networks, 32; patterning of, 57; robustness of sampling of, 102–6 interaction specificity, 11 interaction strength, 34, 54–56, 61–63 interdependent networks, 111 Internet: degree distributions in, 22–23; epidemiological models of, 39; heterogeneity of, 39 invader complexes, 125 invariance, 34 invasive species, 124–26 isolated species, 47–48 Jaccard’s similarity index, 150 Janzen, Daniel, xi

203

keystone species, 36 Knuth, P., 1 Köllreuter, J. G., 1 Königsberg bridges problem, 17–19 Kropotkin, Peter, 4 lattice networks, 31, 33, 100 legitimate pollinators, 8 Library, 176–77 Library vegan, 178 local context dependency, 13 local stability in proximity of a steady state, 113. See also dynamic stability log-normal distributions, 143 long-tongued fly-monocot plant mutualisms, 5 Lotka-Volterra models, 3 match/mismatch hypothesis, 69, 70 mean field mutualism, 171 measures of network structure, 15; connectivity distributions in, 21–23; graph theory in, 19–20; modularity function in, 25–29, 35; motifs (subgraphs) in, 29–31; small-world property in, 24–25, 26; strength distribution in, 24. See also architecture of networks mechanisms of mutualistic networks, xii metacommunity models, 87, 98, 99; of biological invasions, 124–26; of dynamic stability, 115; of habitat loss, 121–24; sampling parameters in, 100–102 metapopulation theory, 32–34, 87, 98 minimum spanning tree, 33 mobility, 81 modularity function, 25–29, 35, 50–54, 63; classification of species’ roles in, 52; definition of, 151–52; measures of, 150–53; nestedness of networks and, 52–54; simulated annealing (SA) in, 28 module-finding algorithm, 151–53 Monte Carlo simulations, 150 moth-orchid mutualisms, 5 motifs, 29–31; dynamic implications of, 30–31; structural studies of, 29–30 multispecific interactions, 6; coevolution in, 12–14, 126–32; diversification in, 12; in pollinating mutualisms, 10; in seed-dispersal mutualisms, 8–10; theory of complex networks and, 13–14 Mutual Aid (Kropotkin), 4

204

mutualistic interactions (mutualism), 6–12, 136–37; community context of, 5–6, 12–14; Darwin’s demonstration of, 1–3; early studies of, 5–6; five major groups of, 6; historical overview of, 1–6; origins of, 4–5; present relevance of, 5 mutualistic plant-animal networks: analytical theory of, 167–74; bipartite character of, 60–61; classification of species’ roles in, 52; coextinction cascades in, 108–13; cohesion in, 54; comparisons of, 60–62; dynamic stability of, 113–20; ecological and evolutionary mechanisms in, 64–86; generalist species in, 46, 47, 59; global change and, 120–26; global interaction patterns of, 84–85; heterogeneity of, 39–44, 62; spatial and temporal dynamics of, 87–106; specialists in, 46, 47, 59; species identities in, 45–48; stability and persistence in, 107; structure of, 42–63, 107; topology of interaction in, 68 nectar thieves, 8 Nei’s distance, 33 nestedness, 44–48, 52–54, 62; asymmetric specialization in, 47–48, 59, 124; distribution in space of, 98–100; in food webs, 61–62; measures of, 147–49; in nonbiological networks, 133, 134; robustness to sampling effort of, 104–6; soft mean field model of, 171–74; species abundance and, 75–76; species driving coevolution and, 47; species extinction and, 111; species position and, 80–81; species richness and, 102, 117–19; year-to-year variability and, 97 nestedness temperature calculator (NTC), 147–48, 161, 177 NetCarto, 178 network diameter, 97 network dynamics. See spatial and temporal dynamics networks: architecture of, xi–xii, 15–41; buildup models of, 31–32, 37, 41, 65; definition of, 17; interdependence of, 111; motifs as building blocks of, 29–31, 52, 127; nodes of interactions in, 15, 16, 18, 19, 83–84; in nonbiological systems, 132–34; quantifiable components of, 15; robustness to perturbation of, 8, 20, 23,

INDEX

41, 107; spatial and temporal dynamics of, 87–106; statistical measures of complexity of, 15, 19–20; theory of, xi, 13–14, 17–21, 127, 136; three classes of, 23. See also ecological networks; mutualistic plant-animal networks NetworkX, 175 neutral theory of biodiversity, 76–77, 166 niche conservatism, 66 niche intervality, 48–49 niche model, 37, 158 niche variation, 12 node overlap and decrease filling (NODF), 148–49 nonbiological systems, 132–34 noncoevolutionary events, 131 null models, xi–xii, 153, 160–66 one-mode networks, 20, 21, 65, 152; modularity function in, 28; in small-world models, 48–50 On the Origin of Species (Darwin), 2–3 orchid studies, 1–3, 5 pairwise paradigms, 5–6 PAJEK, 175 parasitoids, 37–38 Pareto’s distributions, 43 participation coefficient of connectivity, 52 passive sampling, 74–75 Patefield’s algorithm, 165–66 path analysis, 166 peripheral species, 83–84, 96 phenophases, 89; daily dynamics of, 91; seasonal dynamics of, 94 phenotype differences model, 131–32 phenotype matching model, 131–32 phylogenetic covariation patterns, 84–85 phylogenetic niche conservatism, 77 phylogenetic signals, 66, 77–85; analysis using, xi, 154–59; ecological correlates of, 81–84; global structure of interactions of, 84–85; interaction pattern of, 77–79, 84–85, 154–59; pattern buildup of, 79; shared histories in, 77; species position and, 79–84 plant-animal mutualistic networks. See mutualistic plant-animal networks pollination networks, 6–12; global change and, 120; mutualisms in, 6, 10–11; small-world property of, 25

INDEX

population cycle of a higher plant, 8, 9 Pound, R., 3 power-law distributions, 23, 42–44, 143; in daily interaction matrices, 92–93; truncated variants of, 43–44, 64–65 predictability of target quality, 11 preferential attachment mechanisms, 31–32, 64–65, 90; daily interaction matrices of, 91, 92–93; forbidden links and, 65, 67–71; information filtering and, 65; species-specific constraints of, 92 Prisoner’s Dilemma, 40 protective mutualisms, 6 quality components, 9, 10–11 quantity components, 9, 10–11 random-fill procedure, 162 rates of evolution, 128 regeneration cycle of a higher plant, 8, 9 relative nestedness, 104 R libraries, 176–77 robustness to perturbation, 8, 20, 23, 41, 107, 108; coextinction cascades and, 108–13; foraging adaptation and, 111–12; nested structures and, 111, 117–19; species richness and, 117–19; structural stability and, 169–74 robustness to sampling effort, 102–6 sampling bias, 75–76, 106 sampling effort, 102–6 scale-free networks, 23, 64, 90, 108–10 scaling functions, 44, 45 seasonal dynamics, 93–96 seed banks, 9 seed dispersal mutualisms, 6, 8–11 shared phylogenetic history, 77 simulated annealing (SA), 28, 151, 152 single-scale networks, 23 SIR equations, 38 six degrees of separation, 25 size mismatching, 69, 71, 72–73, 74 small-world networks, 24–25, 26, 48–50, 62, 63 social networks, 39–41, 67–69 socioeconomic cooperative networks, 133–34 soft mean field model, 171–74 software, 175–78 SONIA, 177

205

spatial and temporal dynamics, xii, 87–106; of bipartite networks, 89; in daily interactions, 88–96; in distribution of networks, 98–102; habitat loss and, 121–24; mosaic theory and, 87, 98, 99; robustness to sampling effort and, 102–6; in seasonal variability, 93–96; species shift and, 97–98; in switching behaviors, 94–96; in temporal uncoupling, 69, 70, 72–73, 74; in year-to-year variability, 96–98 spatial networks, 20, 32–34, 35 specialist species, 46, 47–48, 59; extinction rates of, 110–11; habitat loss and, 122–24 species abundance, 71–77, 166; sampling bias and, 75–76; species position and, 80–81 species accumulation curves (SAC), 102–6 species competition, 115–20 species degree, 59, 62, 65, 71, 83–84 species position, 79–84; ecological correlates of, 81–84; species abundance and, 80–81 species richness, 102; coevolutionary events and, 131; coextinction cascades and, 111–12; dynamic stability and, 115–20; global change and, 120–26; soft mean field model of, 171–74. See also biodiversity species shift, 97–98 species strength, 58–59, 62–63 specificity, 11 spectral algorithms, 152 Sprengel, C. K., 1, 2 stability and persistence, 36–37, 61–62, 107; analysis of, 167–74; asymmetry and, 113–15; dynamic stability and, 113–20, 167–69; structural stability and, 169–74. See also robustness to perturbation statnet bundle, 176–77 strength distribution, 24 strong mutualism, 167 structural stability, 169–74 structure of mutualistic plant-animal networks, 42–63, 107; consequences of, 107–35; degree distributions in, 42–44, 58–59, 60–61; impact on competition and biodiversity of, 169–74; modularity function in, 50–54, 63, 150–53; nestedness in, 44–48, 61–62, 97, 147–49; small-world properties in, 48–50, 62, 63;

206

structure of mutualistic plant-animal networks (cont’d); stability and persistence in, 61–62; weightedness in, 54–59, 62–63. See also architecture of networks subgraphs, 29–31 supergeneralist species, 125, 128–31 supertrees, 154–55 swap algorithm, 162–63 switching behavior, 94–96 symbiosis, 3. See also; mutualistic interactions Tansley, Arthur, 3 target-site suitability, 11 temporal dynamics. See spatial and temporal dynamics temporal suitability, 11 temporal uncoupling, 69, 70, 72–73, 74 theory of complex networks. See complex network theory topological stability, 107, 113 Tragedy of the Commons, 134 trait complementarity, 128 trait evolution models, 128–31 Tree of Life, 137 tripartite networks, 20

INDEX

truncated power-law distributions, 43–44, 64–65, 90, 143 truncated scale-free networks, 23 two-mode networks, 20–21, 65, 152 unipartite graphs, 20–21 unipartite networks, 139, 153 Volterra, M. V., 3 Wallace, Alfred Russell, 1 weak mutualism, 167, 169, 171–74 Web of Life, 137 weighted networks, 21, 54–59, 62–63; asymmetric dependencies in, 55, 56–58, 62, 113–15, 124; interaction strength and, 54–56, 61–62; species degree in, 59, 62; species strength in, 58–59, 62–63 within-module degree of connectivity, 52 Witt, A. A., 3 yearly variations, 96–98 Ythan Estuary food web, 18 yucca moths, 5 Zackenberg Research Station, Greenland, 89–90, 91 Zipf’s distributions, 43

MONOGRAPHS IN POPULATION BIOLOGY EDITED BY SIMON A. LEVIN AND HENRY S. HORN

1. The Theory of Island Biogeography, by Robert H. MacArthur and Edward O. Wilson 2. Evolution in Changing Environments: Some Theoretical Explorations, by Richard Levins 3. Adaptive Geometry of Trees, by Henry S. Horn 4. Theoretical Aspects of Population Genetics, by Motoo Kimura and Tomoko Ohta 5. Populations in a Seasonal Environment, by Steven D. Fretwell 6. Stability and Complexity in Model Ecosystems, by Robert M. May 7. Competition and the Structure of Bird Communities, by Martin L. Cody 8. Sex and Evolution, by George C. Williams 9. Group Selection in Predator-Prey Communities, by Michael E. Gilpin 10. Geographic Variation, Speciation, and Clines, by John A. Endler 11. Food Webs and Niche Space, by Joel E. Cohen 12. Caste and Ecology in the Social Insects, by George F. Oster and Edward O. Wilson 13. The Dynamics of Arthropod Predator-Prey Systems, by Michael P. Hassel 14. Some Adaptations of Marsh-Nesting Blackbirds, by Gordon H. Orians 15. Evolutionary Biology of Parasites, by Peter W. Price 16. Cultural Transmission and Evolution: A Quantitative Approach, by L. L. Cavalli-Sforza and M. W. Feldman 17. Resource Competition and Community Structure, by David Tilman 18. The Theory of Sex Allocation, by Eric L. Charnov 19. Mate Choice in Plants: Tactics, Mechanisms, and Consequences, by Nancy Burley and Mary F. Wilson 20. The Florida Scrub Jay: Demography of a Cooperative-Breeding Bird, by Glen E. Woolfenden and John W. Fitzpatrick 21. Natural Selection in the Wild, by John A. Endler 22. Theoretical Studies on Sex Ratio Evolution, by Samuel Karlin and Sabin Lessard 23. A Hierarchical Concept of Ecosystems, by R.V. O’Neill, D.L. DeAngelis, J.B. Waide, and T.F.H. Allen 24. Population Ecology of the Cooperatively Breeding Acorn Woodpecker, by Walter D. Koenig and Ronald L. Mumme 25. Population Ecology of Individuals, by Adam Lomnicki 26. Plant Strategies and the Dynamics and Structure of Plant Communities, by David Tilman 27. Population Harvesting: Demographic Models of Fish, Forest, and Animal Resources, by Wayne M. Getz and Robert G. Haight 28. The Ecological Detective: Confronting Models with Data, by Ray Hilborn and Marc Mangel 29. Evolutionary Ecology across Three Trophic Levels: Goldenrods, Gallmakers, and Natural Enemies, by Warren G. Abrahamson and Arthur E. Weis

30. Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions, edited by David Tilman and Peter Kareiva 31. Stability in Model Populations, by Laurence D. Mueller and Amitabh Joshi 32. The Unified Neutral Theory of Biodiversity and Biogeography, by Stephen P. Hubbell 33. The Functional Consequences of Biodiversity: Empirical Progress and Theoretical Extensions, edited by Ann P. Kinzig, Stephen J. Pacala, and David Tilman 34. Communities and Ecosystems: Linking the Aboveground and Belowground Components, by David Wardle 35. Complex Population Dynamics: A Theoretical/Empirical Synthesis, by Peter Turchin 36. Consumer-Resource Dynamics, by William W. Murdoch, Cheryl J. Briggs, and Roger M. Nisbet 37. Niche Construction: The Neglected Process in Evolution, by F. John Odling-Smee, Kevin N. Laland, and Marcus W. Feldman 38. Geographical Genetics, by Bryan K. Epperson 39. Consanguinity, Inbreeding, and Genetic Drift in Italy, by Luigi Luca Cavalli-Sforza, Antonio Moroni, and Gianna Zei 40. Genetic Structure and Selection in Subdivided Populations, by Franois Rousset 41. Fitness Landscapes and the Origin of Species, by Sergey Gavrilets 42. Self-Organization in Complex Ecosystems, by Ricard V. Solé and Jordi Bascompte 43. Mechanistic Home Range Analysis, by Paul R. Moorcroft and Mark A. Lewis 44. Sex Allocation, by Stuart West 45. Scale, Heterogeneity, and the Structure of Diversity of Ecological Communities, by Mark E. Ritchie 46. From Populations to Ecosystems: Theoretical Foundations for a New Ecological Synthesis, by Michel Loreau 47. Resolving Ecosystem Complexity, by Oswald J. Schmitz 48. Adaptive Diversification, by Michael Doebeli 49. Ecological Niches and Geographic Distributions, by A. Townsend Peterson, Jorge Soberón, Richard G. Pearson, Robert P. Anderson, Enrique Martínez-Meyer, Miguel Nakamura, and Miguel Bastos Araújo 50. Food Webs, by Kevin S. McCann 51. Population and Community Ecology of Ontogenetic Development, by André M. de Roos and Lennart Persson 52. Ecology of Climate Change: The Importance of Biotic Interactions, by Eric Post 53. Mutualistic Networks, by Jordi Bascompte and Pedro Jordano