Complexity in Landscape Ecology [2 ed.] 3030467724, 9783030467722

Table of contents :
Preface to the Second Edition
Contents
Chapter 1: Complexity and Ecology
1.1 Introduction
1.2 What Is Complexity?
1.2.1 Variety and Form
1.2.2 The Chicken and the Egg
1.3 What Makes Ecosystems Complex?
1.3.1 Measuring Diversity
1.3.2 The Origins of Complexity
1.4 Why Study Ecological Complexity?
1.5 The Complexity Paradigm
1.5.1 Scientific Paradigms
1.5.2 A New Ecology for a New Age?
References
Chapter 2: Seeing the Wood for the Trees: Emergent Order in Growth and Behaviour
2.1 Plant Growth and Form
2.1.1 Factors Influencing Growth
2.1.2 Branches and Leaves
2.1.3 Overall Plant Form
2.1.4 Self-Organisation Versus Constrained Growth
2.2 Animal Behaviour
2.2.1 Searching for Food
2.2.2 Territory
2.2.3 Social Networks
2.2.4 Animal Intelligence
2.3 Multiagent Systems
2.3.1 Turtle Geometry
2.3.2 From Turtles to Agents
2.3.3 The Boids and the Bees
References
Chapter 3: Complexity in Landscapes
3.1 The Eye of the Beholder
3.1.1 Geographic Information Systems
3.1.2 The Game of Life
3.1.3 Cellular Automata Models of Landscapes
3.2 Sampling and Scale
3.3 Complexity in Spatial Processes
3.4 Complexity in Spatial Patterns
3.4.1 Fractal Dimensions
3.4.2 Fractals in Nature
3.4.3 Measuring Landscape Complexity
3.5 Are Landscapes Connected?
3.5.1 Connectivity in a Grid
3.5.2 Why Is a Starfish Like an Atomic Bomb?
References
Chapter 4: Oh, What a Tangled Web … Complex Networks in Ecology
4.1 The Roots of Complexity Theory
4.2 The Network Model
4.2.1 Interactions and Connectivity
4.2.2 Networks
4.2.3 Networks Are Everywhere
4.2.4 The Connectivity Avalanche
4.2.5 Phase Changes and Criticality
4.2.6 The Order of Things
4.3 Self-Organisation
4.3.1 Emergent Properties
4.3.2 Modules and Motifs
4.3.3 The Shape of Complexity
4.4 Networks of Networks
References
Chapter 5: The Imbalance of Nature … Feedback and Stability in Ecosystems
5.1 Feedback
5.1.1 Negative Feedback Promotes Stability
5.1.2 Positive Feedback Promotes Self-Organization
5.2 The Big Get Bigger
5.3 Who Eats Whom?
5.3.1 Equilibrium and Stability
5.3.2 Transients and Attractors
5.3.3 Sensitivity to Initial Conditions
5.3.4 The Onset of Chaos
5.3.5 Fractals
5.4 Is There a Balance of Nature?
5.4.1 Succession
5.4.2 Ecosystems in Balance?
5.4.3 Does a Balance Really Exist?
References
Chapter 6: Populations in Landscapes
6.1 One Population or Many?
6.2 Spatial Distributions
6.3 Patches, Edges and Zones
6.3.1 Salt of the Earth
6.4 To See the World in a Grain of Pollen
6.5 Galloping Trees?
6.6 Phylogeography
References
Chapter 7: Living with the Neighbours: Competition and Stability in Communities
7.1 Invasions and Persistence
7.2 Disturbance and Competition
7.3 Ecological Communities
7.3.1 Do Ecological Communities Exist?
7.4 Networks of Interactions
7.4.1 Food Webs
7.4.2 Networks
7.4.3 The Paradox of Stability
7.4.4 Stability and Food Webs
7.4.5 Resilience
References
Chapter 8: Adaptation in Landscapes
8.1 Genes and Selection
8.1.1 Evolutionary Trade-Off
8.2 Genetics in Heterogeneous Landscapes
8.2.1 Adaptation on a Gradient
8.2.2 Fragmentation and Drift
8.2.3 Friends and Relations
8.3 Catastrophes, Criticality and Macroevolution
8.3.1 Mass Extinctions
8.3.2 Landscape Phases and the Origin of Species
References
Chapter 9: Virtual Worlds: The Role of Simulation in Ecology
9.1 Virtual Experiments
9.1.1 From Landscapes to Virtual Worlds
9.1.2 The Need for Simulation
9.1.3 A World Inside a Computer?
9.1.4 The Limits to Growth Model
9.1.5 Just So Stories?
9.2 What Is Artificial Life?
9.2.1 Tierra
9.2.2 Daisyworld
9.3 From Virtual to Real
9.3.1 Swarm
9.3.2 SmartForest
9.3.3 NetLogo
9.3.4 Computer Games
9.4 Virtual Reality
References
Chapter 10: Digital Ecology: New Technologies Are Revolutionizing Ecology
10.1 Information and Complexity
10.2 From Field Work to Ecotechnology
10.2.1 Traditional Ecology
10.2.2 New Sources of Data
10.2.3 Monitoring
10.3 Sharing Ecological Information
10.3.1 Data Warehouses and Repositories
10.3.2 From Informatics to e-Ecology
10.3.3 Many Hands Make Light Work
10.3.4 Quality not Quantity
10.3.5 Crowd Projects
10.4 Coping with Complex Ecological Information
10.4.1 Putting IT to Work
10.4.2 Applying Geographic Information
10.4.3 Complexity in Land-Use Planning
10.4.4 Serendipity
10.4.5 Modelling Species Distributions
10.5 Biodiversity Information
References
Chapter 11: The Global Picture: Limits to Growth Versus Growth Without Limits
11.1 Humans and the Global Environment
11.1.1 The Fall of Civilisations
11.1.2 Were the Elders Wise?
11.2 Global Climate Change
11.2.1 The Runaway Greenhouse
11.2.2 Climate Change and Denial
11.2.3 Ecological Effects of Global Warming
11.3 An Environmental Crisis
11.4 Globalisation
11.5 The Changing Nature of Conservation
11.6 The Future
11.6.1 The Challenge of Economics
11.6.2 Some Final Lessons
References
Index

Citation preview

Landscape Series

David G. Green Nicholas I. Klomp Glyn Rimmington Suzanne Sadedin

Complexity in Landscape Ecology Second Edition

Landscape Series Volume 22

Series editors Jiquan Chen, Department of Geography, Environment, and Spatial Sciences, Michigan State University, East Lansing, USA Janet Silbernagel, Department of Planning and Landscape Architecture,  University of Wisconsin-Madison, Madison, USA

Springer’s innovative Landscape Series is committed to publishing high quality manuscripts that approach the concept of landscape from a broad range of perspectives. Encouraging contributions that are scientifically-grounded and solutions-oriented, the series attracts outstanding research from the natural and social sciences, and from the humanities and the arts. It also provides a leading forum for publications from interdisciplinary and transdisciplinary teams. The Landscape Series particularly welcomes contributions around several globally significant areas for landscape research: Ecosystem processes linked to landscapes and regions Regional ecology (including bioregional theory & application) Coupled human-environment systems / interactions (CHES) Ecosystem services Global change science and adaptation strategies Volumes in the series can be authored or edited works, cohesively connected around these topics and tied to global initiatives. Ultimately, the Series aims to facilitate the application of landscape research to practice in a changing world, and to advance the contributions of landscape theory and research to the broader scholarly community. More information about this series at http://www.springer.com/series/6211

David G. Green • Nicholas I. Klomp   Glyn Rimmington • Suzanne Sadedin

Complexity in Landscape Ecology Second Edition

David G. Green Faculty of Information Technology Monash University Clayton, VIC, Australia Glyn Rimmington Wichita State University Wichita, KS, USA

Nicholas I. Klomp Central Queensland University Rockhampton, QLD, Australia Suzanne Sadedin Fremont, CA, USA

ISSN 1572-7742     ISSN 1875-1210 (electronic) Landscape Series ISBN 978-3-030-46772-2    ISBN 978-3-030-46773-9 (eBook) https://doi.org/10.1007/978-3-030-46773-9 © Springer Nature Switzerland AG 2006, 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface to the Second Edition

We published the first edition of this book in 2006. However, the project really goes back to the early 1990s, when we recognised the need for a book to introduce the principles of complexity to ecologists. The changes since then have been enormous. For example, when we published the first edition, complexity-based models in ecology were rare; now they are mainstream. Likewise, monitoring, data analytics and other new technologies are changing the way we do ecology. The increase in relevant publications has been phenomenal. There are some structural changes since the first edition. To make the subject matter clearer, each chapter now begins with a title page, which includes a subtitle, along with key words and a brief abstract of the main topics covered. We have reorganised each chapter to highlight important and emerging concepts. Perhaps the most obvious changes are the figures, which are essential for conveying concepts in complexity. We have added 30 figures, updated others, and rendered all of them in colour. We also plan to provide simulation models online of selected figures, so readers can experiment with complexity themselves. We have expanded every topic and added new ones not covered in the first edition. We have also added many recent references and contemporary examples. Despite these additions, the second edition is still only an introduction to the field. We do not claim to provide a comprehensive review of recent research. In the interests of keeping the book to a reasonable length, each topic and chapter is introductory in nature. For example, researchers have published work on literally hundreds of metrics to quantify patterns of complexity in landscapes. To cover them all would take an entire book in itself. Many of those metrics were developed for specialised purposes that are beyond the scope of this book. We can give only a few examples of metrics that we think best reflect the idea of complexity. The same limitation applies to almost every topic we cover. This second edition aims to be a useful introduction to complexity in ecology. With this in mind, we have tried to make the book accessible to a wide, general audience. Most notably, we have tried to keep mathematics and other technical matters in the background. This is challenging because research in the field has become increasingly sophisticated and technical. Another challenge we faced was to update v

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Preface to the Second Edition

the case studies. The story of deforestation in the Amazon, for instance, is an on-­ going tragedy that is continually documented. Similarly, alarming reports keep appearing about the impact of human activity on biodiversity. We are grateful to many people who contributed to the second edition. Dr François Petitjean provided satellite images. Benjamin O’Leary tracked down many references for the bibliography. Associate Professor Alan Dorin provided information about foraging bees. Associate Professor David Dowe provided useful suggestions about complexity measures. Dr Marc Cheong proofread the draft manuscript and provided many useful suggestions. Ira (Maria) Djaja wrote programs for several figures and the demonstrations that we will provide online. Dr Laura Trouille provided helpful information about the Zooniverse project. Clayton, VIC, Australia Rockhampton, QLD, Australia Wichita, KS, USA Fremont, CA, USA

David G. Green Nicholas I. Klomp Glyn Rimmington Suzanne Sadedin

Contents

1 Complexity and Ecology��������������������������������������������������������������������������    1 1.1 Introduction��������������������������������������������������������������������������������������    2 1.2 What Is Complexity?������������������������������������������������������������������������    4 1.2.1 Variety and Form������������������������������������������������������������������    4 1.2.2 The Chicken and the Egg������������������������������������������������������    5 1.3 What Makes Ecosystems Complex? ������������������������������������������������    7 1.3.1 Measuring Diversity��������������������������������������������������������������    7 1.3.2 The Origins of Complexity ��������������������������������������������������    9 1.4 Why Study Ecological Complexity?������������������������������������������������   12 1.5 The Complexity Paradigm����������������������������������������������������������������   13 1.5.1 Scientific Paradigms��������������������������������������������������������������   14 1.5.2 A New Ecology for a New Age?������������������������������������������   15 References��������������������������������������������������������������������������������������������������   18 2 Seeing the Wood for the Trees: Emergent Order in Growth and Behaviour��������������������������������������������������������������������������������������������������   21 2.1 Plant Growth and Form��������������������������������������������������������������������   22 2.1.1 Factors Influencing Growth��������������������������������������������������   22 2.1.2 Branches and Leaves������������������������������������������������������������   25 2.1.3 Overall Plant Form����������������������������������������������������������������   31 2.1.4 Self-Organisation Versus Constrained Growth ��������������������   33 2.2 Animal Behaviour ����������������������������������������������������������������������������   35 2.2.1 Searching for Food����������������������������������������������������������������   35 2.2.2 Territory��������������������������������������������������������������������������������   37 2.2.3 Social Networks��������������������������������������������������������������������   38 2.2.4 Animal Intelligence��������������������������������������������������������������   38 2.3 Multiagent Systems��������������������������������������������������������������������������   39 2.3.1 Turtle Geometry��������������������������������������������������������������������   39 2.3.2 From Turtles to Agents���������������������������������������������������������   40 2.3.3 The Boids and the Bees��������������������������������������������������������   42 References��������������������������������������������������������������������������������������������������   45 vii

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3 Complexity in Landscapes����������������������������������������������������������������������   47 3.1 The Eye of the Beholder ������������������������������������������������������������������   48 3.1.1 Geographic Information Systems ����������������������������������������   48 3.1.2 The Game of Life������������������������������������������������������������������   50 3.1.3 Cellular Automata Models of Landscapes����������������������������   52 3.2 Sampling and Scale��������������������������������������������������������������������������   54 3.3 Complexity in Spatial Processes������������������������������������������������������   56 3.4 Complexity in Spatial Patterns����������������������������������������������������������   58 3.4.1 Fractal Dimensions ��������������������������������������������������������������   59 3.4.2 Fractals in Nature������������������������������������������������������������������   60 3.4.3 Measuring Landscape Complexity ��������������������������������������   61 3.5 Are Landscapes Connected? ������������������������������������������������������������   64 3.5.1 Connectivity in a Grid����������������������������������������������������������   64 3.5.2 Why Is a Starfish Like an Atomic Bomb?����������������������������   66 References��������������������������������������������������������������������������������������������������   69 4 Oh, What a Tangled Web … Complex Networks in Ecology ��������������   73 4.1 The Roots of Complexity Theory ����������������������������������������������������   74 4.2 The Network Model��������������������������������������������������������������������������   76 4.2.1 Interactions and Connectivity ����������������������������������������������   76 4.2.2 Networks ������������������������������������������������������������������������������   76 4.2.3 Networks Are Everywhere����������������������������������������������������   78 4.2.4 The Connectivity Avalanche ������������������������������������������������   79 4.2.5 Phase Changes and Criticality����������������������������������������������   80 4.2.6 The Order of Things��������������������������������������������������������������   82 4.3 Self-Organisation������������������������������������������������������������������������������   84 4.3.1 Emergent Properties��������������������������������������������������������������   84 4.3.2 Modules and Motifs��������������������������������������������������������������   86 4.3.3 The Shape of Complexity ����������������������������������������������������   88 4.4 Networks of Networks����������������������������������������������������������������������   91 References��������������������������������������������������������������������������������������������������   95 5 The Imbalance of Nature … Feedback and Stability in Ecosystems��������������������������������������������������������������������������������������������   97 5.1 Feedback ������������������������������������������������������������������������������������������   99 5.1.1 Negative Feedback Promotes Stability ��������������������������������   99 5.1.2 Positive Feedback Promotes Self-Organization��������������������  100 5.2 The Big Get Bigger��������������������������������������������������������������������������  101 5.3 Who Eats Whom? ����������������������������������������������������������������������������  104 5.3.1 Equilibrium and Stability������������������������������������������������������  105 5.3.2 Transients and Attractors������������������������������������������������������  107 5.3.3 Sensitivity to Initial Conditions��������������������������������������������  108 5.3.4 The Onset of Chaos��������������������������������������������������������������  109 5.3.5 Fractals����������������������������������������������������������������������������������  112 5.4 Is There a Balance of Nature?����������������������������������������������������������  112 5.4.1 Succession����������������������������������������������������������������������������  113

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5.4.2 Ecosystems in Balance?��������������������������������������������������������  114 5.4.3 Does a Balance Really Exist? ����������������������������������������������  115 References��������������������������������������������������������������������������������������������������  116 6 Populations in Landscapes����������������������������������������������������������������������  119 6.1 One Population or Many? ����������������������������������������������������������������  120 6.2 Spatial Distributions��������������������������������������������������������������������������  122 6.3 Patches, Edges and Zones ����������������������������������������������������������������  124 6.3.1 Salt of the Earth��������������������������������������������������������������������  125 6.4 To See the World in a Grain of Pollen����������������������������������������������  126 6.5 Galloping Trees? ������������������������������������������������������������������������������  129 6.6 Phylogeography��������������������������������������������������������������������������������  131 References��������������������������������������������������������������������������������������������������  133 7 Living with the Neighbours: Competition and Stability in Communities��������������������������������������������������������������������������������������������  137 7.1 Invasions and Persistence������������������������������������������������������������������  138 7.2 Disturbance and Competition ����������������������������������������������������������  140 7.3 Ecological Communities������������������������������������������������������������������  142 7.3.1 Do Ecological Communities Exist?��������������������������������������  144 7.4 Networks of Interactions������������������������������������������������������������������  145 7.4.1 Food Webs����������������������������������������������������������������������������  145 7.4.2 Networks ������������������������������������������������������������������������������  147 7.4.3 The Paradox of Stability ������������������������������������������������������  148 7.4.4 Stability and Food Webs ������������������������������������������������������  149 7.4.5 Resilience������������������������������������������������������������������������������  152 References��������������������������������������������������������������������������������������������������  153 8 Adaptation in Landscapes����������������������������������������������������������������������  157 8.1 Genes and Selection��������������������������������������������������������������������������  158 8.1.1 Evolutionary Trade-Off��������������������������������������������������������  160 8.2 Genetics in Heterogeneous Landscapes��������������������������������������������  161 8.2.1 Adaptation on a Gradient������������������������������������������������������  163 8.2.2 Fragmentation and Drift��������������������������������������������������������  164 8.2.3 Friends and Relations������������������������������������������������������������  167 8.3 Catastrophes, Criticality and Macroevolution����������������������������������  169 8.3.1 Mass Extinctions������������������������������������������������������������������  169 8.3.2 Landscape Phases and the Origin of Species������������������������  171 References��������������������������������������������������������������������������������������������������  174 9 Virtual Worlds: The Role of Simulation in Ecology ����������������������������  177 9.1 Virtual Experiments��������������������������������������������������������������������������  179 9.1.1 From Landscapes to Virtual Worlds��������������������������������������  180 9.1.2 The Need for Simulation������������������������������������������������������  181 9.1.3 A World Inside a Computer?������������������������������������������������  182 9.1.4 The Limits to Growth Model������������������������������������������������  183 9.1.5 Just So Stories? ��������������������������������������������������������������������  184

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9.2 What Is Artificial Life? ������������������������������������������������������������������  185 9.2.1 Tierra��������������������������������������������������������������������������������  187 9.2.2 Daisyworld����������������������������������������������������������������������  187 9.3 From Virtual to Real�����������������������������������������������������������������������  190 9.3.1 Swarm������������������������������������������������������������������������������  190 9.3.2 SmartForest����������������������������������������������������������������������  191 9.3.3 NetLogo ��������������������������������������������������������������������������  191 9.3.4 Computer Games ������������������������������������������������������������  192 9.4 Virtual Reality��������������������������������������������������������������������������������  192 References��������������������������������������������������������������������������������������������������  194 10 Digital Ecology: New Technologies Are Revolutionizing Ecology ������  197 10.1 Information and Complexity����������������������������������������������������������  198 10.2 From Field Work to Ecotechnology������������������������������������������������  199 10.2.1 Traditional Ecology����������������������������������������������������������  199 10.2.2 New Sources of Data��������������������������������������������������������  201 10.2.3 Monitoring ����������������������������������������������������������������������  202 10.3 Sharing Ecological Information������������������������������������������������������  203 10.3.1 Data Warehouses and Repositories����������������������������������  203 10.3.2 From Informatics to e-Ecology����������������������������������������  204 10.3.3 Many Hands Make Light Work����������������������������������������  205 10.3.4 Quality not Quantity��������������������������������������������������������  207 10.3.5 Crowd Projects����������������������������������������������������������������  208 10.4 Coping with Complex Ecological Information������������������������������  210 10.4.1 Putting IT to Work ����������������������������������������������������������  210 10.4.2 Applying Geographic Information����������������������������������  213 10.4.3 Complexity in Land-Use Planning����������������������������������  215 10.4.4 Serendipity ����������������������������������������������������������������������  216 10.4.5 Modelling Species Distributions��������������������������������������  217 10.5 Biodiversity Information����������������������������������������������������������������  219 References��������������������������������������������������������������������������������������������������  221 11 The Global Picture: Limits to Growth Versus Growth Without Limits ����������������������������������������������������������������������������������������  225 11.1 Humans and the Global Environment��������������������������������������������  226 11.1.1 The Fall of Civilisations��������������������������������������������������  227 11.1.2 Were the Elders Wise? ����������������������������������������������������  229 11.2 Global Climate Change������������������������������������������������������������������  231 11.2.1 The Runaway Greenhouse ����������������������������������������������  231 11.2.2 Climate Change and Denial ��������������������������������������������  232 11.2.3 Ecological Effects of Global Warming����������������������������  233

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11.3 An Environmental Crisis����������������������������������������������������������������  236 11.4 Globalisation ����������������������������������������������������������������������������������  239 11.5 The Changing Nature of Conservation ������������������������������������������  242 11.6 The Future��������������������������������������������������������������������������������������  243 11.6.1 The Challenge of Economics ������������������������������������������  243 11.6.2 Some Final Lessons ��������������������������������������������������������  246 References��������������������������������������������������������������������������������������������������  248 Index������������������������������������������������������������������������������������������������������������������  253

Chapter 1

Complexity and Ecology

Abstract  Complexity is the richness and variety often seen in large systems. Species diversity is often used to represent complexity in ecosystems, but true complexity arises from the enormous number of ways to order combinations of objects. To manage the natural world successfully, we need to understand ecological complexity. Keywords  Complexity · Diversity · Information theory · Scientific paradigms

Many severe environmental problems have complex causes. This photo shows a section of the Murray River near Albury in New South Wales, Australia, declared the country’s most endangered heritage site in 2002

© Springer Nature Switzerland AG 2020 D. G. Green et al., Complexity in Landscape Ecology, Landscape Series 22, https://doi.org/10.1007/978-3-030-46773-9_1

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1.1  Introduction Covering an area of more than six million square kilometres, the Amazon Basin dominates northern Brazil and forms a large part of the South American continent. The richness of its biodiversity, and the hostility of its natural environment, mean that even today we can form no clear picture of the Amazon rainforest ecology. Yet even fragmentary glimpses reveal that the Amazon forms an extravagance of nature beyond the wildest imaginings of taxonomists. When biologist T. L. Erwin examined a single species of Amazonian tree, he found 163 unique species of beetle living in its canopy alone [11]. Comparing sites seventy kilometres apart, Erwin found only a 1% overlap in the beetle species present [12]. Brazil has more species of flowering plants and amphibians than any other country, and ranks in the top four countries on earth for mammals, birds, butterflies and reptiles [13]. Up until the mid-twentieth century, the Amazon rainforest remained shrouded in mystery, isolated by its sheer size and inaccessibility. But a rapidly growing technological society, with a huge appetite for raw materials, could not leave such a massive resource untouched forever. The forest seemed to promise fertile croplands, and the trees themselves became valuable exports under the chainsaw. In 1960, the Brasilia-Bellem road opened, making the Amazon accessible for commercial exploitation. The road sliced the rainforest open, bringing swarms of loggers and farmers eager to claim a share of the wealth. During the next two decades, the human population of the region swelled to more than 17 million. Soybean farms and cattle ranches proliferated. In the 1980s, the rate of deforestation across Amazonia exceeded 22,000 square kilometres per year (or about 4  ha per minute). During the late 1990s, the Brazilian government began efforts to reduce the rate of land clearing. However, their attempts were frustrated by the sheer scale of the problem, with illegal operations being responsible for up to 80% of logging in Amazonia. By 1998, over half a million square kilometres of Brazilian rainforest had been cleared, and destruction continued to accelerate. During 2004 alone, over 26,000 square kilometres were destroyed, an area roughly the size of Belgium [14]. The assumption that clearing lush rainforest would yield prime agricultural land proved to be a tragic mistake. Beneath a thin surface layer of rich soil, farmers mostly find infertile wasteland. After a season or two of good crops, the soil is used up, erosion is rife and rainfall declines. Farmers are forced to douse the land with chemical fertilizers, and leave fields fallow for many years, in order to maintain any yield at all. For poverty-stricken smallholders, often faced with an immediate need for cash crops to service World Bank loans, such management techniques are unfeasible. Consequently, millions of displaced farmers move ever deeper into the rainforest, clearing more and more forest to eke out a few extra years’ worth of crops. Ironically, the reason for the land’s failure lies in the extraordinary efficiency of the rainforest itself. In the dense, lush ecosystem of the forest, almost nothing is wasted. Nutrients that reach the ground decompose quickly and are recycled into fast-growing plants. Little is left in the soil: when the trees vanish, so do the raw

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materials of life. Even the rain itself depends on the trees: approximately 50% of the Amazon’s rainfall is recycled through the forest [13]. From an ecological viewpoint, the exploitation of the Amazon has been an unmitigated disaster. Logging creates networks of roads, encouraging further migration of farmers into untouched forest. Their slash-and-burn methods break up the forest, gradually turning it into isolated fragments. This fragmentation leads to sharp increases in fires, in hunting, and in soil erosion, as well as invasions of grasses, vines and exotic species. All of these changes spell trouble for native plants and animals. The impact of clearing the Amazon rainforest is not confined to loss of biodiversity; it also involves loss of carbon into the atmosphere. By early in the twenty-first century, it was clear that the loss was playing a significant role in global greenhouse gas emissions. This raised concerns about the effects that deforestation in Amazonia, and elsewhere, was having on global climate change [15, 22, 29]. The story of development in the Amazon Basin is a dramatic example of how simple assumptions about ecological systems can lead to disastrous mistakes in land management. Almost always, problems arise because the complexity of landscapes and ecosystems defeats our efforts to understand them as simple systems of cause and effect. In the case of the Amazon, building a road into the region initiated a cascade of mutually reinforcing processes. The underlying error in this ongoing catastrophe is “cause and effect” thinking: assuming that the forest ecosystem is a direct effect of suitable climate and soil conditions, rather than a complex, dynamic process in itself. We can see this failure to grasp ecological complexity in early attempts to understand the role of landscapes. When people first began using computer models to study ecosystems, spatial interactions were largely ignored. Local interactions between individuals were assumed to be minor effects that would average out over time and space. Understanding the influence of landscape was therefore seen as easy. To account for a hillside, for instance, all you needed to know was what happened at the top of the hill, at the mid-slope and in the swale. The assumption was that the differences in environmental conditions from place to place were the only factors that influenced the outcome, so accounting for them would tell you all you needed to know about the role of landscapes in ecology. Unfortunately, the assumption that local effects will average out over time and space is not only incorrect; it is in many cases drastically misleading. Interactions do matter, and local interactions can blow up to have large-scale effects. In ecological systems, many of these interactions are not simple, one-way cause and effect relationships, but complex feedback relationships (see Chap. 5). Only by explicitly studying these interactions can we explain many of the patterns and processes that occur in landscapes. The landscape, the Earth’s surface, is the stage on which ecology is played out. It comprises the landforms, the soils, the water and all the other physical features that influence the organisms that make up an ecosystem. And just as the landscape constrains and influences the ecology of a region, so too the ecosystems interact with and affect the landscape.

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This book is about the profound but often subtle ways in which interactions affect both ecosystems and landscapes. Our aim is to help readers to understand the nature of complexity in the context of landscape ecology. In the chapters that follow, we will explain what complexity is and what recent research has been learnt about it. We will also look at some of the many ways in which complexity turns up in ecosystems and in landscapes. As we shall see in Chap. 3, the landscape itself can be complex. In subsequent chapters we will look both at the many processes that make ecosystems complex, and at the ways in which the interplay between landscapes and ecosystems creates complexity of its own. Finally, we will explore the relationship between landscape ecology, complexity and the information revolution. Besides describing some of the key ideas and the insights that flow from them, we will also introduce some of the techniques that are emerging to deal with ecological complexity in practice.

1.2  What Is Complexity? Like life itself, complexity is a phenomenon that is well known, but difficult to define. A general definition is difficult because the term complexity appears in different guises in different fields. In computer science, for instance, it usually refers to the time required to compute a solution to a problem. In mathematics, it is usually associated with chaotic and other nonlinear dynamics.

1.2.1  Variety and Form Here we will take “complexity” to mean the richness and variety of form and behaviour that is often seen in large systems [3, 4]. Complexity is not the same as size. For example, a herd of zebras feeding on a grassy plain do not behave in the same way as billiard balls on a table. If you strike one of the billiard balls, it will roll around hitting the other balls. Eventually its motion and energy will dissipate through repeated collisions between the balls. This is simple behaviour. Although the balls interact, their energy soon averages out and they stop moving. On the other hand, if one zebra starts running, then the entire herd is likely to panic, creating a stampede. What is more, the stampede is not random. The running zebras avoid colliding with each other, but remain packed close together and head in the same direction. This is complex behaviour. The stampede emerges out of interactions between the zebras. Later, we will see that networks of interactions characterize most instances of complexity (see Chap. 4). The property that is most closely associated with complexity is emergence. This idea is captured by the popular saying: the whole is greater than the sum of its parts. Emergence takes many forms. A forest emerges from the interactions of millions of individual plants, animals and microbes with each other and with the landscape. A

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forest fire emerges from the spread of ignitions from one plant to another. A flock of birds emerges from the individual behaviour of many individual birds interacting with one another. The organisation of an ant colony emerges from the joint behaviour of many individual ants interacting with each other and with the colony environment. To understand complexity in ecosystems, we need to learn how large-scale properties like these emerge from interactions between individuals. To summarize the above discussion, we can draw some general distinctions between different kinds of systems. Simple systems are predictable, within minimal interactions between individual components, which behave as separate individuals. Complicated systems are larger and have some interactions, but are reducible to simple component systems, and therefore predictable. Complex systems, however, are often unpredictable, rich in interactions, and large-scale properties of behaviour often emerges from those interactions. The most common way of coping with complexity, in both human and natural systems, is to avoid it by breaking a system into distinct parts (modules) and by reducing the interactions between them. We do this in human organization (e.g. company, division, branch, department), in technology (e.g. modular construction). It is also common in living systems, as we shall see later for growth (e.g. branches) and behaviour (e.g. territories).

1.2.2  The Chicken and the Egg Which came first, the chicken or the egg? This famous conundrum exposes a gap in our intuition.1 It is natural to assume that each cause has a simple effect, and vice versa. So an egg “causes” the chicken that hatches from it. Conversely, the chicken “causes” the egg that it lays. The chicken or egg question invites an answer in terms of simple causality. In reality, however, both the egg and the chicken are manifestations of a complex process.2 Ilya Prigogine expressed this mental transition from static causal models to dynamic systems models concisely in the title of his book, From Being to Becoming [35]. Similarly, Barry Richmond talked about the need for structured thinking over simple causal thinking [37]. Many situations in landscape ecology are like the chicken and egg problem. Look again at the story that began this chapter. To understand what happened in the Amazon rainforests, it is necessary to realise that a rainforest is not a fixed object, but an on-going process. Traditional cause and effect classifications characterize a rainforest as a forest that grows in areas of high rainfall and soil that is rich in 1  Philosophers have long been aware of problems with simple causality. For example, they distinguished between “proximate” and “final” causes. What causes a wild fire? A lightning strike may be the proximate cause, but hot weather and lack of rain also contribute as more final causes. These climatic conditions may themselves be caused by global warming caused by carbon dioxide emissions caused by oil and coal burning caused by humans. And so on. 2  Evolutionary biologists would say that the egg came first. Fishes, amphibians and reptiles all laid eggs many millions of years before birds, and chickens, evolved.

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nutrients. Based on this model, it seems reasonable to conclude that if you cut down a rainforest for timber, then the high rainfall and rich soil will cause rainforest to grow back again in a few years. Not so! The truth is that the rainforest is a complex system. The species richness, the lush soils, and the high rainfall are all mutually dependent. They are each the product of a long feedback process. Higher-order or systems thinking is needed to understand the rainforest, which contains a vast network of feedback loops, flows and accumulations. The tendency to think in terms of simple cause and effect leads to many problems in conservation. People see a local problem and seek a simple, local solution. They are often unaware of the spatial interactions that may be involved and do not realise what effects their local actions may have elsewhere. The Murray River is Australia’s largest river system. Its catchment, which and encompasses some of the country’s most productive agricultural land, covers 14% of the continent receiving water from 41 tributaries across four states. The health of this river is of critical importance to the country’s economy. However, in 2002, the National Trust of Australia was forced to declare the entire Murray River to be the country’s most endangered heritage site: “Today no water flows into the sea from the Murray River. This once magnificent river now regularly fails to reach the sea.” In 2002 the National Trust of Australia gave the following statement: The significant threats to the health of the Murray remain largely unaltered. However, the community engagement process, along with the work of community organisations have raised the profile of the Murray issue considerably. National action is required, and the National Trust urges the Council of Australian Governments to commit to a national approach to ensure that healthy flows are restored to the Murray, and that the community is fully informed and engaged in the conservation of our most precious resource – water [27].

Many factors motivated the Heritage Trust’s decision. These issues included diversion of river water for irrigation causing floodplains and wetlands to dry up, dams causing permanent inundation of other floodplains, increasing pollution and sedimentation within the river itself, declining rainfall, salinization, desertification, and the spread of introduced species such as European carp throughout the system. The ecological consequences of these stresses were already severe. Populations of native fish, invertebrates and reptiles went into decline. Many species of waterbird became locally extinct [20]. The problem of the Murray River is an example of the Tragedy of the Commons [17]. No single person or organisation was responsible for the dying Murray. It was a complex problem that emerged as the accumulated result of local actions and interactions all along the river. Essentially, the crisis arose because people tend to think of problems in terms of simple cause and effect. Farmers, councils, and government agencies all acted to solve local environmental challenges without understanding the wider impacts that their actions then have on the river system as a whole. A drought that lasted from the mid-1990s until 2009, highlighted the problem and the need for action. By 2004, Government policies had shifted from exploiting water resources to “restoration of environmental assets” [23].

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1.3  What Makes Ecosystems Complex? What does it mean to say that an ecosystem is complex? What does it mean to say that one ecosystem is more complex than another? Can we measure the degree of complexity we find in nature? When talking about complexity in the living world, the discussion usually begins with diversity. The reasoning is that the more species present, the more complex an ecosystem must be. Early in the twentieth century, ecologists grew increasingly interested in the notion of ‘diversity’ [32]. Some ecosystems, especially rainforests and coral reefs, seemed to be teeming with a great variety of species. Others, such as the Arctic wastes, seemed to have very few inhabitants.

1.3.1  Measuring Diversity As ecological theories became quantitative during the twentieth century, ecologists needed ways to make this idea of diversity more precise. The first and most obvious measure is the number of species, which we can think of as measuring biological richness. Taken on a large scale, the number of species continues to be the most widely used indicator of biodiversity. Scientists use the term “diversity” in several ways — genetic variability, population and ecosystem diversity — but even if we take it to mean the number of plant and animal species, the global picture is enormous. Taxonomists have described about 300 thousand plant and 1.9 million animal species [33]. Discoveries of new species lead to around 13,000 new descriptions per year [20]. The total number of species is estimated to be somewhere between five million and 30 million [25]. Most of the vertebrates and flowering plants are described, while invertebrates, bacteria and other microscopic creatures are largely undescribed [41, 45]. At the current pace of taxonomic research, it would take at least another 300 years merely to document the specimens already in collections [16]. In landscape ecology, as in every area of science, there is a close relationship between theory and data. The theories that scientists develop are limited by the data available to them. Conversely, the data that scientists collect are determined by the theories that they wish to test and by the experiments, or observations that they carry out. We can look at the concept of diversity to see how this interaction between theory and experiment happens. By expressing such a fundamental idea in quantitative terms, ecologists had a tool with which they could state ecological hypotheses in precise fashion. This ability in turn made it possible to devise precise tests of those hypotheses. It also made clear what kinds of data they needed to collect. At the ecosystem level, the idea of species richness does not account for everything: some communities have a more varied mix of species than others in which one or two species are super abundant. Ecologists therefore found it necessary to

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introduce other measures to express different interpretations of diversity, such as uneven population sizes and habitat variety. Information theory gives us one approach to measuring diversity [32]. Imagine you are an explorer trying to describe an ecosystem in a message. The more complex the ecosystem is, the longer your report will need to be. A basic report might just list the species you find, so the number of species, the ecosystem’s richness, provides a very crude indicator of the ecosystem’s complexity. The more species, the longer the list. Adopting this idea of a description, one way of measuring complexity is to use the length of a message that describes a system. Solomonoff, Kolmogorov and Chaitin all independently suggested this idea [6, 21, 40]. They defined the complexity of a string X to be the length of the shortest string Y that produces X when used as an input to a fixed program. This model implicitly treats complexity in nature in terms of computation. That is, it equates the complexity of a system with the input we give to a computer program, which then outputs a complete description of the system. A drawback of the above definition is that it presupposes we use a fixed program for all cases. However, the program itself may be simple, or highly complex. To deal with this problem, Papentin [30] proposed that complexity really comprises two component strings: primary order (or ordered complexity), which is the set of rules describing pattern in a system, and secondary order, which describes the entropy, or random components. Another way of viewing this is to say that we can describe any complex pattern as a program (the pattern) and its input (the random elements). For instance, the alternating string bwbwbwbw … could be interpreted as a repeating pattern P…, with input P = bw. Another alternative is to include the program as part of the message. Wallace (1990) proposed the idea of minimal message length (MML), which combined the program plus its input string [43, 44]. The problem underlying all of these approaches is the requirement of finding the minimal description. How do you know that a description is the shortest one possible? Ecologists have generally avoided this difficulty by focussing on one feature of an ecosystem: its diversity, which is the mix of species that goes to make up an ecosystem. However, even such a simple idea raises difficulties. Even just listing species conveys the impression that there are equal numbers of each species. In most ecosystems, some species are abundant while others are rare. To indicate the variety, the message would need to include the numbers of each species. Ecologists usually use Shannon’s information measure to represent variety.3 This measure, also known as the Shannon-Weaver index of diversity, takes values that range from zero if there is just one species (all others very rare) to log(N) if there are N species, all equally common. There are problems with using variety as a measure of diversity. For instance, as measured above, diversity will vary from place to place, from one habitat to another,

 Shannon’s measure is =∑ipi log pi, where pi is proportion of species i.

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within an ecosystem. To deal with this, Whittaker introduced three different kinds of diversity: the average diversity within habitats (alpha diversity), the variation between different habitats (beta diversity) and their product is the total diversity in a landscape (gamma diversity) [32]. As a research tool, diversity indices can be useful in revealing biological relationships. They have also helped ecologists to study connections between a community’s diversity, its environment, and its internal composition and processes. To take a classic example, separate studies in Australia and in Chile showed that the diversities of bird species showed significant correlation to foliage height diversities [7, 36]. This relationship implied that birds chose habitats horizontally in different vertical layers of the forest, as well as in different localities. Having been formally defined, diversity has come to be seen as a fundamental property of communities that can be studied in its own right. Like mass or energy in physics, it is an abstract concept that can be discussed independently of particular communities, allowing us to uncover general rules of ecology. Thus, for example, we know that diversity usually increases during succession, that communities in harsh environments (for example, mountain tops) are usually less diverse than comparable ones in more benign surroundings. Diversity studies have also provided evidence bearing on controversial areas of ecological theory, such as island biogeography and the relationship between stability and complexity in ecosystems. People have often used diversity as an indicator of ecological complexity. The reasoning is that high species diversity implies a greater richness of interactions between species, which in turn leads to more complex behaviour. Likewise, high habitat diversity implies a rich and complex range of interactions between sites. The story of ecology’s attempts to deal with diversity therefore reflects, in part, its attempts to come to grips with complexity. As we shall see in Chap. 7, the diversity-­ stability issue forced ecologists to re-examine many traditional assumptions. It also forced scientists to focus on the nature of complexity by looking in particular at the underlying structures of food webs.

1.3.2  The Origins of Complexity The notion of diversity has long been a source of contention in ecology. Although the number of species is impressive, it is not enough to make the living world complex. We need to be careful to distinguish complexity from mere complication. Adding more species to an ecosystem makes it more complicated, but not necessarily more complex, that is, richer in its overall behaviour. Real complexity stems from the enormous variety of ways in which species combine and interact. Interactions between pairs of species can take many forms, such as predation, parasitism and competition. When there are many species present, the number of interacting pairs can be very large. For instance, suppose that 100 species inhabit a region; then there are 4950 possible pairs of interacting species.

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Species do not interact in pairs alone: the effect of one species on another may be altered by the presence of a third. When we look at possible combinations of multiple species, the possibilities blow out to astronomical proportions (Fig.  1.1a). There are more than 1.7 × 1013 ways in which communities of 10 species could form from a pool of 100 species.4 This number is about the same as the distance from the Earth to the sun in centimetres. For communities of 50 species at a time, this number rises to over 10120 combinations: a number greater than the number of atoms in the universe! Within any given ecosystem, this complexity can increase by several orders of magnitude if we consider the possible ways of organizing them (Fig. 1.1b) and the interactions between organisms (biotic components) and their environment (abiotic components). These interactions determine the way an ecosystem functions. Every species is unique, and its behaviour forms a unique interaction with its environment. Astronomical as some of the above numbers may seem, the reality is even more extreme. The very concept of interactions occurring between species is itself a simplification. Each species consists of unique individuals, and ecological interactions occur not between whole species, but between individuals belonging to each species. These interactions are governed partly by the spatial arrangement of individuals, making the landscape itself a major source of complexity. The earth’s land surface totals about 150 million square kilometres, but resources are patchy, both in space and time, so organisms are not spread evenly across the landscape (nor in the oceans). However, understanding landscape processes can make conservation easier. The identification of ‘hotspots’, or areas of unusual

Fig. 1.1  The origins of complexity. (a) The increase (orders of magnitude) in the number of ways of drawing combinations of 10 species as the size of the pool from which they are drawn increases. The vertical axis shows the number of zeroes in the number, so the label 5 indicates the number 100,000 and 50 indicates the number 1 followed by 50 zeroes. (b) The number of ways we can arrange a set of objects in order, and the number of ways to form networks from a given set of nodes  The number 1 × 1013 (ten trillion) is written as 1 followed by 13 zeroes.

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ecological importance, allows a more concentrated effort to achieve greater conservation results [34]. At least 35 such hotspots are known to exist around the world [9, 26]. They contain the highest concentrations of endemic species (species found nowhere else) and are also experiencing significant loss of habitat. Packed within just 2.3% of the land surface area on earth, these hotspots contain about 50% of all threatened vascular plant species and 43% of the threatened species of amphibians, birds, reptiles and mammals. Given the richness of ecological interactions, we cannot hope for a simple and general understanding of the behaviour of ecosystems, such as we have of (say) the behaviour of gases. However, there is another way to approach the problem. We can view ecological interactions as a network of connections, recognising that although complex systems are diverse, they also tend to share certain internal structures and processes that lead to consistent behaviours. We mentioned earlier that the landscape itself is a source of complexity in ecosystems. Spatial features, such as soil types, temperature and humidity gradients, topography, wind and water currents, connect sites in a landscape in many different ways. These connections form diverse patterns, which influence the distributions of plants and animals. Spatial processes, such as seed dispersal and migration, provide mechanisms by which different sites in a landscape interact ecologically (see Chaps. 3, 6, 7 and 8). In later chapters, we will explore the major factors that govern these spatial processes. If patterns in space are important, then so too are patterns in time. Apparently random events, such as drought, flood and fire, each cause massive and unpredictable mortality, but also provide opportunities. Both the frequency and magnitude of environmental variations can strongly influence ecological structures. It is important to remember that the ecosystems can affect landscape characteristics, such as soil stability or rainfall and temperature. Preserving Amazonian forests prevents soil erosion, whereas large scale deforestation can change rainfall and temperature patterns. Understanding the interaction between feedback processes and temporal and spatial variation within ecosystems has helped to resolve some key questions that cannot easily be approached empirically. For example, ecologists noticed that ecosystems with very large numbers of species, such as the Amazon rainforest, tended to be very stable, with low extinction rates and little variation in population sizes over time. In contrast, they saw ecosystems with only a few species, such as Arctic tundra, experiencing much larger fluctuations in population sizes and frequent extinctions. This led ecologists to believe that complexity in ecosystems enhances their stability, making them more robust to disturbances. However, during the 1970s a series of models showed that exactly the opposite was likely to be true [24]. Simulation models in which sets of interacting populations are formed randomly predicted that systems with greater numbers of species are more likely to collapse, simply because there is a greater chance of forming positive feedback loops [42]. This finding suggests that the arrow of causality points in the opposite direction. Rather than complexity creating stability, stability permits complexity. It is no coincidence, then, that stable and complex ecosystems occur in

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areas of minimal seasonal variation near the equator, while instability and simplicity are associated with the extreme seasonality of the polar regions. Similarly, one of the most important results to spring from ecological studies is that external environmental factors alone do not fully explain the distributions of organisms. For instance, competition between species often truncates distributions along an environmental gradient [31]. An epidemic may reduce the size of a population to a point where inbreeding decreases survival and reproduction [38]. All of these results imply that ecosystems are not controlled in a simple fashion by either external or internal factors, but by the interaction of many biotic and abiotic factors [5]. As the above discussion implies, another possible measure of complexity in an ecosystem is its connectivity, which is the richness of interactions between the populations present. Later we shall see how this applies to landscapes (Chap. 3) and to food webs (Chap. 7). We will also look at other measures that reflect structural properties of complex networks (Chap. 4).

1.4  Why Study Ecological Complexity? In September 1993, a group of eight men and women emerged into the open air, after living for two years in a closed, artificial habitat. Their home for all that time had been an interconnected series of domed greenhouses located in Arizona’s Sonoran Desert, about 30 miles north of Tucson. Called Biosphere 2, the $150 million experiment had two main goals. Firstly, NASA, the USA’s National Aeronautics and Space Administration, wanted to test the viability of colonies in space. Secondly, isolated ecosystems provided a chance to learn more about issues for the long-term maintenance of Biosphere 1, the Earth itself. The two-year experiment provided many valuable lessons. Chief among these lessons was the realization that managing a closed ecosystem was a far more complex and delicate problem than anyone had imagined: [1, 2, 8]. Artificial biospheres of the scale and complexity of Biosphere 2 can only work with coordinated rigorous design at each level of ecology: biospheres, biomes, bioregions, ecosystems, communities, patches, phases, physical-chemical functions, guilds, populations, organisms, and cells (both eukaryotic and prokaryotic). The coral reef proved remarkably responsive to changes in atmospheric composition, light levels and climatic conditions, requiring skilled and frequent management intervention.

Certain differences from the Earth’s biosphere caused problems in the artificial habitat. For instance, the ratio of carbon held in living biomass to carbon held in the atmosphere is about 1:1 on Earth as a whole, but in Biosphere 2 it was about 100:1. Likewise, the residence time for carbon dioxide in the atmosphere was about 3–4  days in Biosphere 2, hundreds of times less than for the Earth’s biosphere, which has a vast buffer of air. This meant that the carbon balance was highly sensitive to changes in any part of the ecology.

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More subtle issues arose from the self-organisation that occurred within the system: Examination of the ecosystem within Biosphere 2, after 26 months... [showed] features of self-organization....the self-organizing system appeared to be reinforcing the species that collect more energy … [28]

Unexpected difficulties emerged, and the inhabitants had to intervene to prevent runaway problems from taking hold. For instance, oxygen levels dropped dangerously low, requiring repeated input from the outside world [39]. Researchers suspected that rich soils chosen to promote plant growth were causing uncontrolled growth of soil microbes, whose respiration consumes oxygen. But that created a further puzzle: if soil microbes were responsible, why did atmospheric carbon dioxide levels remain constant? The missing factor was eventually identified as unsealed structural concrete absorbing carbon dioxide. If a stable environment proved unattainable even in the small, simple and carefully planned world of Biosphere 2, then how much greater is the problem of managing the Earth’s ecosystems? We need to study ecological complexity. The solution of many crucial environmental problems hinges on our understanding of complexity. With modern global trade systems, the human world is now so interconnected that potentially everything can affect everything else. The echoes of human activity are felt everywhere, even in regions usually regarded as wilderness. Furthermore, we cannot deal with ecosystems in isolation. To manage ecosystems and species in one area, we need to understand their place in the larger environmental context on a global scale. Conservation in many countries is really a question of economics, and chains of events link processes that cross national boundaries. Deforestation in the Amazon was driven in part by demand for cheap hamburgers in the United States. In 1973, an oil embargo by OPEC nations raised fuel prices worldwide, causing greenhouse gas emissions to drop briefly. In 1991, war in Kuwait caused massive increases in greenhouse emissions when oil wells were set on fire. The problem is urgent. Environmental management has become a race between conservation and exploitation. On the one hand, we need to develop an understanding of ecosystems and to set in place the necessary agreements, infrastructures and practices. On the other hand, environmental exploitation and degradation (especially land clearance and carbon dioxide emissions) are now reaching into environments everywhere. To understand ecology, we need to understand complexity. In the next chapter, we shall look at the sources of complexity in ecology and at some of the ways in which ecologists have sought to understand it.

1.5  The Complexity Paradigm The complexity revolution is aptly named … for it changes the way that scientists approach their subject matter. It has given us new tools and concepts, as well as fresh explanations for age-old puzzles [10].

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1.5.1  Scientific Paradigms Scientific research does not happen in a vacuum. The way in which people look at the world influence the questions that scientists ask, the experiments they perform, the data they collect, the way they interpret their results, and ultimately, the kinds of theories they develop. The prevalence of such influences led Thomas Kuhn and others to argue that science is dominated by paradigms (Kuhn 1962; Popper 1968). Paradigms are bodies of theory and methods that together define a particular way of looking at the world. The new ideas and methods that have been developed to deal with complexity in landscape ecology amount to a new paradigm. In later chapters, we will describe many ideas and methods associated with this paradigm in the course of discussing the effects of complexity on landscapes and ecosystems. This is not to imply that current paradigms need to be replaced or forgotten, but rather that they need to be complemented by different approaches to deal with different issues. In particular, complexity theory recognises that: • interactions lead to richness and variety in ecosystems and landscapes; • ‘global’ phenomena (e.g. across an entire ecosystem) often emerge out of local interactions and processes; • many processes are based on qualitative effects that need to be captured by symbolic rules, rather than by numeric formulae; • no single model of a complex system can be all-embracing, so different models may be needed for different purposes, whether it is to explain, to predict, or to control. However, we have not yet sketched out how the complexity paradigm works in practice. By this stage, for instance, any field ecologist will be asking how they can come to grips with ecological complexity. What questions do you ask? What experiments can you do? What data should you collect? And once you get those data, how do you go about interpreting them? Some of the differences between the complex systems paradigm and traditional paradigms are evident if we compare simulation with deterministic approaches. Just like a mathematical equation, simulation is a tool that scientists can use to represent patterns and processes in nature. The difference is that equations are solved, whereas simulations are played out. Because complex systems are often inherently unpredictable, we examine scenarios instead of making forecasts. Instead of solving equations, we perform sensitivity analyses. Instead of using analytic methods to find optimal solutions, we use genetic algorithms and other adaptive methods [18]. Instead of plotting graphs, we use visualisation or virtual reality. Finally, because relationships abound in complex systems, simple relationships may not be evident, so exploratory methods, such as data mining, are increasingly important in research. Many common kinds of questions associated with complexity involve patterns of connections. In Chap. 7, for instance, we will see that food web studies go beyond

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simplified or aggregated classifications, such as trophic levels,5 or predator-prey interactions, and look in detail at the entire pattern of interactions within an ecosystem. Likewise, many studies of landscape habitats and species distributions now include analysis of spatial connectivity and its effects.

1.5.2  A New Ecology for a New Age? Many people still alive can remember the early days of computing. In the 1950s, the world’s most powerful computers were huge machines. Built using thousands of tubes and valves, a powerful computer could fill a large room. Today you can place in your pocket a computer that is about a million times faster, that holds a million times as much data, and is thousands of times smaller than those early machines. Along with the increase in raw power, the range and diversity of computers has blossomed. Not only does virtually every home and business in the western world own a computer, but we also have computers in our cars, watches, phones, TVs, and many other everyday devices. The above figures highlight the magnitude of the information revolution that has transformed society since the mid-twentieth century. The revolution is also transforming science. In the 1950s, data as a commodity were rare and expensive; whereas today datasets are abundant and cheap. This transition marks a huge change in the way science is done. Computers today provide landscape ecology with a host of tools, such as geographic information systems (GIS) to work with landscapes, databases and data warehouses to store environmental information, and virtual reality to simulate environments. Advances in communication mean that these resources are now available online, so that everyone, everywhere potentially has access to ‘e-science’ (See Chap. 10). The information revolution is also changing the way we perceive the world around us. Every age projects its preoccupations onto the world around it. The Inuit people, for example, have 16 words for snow [19]. In the course of the Industrial Revolution, especially in the nineteenth century, machines and mechanistic views of the world dominated people’s thinking. As a result, ideas based on machines dominated science. The world was seen as a great machine. The planets moved around the sun like clockwork. Living things were seen as factories, with a power source and systems for transport, waste disposal, and central control. In the information revolution, a new view of the world has emerged. Nature today is often seen as a great computer whose programming we have to discover. This idea of natural computation fits well with biology. People often compare the genetic code to a computer program because it contains the “recipe” for building new organisms. Ribosomes, the cellular machines that make proteins, are

5  In traditional ecology, populations that comprise an ecosystem fit into a trophic pyramid, with producers (e.g. plants) on the bottom layer and top predators at its peak.

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information processors in that they “read” genetic information transcribed from DNA and output proteins according to its instructions. Sensory perception involves processing data input. Animal communication involves the transfer of information from one animal to another, and the subsequent interpretation (processing) of that information. Animals often act as though they are programmed to behave in certain ways. Not only can nature be regarded as computation, it is no exaggeration to say that advanced computing is sometimes indistinguishable from biology. This is happening because computing deals increasingly with problems that resemble biological or ecological systems. For example, in the drive to increase processing power, computers have gone from single chips to multi-processors, then to distributed processing, and eventually to swarms of processing agents. To solve the increasingly complex challenges posed by such problems, computing has looked to living systems, which have evolved ways of coping with complexity. The result has been a proliferation of biologically inspired ideas borrowed by computing. These include such ideas as cellular automata, genetic algorithms, neural networks, artificial intelligence and swarm intelligence. We shall see more about these ideas in later chapters. While it is risky to take computing analogies too literally, the idea of natural computing does help to explain certain aspects of ecology, as well as many other natural phenomena. What is stimulating these new ways of viewing nature is the recognition that new methods are needed if we are to understand the complexity of the living world. By themselves, traditional ways of doing science are not enough to allow us to fully understand complex systems and processes. Trying to come to grips with complexity poses a challenge for science. The reason for this is that complexity is about the way the world is put together. However, the traditional way of doing science is to do the exact opposite: that is, to take things apart and examine each piece in isolation. This is known as the reductionist approach. The reductionist approach has served science well. The wonders of modern technology are tangible proof of the success of the method. The principle is simple: “divide and rule”. If you want to understand something complex, then break it down into its component parts and figure out how they work. If necessary, keep dividing until you reach a level that is simple enough to understand. This is an immensely powerful idea. Over the course of four centuries, physicists and chemists have dissected matter down into its components: molecules, atoms, protons, electrons and, more recently, bosons and fermions. At every step of the way, they made important discoveries that have helped us to understand aspects of nature as diverse as the origins of the universe and the structure of DNA.  These discoveries led to all manner of practical applications, from aeronautics to biotechnology. There is a well-established logic behind the reductionist approach. For instance, if you want to know how a plant performs under different conditions, then put it in growth chambers and take precise measurements of respiration, photosynthesis and growth. What you find are clear patterns of cause and effect. If you raise the temperature by so many degrees, then you can observe the resulting effects on

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photosynthesis. If you alter day length by a certain amount, then the results are not only repeatable, but also predictable. With this information, not only can you understand why the plant grows better in certain environments, but also you can predict exactly how it would fare in any other environment. Unfortunately, the world does not always divide itself in straightforward ways. When you pull things to bits, you learn about the parts, but unexpected things can happen when you put them back together again. For instance, by experimenting with individual plants in a growth chamber, you may learn how that plant responds to light, water and so on. But when it grows out in the wild, it interacts with many influences, especially other plants. Laboratory results are often not repeatable under field conditions. No laboratory experiment can account for every factor that might affect plants and animals in the field. Interactions do matter. Often, the whole is much greater than the sum of its parts. Learning what happens when you put things back together is what complexity research is all about. Such questions abound in science. How do billions of neurons become organised into a living brain? How does the genetic code control the growth of developing embryos into complete human beings? How do ants and bees manage to construct elaborate nests and societies without the ability to think and plan? In many fields, scientists have found that they needed to develop new ways of dealing with the issues that arise when many things interact with each other. Physicists, for instance, needed to understand what happens when large numbers of particles, such as atoms, or stars, interact. The need to understand the whole system does not mean abandoning the tried and true divide-and-rule approach of reductionism. We need to complement traditional approaches with new methods and ideas. In brief, as well as understanding the parts, we also have to understand how the whole emerges from those parts. Ecology has developed many concepts, such as diversity and food webs, that recognise the inherent complexity of ecosystems (see Chaps. 4 and 10 re ‘diversity’ and Chap re ‘food webs’). What the information science revolution has done is to provide new tools that can deal with complexity in more direct ways than were previously possible. By allowing us to deal with complexity more directly, the information revolution provides new ways of doing ecology, which complement traditional approaches. Features of this new science of complexity include the following: • it is based on the idea of natural computation, interpreting natural processes as computation; • computers play a crucial role, both in their ability to handle large volumes of data, and in their ability to create “virtual worlds”; • for complex systems, no single model can be all-embracing, so different models may be needed for different purposes, including explanation, prediction, and control; • complex systems often behave unpredictably, so we need to study scenarios (that is, to ask ‘what if’ questions).

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In the chapters that follow we will explain these ideas in more detail. Moreover, we will provide an overview of what they can tell us about landscape ecology. We will explain many of the methods used and summarize some of the fresh insights that they are providing about ecosystems and environmental management. Only in the past few decades has science begun to make inroads into understanding complexity. In the early part of the twentieth century, there was a mood among scientists that complete knowledge of how the world works was just around the corner. Physicists had penetrated matter to its core, discovering the atomic building blocks of all matter. At the beginning of the third millennium, there is no such mood of optimism. Instead, there is acute awareness of just how little we yet know. The greatest challenge is to understand the complexity of the living world.

References 1. Allen JP (2000) Artificial biospheres as a model for global ecology on planet earth. Life Supp Bios Sci 7(3):273–282 2. Allen JP, Nelson M, Alling A (2003) The legacy of biosphere 2 for the study of biospheric closed ecological systems. Adv Space Res 31(7):1629–1639 3. Bossomaier TRJ, Green DG (1998) Patterns in the sand. Allen & Unwin, Sydney 4. Bossomaier TRJ, Green DG (2000) Complex systems. Cambridge University Press, New York 5. Bull CM, Possingham H (1995) A model to explain ecological parapatry. Am Nat 145(6):935–947 6. Chaitin GJ (1969) On the length of programs for computing finite binary sequences: statistical considerations. J ACM 16(1):145–159 7. Cody ML (1970) Chilean bird distribution. Ecology 51(3):455–464 8. Cohen JE, Tilman D (1996) Biosphere 2 and biodiversity  – the lessons so far. Science 274(5290):1150–1151 9. Conservation International (2018) Hotspots – targeted investment in nature’s most important places. http://www.conservation.org Accessed December 2018 10. Davies P (1998) Foreword in: Bossomaier TRJ, Green DG patterns in the sand. Allen & Unwin, Sydney, pp vi–x 11. Erwin TL (1982) Tropical forests: their richness in Coleoptera and other arthropod species. Coleopt Bull 36(1):74–75 12. Erwin TL (1988) The tropical forest canopy: the heart of biotic diversity. In: Wilson EO (ed) Biodiversity. National Academy Press, Washington, DC, pp 123–129 13. Fearnside PM (1999) Biodiversity as an environmental service in Brazil’s Amazonian forests: risks value and conservation. Environ Conserv 26(4):305–321 14. Fearnside PM (2005) Deforestation in Brazilian Amazonia: history rates and consequences. Conserv Biol 19(3):680–688 15. Fearnside PM (2016) Brazil’s Amazonian forest carbon: the key to southern Amazonia’s significance for global climate. Reg Environ Chang 18(1):1–15 16. Green DG, Klomp NI (1997) Networking Australian biological research. Aust Biol 10(2):117–120 17. Hardin G (1968) The tragedy of the commons. Science 162(3859):1243–1248 18. Holland J (1975) Adaptation in natural and artificial systems. University of Michigan Press, Ann Arbor 19. Inuit/Inupiaq Source: www.mustgo.com/worldlanguages/inuit/ Accessed 30 Dec 2019

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20. Kingsford RT (2000) Ecological impacts of dams, water diversions and river management on floodplain wetlands in Australia. Austral Ecol 25(2):109 21. Kolmogorov AN (1965) Three approaches to the quantitative definition of information. Probl Inf Transm 1(1):1–7 22. Lawrence D, Vandecar K (2015) Effects of tropical deforestation on climate and agriculture. Nat Clim Chang 5(1):27–36 23. Leblanc M, Tweed S, Van Dijk A, Timbal B (2012) A review of historic and future hydrological changes in the Murray-Darling basin. Glob Planet Chang 80-81:226–246 24. May RM (1973) Stability and complexity in model ecosystems. Princeton University Press, Princeton 25. May RM (1988) How many species are there on earth? Science 241(4872):1441–1449 26. Myers N, Mittermeier RA, Mittermeier CG, da Fonseca GAB, Kent J (2000) Biodiversity hotspots for conservation priorities. Nature 403(6772):853–858 27. NTA (2002) Endangered Places – 2002 Report Card National Trust of Australia Canberra 28. Odum HT (1996) Scales of ecological engineering. Ecol Eng 6:7–19 29. Pachauri RK, Allen MR, Barros VR, Broome J, Cramer W, Christ R, Church JA, Clarke L, Dahe Q, Dasgupta P, Dubash NK (2014) Climate change 2014: synthesis report. Contribution of Working Groups I, II and III to the fifth assessment report of the Intergovernmental Panel on Climate Change. IPCC, Geneva, Switzerland, p 151 30. Papentin F (1980) On order and complexity I General considerations. J Theor Biol 87(3):421–456 31. Pielou EC (1974) Population and community ecology. Gordon & Breach, New York 32. Pielou EC (1975) Ecological diversity. Wiley Interscience, New York 33. Pimm SL, Jenkins CN, Abell R, Brooks TM, Gittleman JL, Joppa LN, Raven PH, Roberts CM, Sexton JO (2014) The biodiversity of species and their rates of extinction distribution and protection. Science 344(6187):1246752 34. Pimm SL, Raven P (2000) Extinction by numbers. Nature 403(6772):843–845 35. Prigogine I (1980) From being to becoming. W H Freeman & Co, San Francisco 36. Recher HF (1969) Bird species diversity and habitat diversity in Australia and North America. Am Nat 103(929):75–80 37. Richmond B (1993) Systems thinking: critical thinking skills for the 1990s and beyond. Syst Dynam Rev 9(2):113–133 38. Schaffer M (1987) Minimum viable populations: coping with uncertainty. In: Soule ME (ed) Viable populations for conservation. Cambridge University Press, Cambridge, pp 69–86 39. Severinghaus JP, Broecker WS, Dempster WF, MacCallum T, Wahlen M (1994) Oxygen loss in biosphere 2 Eos. Trans Am Geophys Union 75(3):33–37 40. Solomonoff R (1964) A formal theory of inductive inference. Inf Control (2):7, 1–22 41. Stork NE (2018) How many species of insects and other terrestrial arthropods are there on earth? Annu Rev Entomol 63:31–45 42. Tregonning K, Roberts A (1979) Complex systems which evolve towards homeostasis. Nature 281(5732):563–564 43. Wallace CS (2005) Statistical and inductive inference by minimum message length. Springer, Berlin 44. Wallace CS, Dowe DL (1999) Minimum message length and Kolmogorov complexity. Comput J 42(4):270–283 45. Wilson EO (1992) The diversity of life. Penguin, London

Chapter 2

Seeing the Wood for the Trees: Emergent Order in Growth and Behaviour

Abstract  The patterns we see in the growth of a plant or the behaviour of animals can appear very complex, but there are often simple rules that underlie what we see. Systems of rules, called L-systems can capture the organisation of branching patterns and other features of growing plants. Simple rules of behaviour can explain many features of animal behaviour; multi-agent simulations use these rules to model community organisation and interaction with the environment. Keywords  Agent based models · Behaviour · Foraging · Intelligence · L-systems · Modularity · Plant growth · Social networks · Territory · Turtle geometry

A rainforest emerges from complex interactions between the many different plants and animals prise it  that comprise it

© Springer Nature Switzerland AG 2020 D. G. Green et al., Complexity in Landscape Ecology, Landscape Series 22, https://doi.org/10.1007/978-3-030-46773-9_2

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The whole is greater than the sum of its parts. As we saw in Chap. 1, this popular saying captures the essence of complexity. In this chapter, we look briefly at examples of processes in which fine scale functions, at the level of individuals, lead to larger scale organisation. There are many other effects that arise from individual interactions besides the examples given here (e.g. foraging strategies), which we will meet in later chapters. Sometimes you are better off not to know too much. If you are trying to understand the workings of a flock of birds, or of a forest ecosystem, then you can only get so far by studying more and more about the individuals. Of course, you do need to know something about the individuals that go to make up the whole—some details matter. The way that living creatures interact, the properties that allow one plant to outgrow another, the relative speed of predator and prey are all characteristics that count on a larger scale. But there are diminishing returns in probing individuals ever more deeply. Once you understand the essential workings of metabolism, for instance, further detail will probably not help you to understand how that animal will fare in its struggle for survival. In trying to understand how self-organisation works, how a large number of individuals become organised into an ecosystem, a certain amount of abstraction is needed. Traditional models tend to gloss over processes that involve interactions at the level of local individuals or elements. Instead they tend to look either at the large scale, or else at the fine details. They tend to take a top-down view of how constraints act on individuals, rather than a bottom-up view of the effects that arise from interactions between individuals. In so doing, they get only half the story. They deal with how plants grow and how animals move. In both cases, the individual-oriented models take bottom-up views that offer us very different insights from top-down, reductionist models. This is not to say that we should abandon the traditional models and use only individual models. Our point is that to gain a complete picture, we really need to look at both.

2.1  Plant Growth and Form 2.1.1  Factors Influencing Growth The shapes of plants play an important part in determining their survival in the landscape and their role within an ecosystem. The main impact of interactions between different plants, and between plants and their environment, lies in the ways they affect growth. The supply of sunlight, water and nutrients all affect the rate of growth. In turn, the nature of a plant’s growth helps to determine the way it interacts with its environment, and with other plants. So, to understand where plants are found in landscapes, and in what numbers, it is essential to understand something about how they grow. Shade-tolerant plants grow slowly, but they can survive under the canopy of other trees. They often cluster leaves around the outside of the canopy to maximise the amount of light they collect. Fast-growing seedlings quickly outgrow their competitors, but they need lots of light so thrive in more open conditions.

2.1  Plant Growth and Form

23

The growth and form of plants helps to determine the ways in which they interact with their environment, and with other plants. Leaves are angled to catch the sun for photosynthesis and growth and also to avoid stress from excess moisture loss or high temperatures. Mean leaf angle affects attenuation rates for light within and below plant canopies (Fig. 2.1). A plant canopy with mean leaf angle of 79° relative to incident radiation, may correspond to clover, while one with a mean leaf angle of 33° could be a cereal crop species, such as wheat or maize. Further, the leaves are positioned by the structural patterns of tree trunks and branches to maximize total photosynthesis rate and to minimize respiration overhead and stresses due to heat and transpiration losses. The shapes of plants vary with horizontal density, vertical stratification and the physical environment. For example, isolated trees may have many branches and an overall ellipsoidal shape to maximize interception of side-lighting and direct sunlight (Fig. 2.2a) [4]. In contrast, trees in closed rainforests tend to be tall with high leaf canopies that shade the plants below (Fig. 2.2b). Different wavelengths affect plant growth and structure in different ways. Visible light affects photosynthesis rates, while red/far-red light affects developmental events, such as onset of flowering. Interception of visible light by leaves changes the spectral composition of the light below, which in turn, inhibits leaf development, both within a canopy and for competing plants (Fig. 2.3) [9]. Light intensity is attenuated by foliage in certain wavelengths. This shading can lead to selection for shade tolerant or shade avoiding species. Shade tolerance is conferred by combinations of different specific leaf area, low leaf angle and different chlorophyll a/b ratios. Photosensor mechanisms for blue light intensity and red/ far-red ratio in shade-avoiding species result in high specific leaf area (cm2 g−1),

I/I0

Legend: each curve represents a particular K value, corresponding mean leaf angles visualised below.

79°

67°

33°

Leaf Area Index Fig. 2.1  Mean leaf angle, show graphically, for corresponding light attenuation coefficients (K), result in different patterns of attenuation of light intensity (I) relative to incident light intensity above the canopy (I0) for increasing leaf area index (ratio of vertically accumulated leaf area, per unit horizontal area)

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Fig. 2.2  Canopy shape and light interception for: (a) an isolated tree; and (b) a tree in a crowded rainforest that experiences little to no side-lighting

Fig. 2.3  Changing spectral composition of light above and below a plant canopy, expressed as irradiance for different wavelengths, shows a decline in photosynthetically active radiation (blue and red) and a decrease in red/far red ratio. (Adapted from [9], p. 2, Fig. 1A&B)

stem internode elongation, less frequent branching and phototropism, or bending, in the direction of more intense blue light. The trunks and branches of trees serve as conduits upward for water and minerals, absorbed by root tips, and downward for carbohydrate that can be used in root growth and development. The below-ground root structure of plants also maximizes the numbers of root tips and associated absorption zones, while minimizing respiratory costs of root branches. To understand the patterns of above- and below-ground components of plants, we need to study how plants grow and develop. This will involve tracking both growth processes and structural or morphological development. Just as positioning of leaves and root tips determines growth rate, the positioning of growing points determines developmental patterns. Positioning of growing points and their response to different parts of the light spectrum ultimately determine

2.1  Plant Growth and Form

25

branching patterns, lengths of trunks, branches and roots as well as numbers and locations of leaves, flowers, fruit and root tips. Plants both respond to the environment and alter it, which in turn can affect other plants. The properties of different parts of the landscape, such as soil depth, moisture and fertility, or prevailing sunlight, temperature and wind, can all select for plants with particular morphological attributes and growth patterns. For example, soils with limited moisture select for those plants that can efficiently take up soil moisture, while minimizing water loss by the leaf canopy. Those plants may have deep root systems and leaves with high water-use efficiency. The latter may be expressed as high CO2 uptake per unit of water lost by transpiration. Southerly slopes (in the southern hemisphere) may select for plants that can maximize interception of sky light from a hemispherically mobile source, and therefore have leaves distributed in an ellipsoidal pattern (Fig. 2.2a). On a geographic scale, latitude, which affects incident sunlight, day length, and temperature; topography, which influences rainfall patterns, wind speed and direction; and geological substrate, which affects soil depth, fertility and water-holding capacity, together will select for different plant growth and structural patterns. Close to the equator, plentiful sunlight, high temperatures, and almost constant day length can select for tropical rainforest species, especially if mountainous topography and atmospheric circulation cells provide high rainfall that follows a monsoonal pattern. Away from the equator, at about 30° (N or S) along the sub-tropical pressure ridge, and on the leeward side of large mountain ranges, such as the Andes, Rockies or Himalayas, there are deserts, which are hot and dry. Plants with deep root systems and characteristics that minimize water loss by transpiration will have an advantage in such environments. Water loss may be reduced by low stomatal density, high leaf angle relative to incident solar radiation (Fig.  2.1), and mechanisms, such as the Crassulacean Acid Metabolism, found in cacti, that makes light harvesting and CO2 absorption asynchronous (energy from light absorption during the day is stored and then used at night to complete the photosynthesis process, when CO2 can be absorbed with minimal water loss). Traditional vegetation models typically ignore details of plant morphology and structure and make use of quantities, such as biomass per unit area. They account for physiological processes, such as photosynthesis, respiration, transpiratory loss of moisture, partitioning of biomass to above- and below-ground components and expression of quantities, such as leaf area index (Fig. 2.2). A powerful combination for better understanding the interplay of growth and structural dynamics is to combine process-based models with L-system models of morphological development.

2.1.2  Branches and Leaves Three important features of plant structural development are modularity, iteration and recursion. These are all common ideas in computing and mean that we can represent the complex organization of plant growth as a form of computation.

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By modularity we mean that plants comprise distinct units, such as stems, branches, and leaves and combinations thereof. Modularity is important because we can model growth and development in terms of repeating components without needing to consider cellular or other lower levels of biological organisation. Our models can comprise formulae for processes, such as photosynthesis and rules for structural development of the modules. By iteration we mean that the same processes repeat over and over. Much of plant growth involves repeating the same steps: leaves sprout from buds, buds form new branches and buds, then the process repeats. By recursion or self-similarity we mean that developmental patterns can recur at increasingly lower levels (patterns repeated within patterns). A fern frond, for instance, is made up of smaller versions of the frond, and those are in turn made up of even smaller versions of the frond, ad infinitum. If you look ever closer, each frond appears to be made up of smaller versions of itself. The pattern recurs. The recursion of these patterns are known as fractals (see Chap. 3). Expression of such patterns can be accomplished using parallel rewriting rules, in which symbols representing plant parts (such as stems, branches or leaves) can be iterated and recurred according to these rewriting rules. Together, these properties – modularity, iteration and recursion – make patterns of plant growth self-similar. A fern frond, for instance, is made up of segments that are smaller versions of the whole. If you look closer still at those segments, each, in turn, appears to be made up of smaller versions of the larger. The pattern repeats itself. In this sense, plant growth patterns form natural fractals. We can capture repetitive growth patterns, such as branching, as rules. To start with a simple example, suppose that we have the following two rules (L-system 2.1) that tell us how to replace one symbol with another: L-system Axiom: Rule 1: Rule 2:

2.1 A A → B B → AB

What Rule 1 means is that if you have the symbol A, then you replace it with a B. Likewise Rule 2 means that if you find the symbol B, then you replace it with the pair of symbols AB. Now suppose that we start with the symbol A. Then by repeatedly applying the above two rules, we can generate strings of symbols of ever increasing length. The following list summarizes the sequence of results at various stages, starting from stage 0, where we have just the initial symbol A.

27

2.1  Plant Growth and Form Stage Stage Stage Stage Stage Stage Stage Stage

0 1 2 3 4 5 6 7

: : : : : : : :

A B AB BAB ABBAB BABABBAB ABBABBABABBAB BABABBABABBABBABABBAB

An interesting feature of this sequence is the lengths of the strings. If we count the symbols at each stage then we obtain the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, ….

Some readers may recognise this as the famous Fibonacci sequence of numbers, which has well-known associations with plant growth. What this model shows is that the sequence is a natural consequence of the iteration process. The above model is an example of a formal language model, in which the rules for rewriting strings define the grammar (syntax) of the language. Formal languages are now widely used to model the organisation of growth processes. Such models are usually called L-systems. They are named after Aristid Lindenmayer, who was one of the first biologists to use syntactic methods to model growth [13]. The power of L-systems comes into play when we assign meaning to the symbols and rules.1 For instance, if the symbols represent branches of a growing tree, the rules would denote the organisation of branching. The central idea of L-systems is parallel rewriting, a technique for constructing complex objects by successively replacing parts of simpler, initial objects using rewriting rules. L-systems are distinct in their application of rewriting rules in a parallel manner. All predecessor symbols are replaced with successor strings simultaneously. This procedure allows us to capture simultaneous processes of expansion, division and initiation of several different cells or plant parts at the same time. L-system 2.2: Axiom: X Rule Rule Rule Rule

1: 2: 3: 4:

X Y A B

→ → → →

F[A]FY F[B]FX X Y

If we apply the growth rules several times, then it leads to the following set of strings:

 The process often involves using several different rules in parallel, not just one at a time.

1

28 Stage 0 1 2 3 4 5

2  Seeing the Wood for the Trees Pattern X F[A]FY F[X]FF[B]FX F[F[A]FY]FF[Y]FF[A]FY F[F[X]FF[B]FX]FF[F[B]FX]FF[X]FF[B]FX F[F[F[A]FY]FF[Y]FF[A]FY]FF[F[Y]FF[A]FY] FF[F[A]FY]FF[Y]FF[A]FY

Fig. 2.4  The branching pattern produced by the L-system given in the text. Notice that the pattern is self-similar; that is, each branch and sub-branch is similar in form to the entire pattern

The square brackets here refer to modules that represent branches. That is, the string within any pair of square brackets represents a pattern of growth within a branch. What emerges is a recurring pattern at different scales of growth, from the whole plant to branches and to twigs. If we apply the growth rules several times, then this leads to longer and longer strings that describe the growing pattern, as seen in this example. Notice how each symbol gets replaced by the symbol, or string, that is given in the rules. If we were to continue the process, then as in growth, the string just keeps on getting bigger and bigger. We can translate the strings into a picture that shows what the resulting structures look like. If we represent each symbol F as a line segment here,2 then we can translate each symbol into a picture element. We need to make an important distinction between the symbols A and B. We take A as denoting a branch to the left of the main stem, and B to denote a branch heading off to the right of the stem. The resulting growth pattern is shown in Fig. 2.4.

 This simple drawing procedure uses turtle graphics, which we look at later in the chapter.

2

2.1  Plant Growth and Form

29

Here we have a few minor confessions to make. The picture in Fig. 2.4 is not quite the pattern generated by the model. If you look closely, you will see that the line segments near the start of each branch are longer than the segments near the tip. This is a slight refinement of the above model, in which we incorporate an expansion factor to allow for continued growth of stem segments after they first appear. Another refinement is to incorporate a brief delay before the expansion of each branch occurs. Finally, the model does not specify the angle at which new stem segments grow relative to the branch where they form. We omitted all such details from the above description to keep the model simple. We can incorporate all of the above details, and more, by refining the L-system. A parametric L-system attaches parameter values to each alphabet symbol. So in the example above, we might replace the constant F by a module F(a, t) in which a is the vertical angle at which the branch grows, relative to the parent branch, and t is its age in terms of growth steps. Parametric L-systems allow models to account for details such as spatial orientation and changes within existing growth modules after they form. They can also specify time delays. The important thing about the model shown above (Fig. 2.4) is that it highlights the modular and iterative nature of plant growth. It also shows that we can capture the apparent complexity of plant growth in a surprisingly small set of simple rules. Using refinements, such as parametric L-systems, we can produce more life-like images of plants, such as Fig. 2.5. Another refinement is a stochastic L-system. In all the examples so far, the outcome at each stage is automatically determined, so we could call the model a deterministic L-system. In contrast, a stochastic L-system would have a choice of two or more alternative rules that could apply at any stage, with each rule having a certain

Fig. 2.5  A 3D rendering of the Tree generated by L-system 2.2. (Reproduced with permission from [22])

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probability of occurring. For instance, in the above model, we could replace (say) rule 3 in L-system 2.2 by a pair of rules: Rule 3a: Rule 3b:

(0.9) A → X (0.1) A → F

Each time the model runs, it would generate a different shape. Stochastic L-systems allow models to represent natural variations. They also provide a way to allow for outside events that affect growth, such as insect damage to buds, leaves or stems. Another important refinement is context-sensitive L-systems. These models employ context-sensitive grammars. That is, the result of a rule depends on the local context in which it is applied.3 Context sensitive L-systems comprise rules such as the following: A A

→ →

X F

Here, the presence of the symbols D1 refers to a developmental trigger set of separate rules that simulate the accumulation of developmental control substances (e.g. nutrients) beyond a critical level, while the symbol refers to any other context. Context-sensitive models make it possible to allow for differences in growth of parts of plants, such as branches being in light or shade. In the following sections, we will look more closely at why these refinements become necessary and how they can affect growth patterns. A further refinement of parametric L-systems is to run a model of the physiological processes in parallel with the L-system model. In such a system, information on physiological quantities from the process model are sent to the L-system model to determine the parameter values. The structure of the L-system model affects environmental variables, such as the amount and quality of incident light reaching each leaf. Hence there is a continual feedback loop between the two models with the behaviour of one constraining the other. In such a system, the steps of the L-system become time intervals that are in synchrony with the process model. The length of branches are determined by the amount of carbohydrate allocated and specific stem length per unit carbohydrate. Similarly, leaf area is determined by carbohydrate allocated to the leaf and the specific leaf area. As leaves are shaded by more leaves, the total light interception declines, as does the photosynthesis rate. As the Red/Far-­ red ratio and blue light intensity change, new branches or stems and leaves may appear with a lower frequency than under full sunlight (see Fig. 2.3).

 Cellular automata, which we discuss in Chap. 3, have context-sensitive rules.

3

2.1  Plant Growth and Form

31

Fig. 2.6  Differences in the growth forms produced by the two L-system models 2.3 and 2.4. Model 2.3 produces a tall thin plant, whereas model 2.4 produces bushy growth with many branches

2.1.3  Overall Plant Form We have seen that formal languages can capture the organisation of many growth processes very simply. Therefore, models based on formal language can help us to understand certain implications of plant growth and form. Consider the growth patterns of the two (hypothetical) trees shown in Fig. 2.6 and governed by the following simple L-systems: L-systems 2.3 and 2.4: Variables: {A}; Constants: {0} Axiom: A 2.3 Rule 1: A → [0]A[0] 2.4 Rule 1: A → [A]A[A]

Here the symbol A denotes a growing tip, and 0 denotes either a terminal or a slow-growing side branch. As before, square brackets indicate modules, which here are branches. In the model, the symbol 0 remains constant. The first few growth stages of these models are given by the following sequences: Stage 0 1 2 3

Model 2.3 A [0]A[0] [0][0]A[0][0] [0][0][0]A[0] [0][0]

Model 2.4 A [A]A[A] [[A]A[A]][A]A[A][[A]A[A]] [[[A]A[A]][A]A[A][[A]A[A]]] [[A]A[A]][A]A[A][[A]A[A]] [[[A]A[A]][A]A[A][[A]A[A]]]

The two tree-like 2D structures in Fig. 2.6 show different growth and development patterns. Suppose that the root systems and soil water and nutrient content are

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Fig. 2.7  The relationship between branch “generations” and time in the two models shown in the previous figure. (a) In model 2.4, the number of growing tips increases exponentially (solid curve), whereas in model 2.3 they are linear. (b) However, the cost of supplying water and nutrients to large numbers of growing tips reduces the rate at which they can grow. So in model 2.3, the tree passes through many more growth stages (making it taller) in the same time as a tree that follows model 2.4

the same for each tree. Further, assume that an equal amount of water and nutrients and carbohydrates from photosynthesis are consumed per unit length of trunk or branch element for each tree. Figure 2.7 shows that model 2.4 grows many more branches (growing tips) than model 2.3, but model 2.3 has to divide those resources among many more growing tips. This means that the plants L-system 2.3 produces would be tall, conical, tree-like structures, whereas L-system 2.4 would produce a short, bushy structure with a broad, elliptical shape, like a low shrub. These different shapes in turn will have implications for the survival and reproduction in different environments. The bushy version may do better, for example, in a windswept desert where its shape offers little resistance to the wind, while the tall tree might thrive as a fast-growing, shade avoiding colonist that monopolises an opening in a forest canopy. Admittedly, the above models are extremely simplistic. For example, the rate of supply to a tree of water and nutrients increases as its root system grows and spreads. Also, not all nutrients and energy go into growth—respiratory overhead and reproduction require many resources too. However, despite being so crude, the models do highlight the consequences of two extreme strategies of plants, which result from different selective pressures during evolution. The point is that a simple change in the pattern of growth leads to a large difference in the ways that the two plants will fare in a landscape and in competition with other trees. Even individuals of the same species may exhibit different development patterns and resulting shape depending on plant density. Dense forests have taller trees with less side branching. If the same species grows in isolation, it may have more side branches and consequently an ellipsoid-shaped leaf canopy (see Fig. 2.1). Growth models of this kind raise the prospect that it may be possible to link patterns of development to genotypic characteristics, and by studying phenotypic

2.1  Plant Growth and Form

33

expression of those characteristics, allow prediction of optimal environmental conditions for each species.

2.1.4  Self-Organisation Versus Constrained Growth Grammatical models of complex growth processes (e.g. L-systems), especially those found in plants and plant tissues, highlight the importance of communication between cells. This means that context-sensitive grammars are often required to describe the processes adequately [19]. This internal growth environment, is not to be confused with responses to the external environment (e.g. light, temperature or soil moisture), which can be incorporated by coupling a process model that connects to parameter values in parametric L-systems. Delays in developmental events in plants relate to triggers that occur when plants have experienced critical quantities of thermal-time (day-degrees), photoperiod, vernalisation (accumulated cold requirement), or changes in red/far-red ratios. We can incorporate these by coupling the L-system developmental model with process models for calculating quantities such as thermal-time or vernalisation and triggering changes at critical levels via parameters in parametric L-systems, which is what we described in L-system 2.3. Delays can also be simulated with sequences of rules as in L-system 2.3 above. Such rules simulate delays due to diffusion of developmental control hormones or critical levels of developmental controls like thermal-time. Of course, the flow of water, nutrients, carbohydrates or control substances, such as abscisic acid, rarely depend solely on diffusion. Many of these flows occur through vessels that are part of special vascular tissue and are driven by concentration gradients that result from loss of water by transpiration, water uptake, or demand for carbohydrates by growing root tips, stems or reproductive organs. For example, a root tip encountering dry soil, may produce abscisic acid that flows in the transpiration stream to the leaf canopy, and when the abscisic acid reaches a particular level, leaf initiation may halt or stomates may close or leaves may be abscised, depending on severity. In this sense, transfer of substances is active, rather than just a matter of molecular diffusion. What we might call traditional growth models typically link structure to external environmental agents. A good example is a biochemical gradient. Biologists look at the diffusion of water and nutrients across a growing structure and assume that the sizes of elements will be a function of the biochemical concentration at any given point. In some instances, these models cut across a lot of unnecessary detail in an L-system model and give a good approximation of the shape that a structure assumes. However, in reality both aspects matter; a complete picture of growth needs to mesh the external constraints with the internal self-organisation. One question that biologists ask about L-system models is whether they are useful. One of the first successes of L-system models was their ability to create detailed and life-like images of plants. Surprisingly, the more realistic the images, the less

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Eucalyptus alpina:



   

/delay=1 year

   

/delay=1 year

Hakea salicifolia:



Fig. 2.8  Branching models for (a) Eucalyptus alpina and (b) Hakea salicifolia. These patterns result from the rules listed. Note the similarities and differences between two sets of rules for completely different species

scientists have seen L-system models as serious models of growth. We suspect that this attitude is influenced by a reaction against the role of L-systems in popular activities such as games and movie-making. L-system modelling has developed to the point where it is possible to develop valid and useful models of many kinds of growth. One application has been in virtual experiments [19]. How close together, for instance, can plants grow before interference seriously affects their ability to survive? But the real potential that we foresee lies in elaboration of the idea introduced earlier of the model of overall growth form. On the one hand, growth models can help us to understand the performance of plants in different kinds of landscapes. On the other hand, if we can identify the similarities and differences between related species in the organisation of growth, then it would give us a deeper understanding of how and why genetic differences arise, and of the mechanisms involved. Consider the two plants shown in Fig. 2.8, for instance. Their growth patterns, from undifferentiated tissue (meristem) to buds, leaves and fruit can be summarized by simple grammars (parametric, and context-sensitive). The grammars emphasize both the similarities and differences between them.

2.2  Animal Behaviour

35

In these models, buds are classified by their type (e.g. “leaf” for leaf-forming buds) and by the angle of their orientation relative to the main stem (indicated by the variable “a”). The symbols “d” and “0” represent numbers associated with the rate of growth. There are similarities between the syntax of the two models: the first two rules of both models are identical, emphasizing the similarity between the two growth patterns. The models differ syntactically in the timing of fruit production. The two models also differ in their semantics (the meaning of the symbols): the shapes of their leaves and fruit are different. A complete picture of plant growth requires models that combine the two approaches: morphological development component simulated with parametric context-sensitive L-systems, and physiological process-based models, that together interact and respond to the external environment. A similar argument also applies to animal growth and development, and animal behaviour.

2.2  Animal Behaviour From a practical point of view, animals are much harder to study than plants. Plants sit still; animals move around. At large spatial scales, the distributions of plants and animals present similar problems, especially how they respond to climate. However, at the local scale, the issues involved are very different. Some of the great challenges of ecology are to identify processes that govern the ways animals behave and to understand the effects that arise from their interactions with each other and with their environment. Many aspects of animal behaviour arise from the need to survive in a complex environment. As we have seen previously with other kinds of complexity, these local interactions often have global effects. In this section, we look at some models of the complexity of animal behaviour, especially phenomena that emerge from interactions between many animals. One of the most common strategies for dealing with complexity is exploit modularity. In general, modules are self-contained, reproducible units. When describing plant growth, we saw that modules enabled plants to grow complex structures from simple units. Here we will see how animals often employ a similar approach in their behavior, To cope with complexity, they break complex problems down into simpler ones.

2.2.1  Searching for Food One of the most basic problems all mobile animals face is to find food. An eagle, flying high over a plain can see large areas all at once. So all they need to do is watch for movement, then swoop. At the other extreme, even a flat grassy field is a vast, complex, unknown environment to an ant. They compensate for their small size by

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Fig. 2.9  Random walks as models of ant foraging. (a) A single “ant” covers very little of the area close to the nest before wandering further afield. (b) As the number of ants increases, the area of coverage grows larger. (c) A swarm of ants will cover the entire area for some distance around their nest

sending out many foragers. Their food gathering can be broken into two phases: exploration (searching) and exploitation (retrieving and using) [2, 14]. Place an exploring ant close enough to food to catch its scent and the ant will move straight to the food. In a barren environment, however, all the ant can do is wander around at random until its senses detect something. We can capture this sort of behaviour as a random walk (Fig. 2.9). The properties of random walks are well-known. For the ants, an important property is how much territory random walkers cover. It turns out that the area covered by a set of random walkers forms a disc around their starting point; they visit every point in that circular area. Outside that circle, the paths formed by different exploring ants branch outwards and the area searched becomes increasingly rough (Fig. 2.9) [12, 25]. Given enough time, a large ant colony of foraging ants will find any food that lies close to their nest. For foraging ants, finding food is only part of the problem. They also need to retrieve as much of the food as they can and bring it back to the nest. To achieve this, ants switch from exploration to exploitation. When scouts find a food source, they lay down pheromone trails, which allow other ants to follow them to the food and carry bits of it back to the nest. Other species use different ways of switching from exploration to exploitation. After foraging bees find food they return to the nest and do the famous waggle dance to let others know the distance and direction of the food. Having explored and made a kill, leopards need to ensure that they can exploit it to the full. They do this by storing their kill in a tree, so they can protect it from other predators. Optimal foraging refers to any foraging strategy that achieves the greatest return for given effort. It usually requires a trade-off between exploration and exploitation [1]. Different environmental conditions affect the balance between the two activities. For instance, simulation studies suggest that if food is concentrated in certain areas, then animals need to spend more time on exploitation. On the other hand, if food is scattered randomly, then exploration requires more effort [5].

2.2  Animal Behaviour

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This idea of a trade-off between two needs is a general principle in solving complex problems. Humans apply this trade-off in many settings, from search and rescue operations to balancing government spending. Trade-offs also occur in evolution (see Sect. 8.1).

2.2.2  Territory A large-scale challenge for animals living in a landscape is how to ensure availability of resources, especially food. If every animal in a population tried to live in the same place, they would soon deplete the food supply. Many strategies have evolved to avoid this problem. Essentially, we can view some of the most common strategies as applying a modular approach to resources over space or time. In many species, individuals or groups claim and defend territories. Each territory holds enough resources to sustain a social unit - food, shelter, and breeding sites, for example. Territory-holding animals will also often engage in ecosystem engineering, modifying their territory to suit them. Since territories are interchangeable in these respects, from an ecological perspective they can function as modules within a landscape. At the other extreme, a species can apply a modular approach in time to a landscape. Many species congregate in large groups for protection against predators. This includes, for instance, most herd animals and schools of fish. However, the problem of living in a herd is that it rapidly exhausts the food available in any particular area. To compensate, they move around the landscape, so that they exploit any one area for a short period only. Of course, necessity imposes many variations on the above strategies. In many regions, seasonal changes make food scarce in winter, forcing animals to migrate to less affected areas. The need to reproduce leads many species to congregate during their breeding season, either to find mates, or for protection against predators. These problems lead to different kinds of trade-offs. For example, prey species that defend territory need to balance defence against escape [20]. In some territorial species, the home range varies seasonally, as the animals respond to the changing opportunities for foraging [24]. In some species, the trade-­ off during foraging between exploration and exploitation can produce switches between territoriality and other behaviours [15]. When the difficulty of exploration is low, animals act as individuals—If exploitation is easy, then there is no need to be concerned about other animals—but if exploitation is difficult, then they compete by scrambling to gather food, rather than by defending a territory. If exploitation is difficult, then individuals defend their territory, but if exploitation is easy, then they are likely to cooperate with others to search more effectively. In short, they defend territory only if finding and exploiting resources are both difficult.

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2.2.3  Social Networks Forming social groups has several advantages for animals, including protection against predators, finding breeding partners, defending territory, and cooperation in foraging and collecting food. The organization of social groups is important in the way populations cope with the above challenges; it also influences the way individuals behave. Some types of social structure are well known. For instance, dominance hierarchies reduce the costs of competition to individuals, as conflicts are automatically resolved by the hierarchy. While social competition absorbs time and effort for individuals, the resulting social hierarchies have advantages, such as helping greater numbers of individuals to survive and maximizing the chances for survival of offspring. More generally, networks of social connections between individuals can have many other implications for the group. For this reason, analysis of the structure and patterns in social networks has become an important field of study [17]. For instance, cooperation in social groups often depends on how well animals know each other. It is common in social groups for individuals to prefer to interact with relatives. Such preferences can influence the way animals arrange themselves in space, and the dynamics of competition in social hierarchies [3]. Most primate species maintain social cohesion by grooming, which rein-forces social bonds between individuals [7, 8]. Since grooming is time-consuming, individuals can only afford to groom a limited number of others. Robin Dunbar [7] has suggested that this sets a natural limit on group size, because social cohesion can only be sustained in a group sufficiently small that all individuals groom one another frequently. Dunbar argues that the evolution of language as a substitute for grooming allowed human societies to achieve larger group sizes (100–150 individuals) [7].

2.2.4  Animal Intelligence How intelligent are animals? It is a difficult question to answer. Most efforts to understand and measure intelligence have focussed on how people, animals, or machines ‘think’. In particular, attempts to measure animal intelligence have often drawn on comparisons with human traits and abilities, such as tool using or the use of language. In the past, many such measures were tainted by anthropocentric bias: an underlying assumption that any test or measure must confirm humans as the pinnacle of intelligence. A deeper reason why measuring animal intelligence is difficult is that there has long been confusion about the nature and definition of intelligence itself. However, a promising advance has been to identify ‘intelligence’ with the ability to solve complex problems. That is, the more complex the problem, the greater the degree of intelligence required [10]. The virtue of this approach is that it has the potential to

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provide a consistent way of measuring intelligence, an approach that is independent of culture and other biases. It also has the potential for a consistent approach to measuring intelligence, not only for humans, animals and machines, but also for the collective intelligence of groups [6]. Advances in robotics, especially the ability to make small, mobile robots, led to increasing interest in computer science about group intelligence and behaviour. The field of swarm intelligence, for instance, concerns coordinating behaviour and problem solving in groups. Studies of behaviour in animal groups have been a rich source of insights for this research [23]. As we will see in the next section, studies of swarm intelligence are likewise providing insights about animal behaviour.

2.3  Multiagent Systems Every era tends to see the world through the lens of its preoccupations. In Victorian times, the Industrial Revolution led people to view the world as a great machine. The human body, for instance, was a machine composed of mechanical systems: the heart was a pump, the digestive system supplied fuel, for instance. In the Information Revolution, a new paradigm emerged that sees nature as a form of computation. As we saw earlier, we can model plant growth in terms of computation: the iteration of simple rules. Similarly, we can treat animals as “agents,” whose behaviour follows simple rules.

2.3.1  Turtle Geometry To model animals in an environment, we need a simple, abstract way of representing animals. Let’s start with a well-known model: a robot. A robot is essentially a computer that can move around, sense its surroundings, and do things. By saying it is a computer, we mean that its behaviour is governed by a program: a set of commands that tell it how to behave. Perhaps the simplest behavioural model based on a robot is turtle geometry. Admittedly, this is a model of geometry, not the behaviour of any real animal. However, it serves to demonstrate a key principle: we can capture many kinds of complex motion with simple rules of behaviour. It also shows how we can begin to build more sophisticated models. In the 1970s, the Stanford mathematician Seymour Papert devised a drawing system called Turtle Geometry [16]. His aim was not to understand turtles, but to help children to learn mathematical ideas by embodying them in concrete form. The idea of turtle geometry is to think of a geometric shape as the trail left by an imaginary turtle as it wanders around on a surface. If we were to attach a pen to a turtle and place it on a table covered with paper, then the pen would draw a line marking where the turtle has been.

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The turtle’s path can be described by a sequence of symbols representing its actions as it moves around. In the simplest case, there are just four possible actions: Step Forward, Step Back, Turn Left, and Turn Right (both turns through 90°). Let’s abbreviate these using the symbols F, B, L, and R, respectively. Forget, for the moment, that a turtle is a complex animal with four legs and so on. All that matters is that it can move forward and do all the rest. The details do not matter. Given the above symbols, we can describe a trail in terms of the sequence of actions that the turtle makes as it moves, such as FFRFFFLFRFF. Strings of symbols like this one form a simple language. Not only can we use the strings to represent a path after the fact; we can also use them to program a robot turtle to follow a certain path, and, in the process, to draw a particular shape. For example, the string FRFRFRFR tells the turtle to draw a square. The above idea becomes powerful when we introduce names for particular actions. For example, suppose that we assign the name SQUARE to the string given above for drawing a square. Then instead of repeating the detailed list of actions each time, all we need do is to give the turtle the command “SQUARE” and it will automatically carry out the sequence FRFRFRFR. In this way, the turtle’s repertoire of behaviour grows. What is more, we can build up ever more complex patterns out of the elements that it has already learned. So for instance, by adding the following two rules, we can teach it to draw an 8 × 8 grid of squares. ROW → 8 SQUARE L 8F R. GRID → 8 ROW.

Of course, to produce really refined behaviour, the turtle needs to be a bit more skilful to begin with. Most versions of turtle geometry allow movements forward by fractions of a step, and turns by any number of degrees. Given these, and a few other enhancements, we can encapsulate a bewildering variety of patterns as turtle programs. Turtle geometry is often used to draw plant growth patterns of the kind that we saw earlier in this chapter.

2.3.2  From Turtles to Agents The grid example above embodies an important idea from computing, and one that we also see in animal behaviour: modularity. Modularity reduces complexity. Complicated behaviour is difficult to learn, just as complicated computer programs are difficult to write. If we were to sit down and try to write out a string to describe the turtle’s complete path in drawing the honeycomb pattern, then it would be a long and difficult process. The chances of making a mistake are great. By giving names to elements of the desired pattern, we carve a big, complex problem up into simpler modules. Once we know that a particular module does its job correctly, then it becomes a building block that we can use again and again (see also Chap. 4).

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Let’s bring the idea of modularity a bit closer to home. What happens when you get up and go to work in the morning? Most people would produce a list something like this: get out of bed have a shower get dressed have breakfast pack my briefcase leave home catch the bus.

The details will vary from person to person, and even from day to day. But we can all identify with this sequence of events. But look more closely. Each event is really a routine, a behavioural module, that encapsulates an entire sequence of activities. Now let’s look at one activity, such as having breakfast. We can likewise carve up this episode into a routine sequence of actions. It might look like this: get out cereal, milk and cutlery fill bowl with cereal and milk heat bowl in microwave oven sit and eat cereal clear breakfast table.

The modularity does not stop there. We could go on. “Heat bowl in microwave oven”, for instance itself involves a sequence of steps, such as “place bowl in oven”. And each of these actions could be broken down further into sequences of hand-eye movements. The point of this example is that our own behaviour is highly programmed. We are so accustomed to thinking in terms of behavioural modules that mostly we do not even realise that we are doing it. In the same way, we can summarize the behaviour of animals in modular terms. To move from geometry to models of real animals, we need a slightly more sophisticated model than Papert’s turtle. The model’s components are usually called agents. The representation of animals as agents is an abstraction. In computing, an agent is any independently acting entity (living or artificial) that can interact with its environment. Depending on the context, an “agent” could mean an animal, a plant, a person, a robot (or drone), a corporation, a piece of intelligent software, or many other things. The simplest agents are like the robots above. Other common attributes often (but not always) assigned to agents are: • computational intelligence (i.e. knowledge expressed as rules), • goal-directed behaviour, and • the ability to interact with other agents. A common finding in agent models is that the behaviour of a large-scale system emerges from the properties and interactions of many individual agents.

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Fig. 2.10  Emergent organisation in a simulated ant colony. The pictures show a cross-section of a landscape, with the sky shown in white and the earth in grey. The ants (black figures) wander around the landscape and pick up any loose sand particles they find. The activity of the ants transforms the initially flat landscapes (left) into a system of hills, valleys and tunnels (right). (Based on [18])

2.3.3  The Boids and the Bees In the early 1980s, two Dutch investigators, Paulien Hogeweg and Brian Hesper, began to look at what happens in systems where many simple agents interact. One of their earliest successes was a simulation model of bumblebee colonies [11]. In this model, the bees are represented as agents that behave according to a set of simple rules. There was no set plan of action for each bee. They simply obeyed what Hogeweg and Hesper called the TODO principle. That is, they moved around, and in doing so, they simply did whatever there was to do in their scheme of behaviour. What Hogeweg and Hesper found was that the social structure that is observed in real colonies also emerged in the model. It is a result of interactions among the virtual bees with each other, and with their environment. The above result is important. It shows that order can emerge in a system without any planning or design at all. Bees are not intelligent planners. They have no concept of what the overall structure of a colony should be. Instead, the order is a by-­ product of the complexity in the system. It emerges out of a multitude of interactions. Subsequent research has extended the bumblebee result to many other kinds of animals, from simulated ants to virtual birds (“boids”). Ants, for instance create nests by wandering around and obeying simple rules, such as, “if you find a stray egg pick it up; if you find a heap of eggs then dump the egg you are carrying” (Fig. 2.10). By obeying this simple rule, the ants quickly sort the eggs into piles.4 As we saw earlier, ants achieve many feats of organisation, such as foraging for food, by following other sets of simple rules. These artificial life studies of bumblebees, ants and other systems have shown that intricate forms of order can emerge from relatively simple interactions between

 This process, known as stigmergy, involves positive feedback (see Sect. 5.1).

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Fig. 2.11  Outputs from the Crown of Thorns starfish model for several scenarios at Davies Reef, off Townsville Queensland. (a) The underlying reef map is a satellite image of Davies Reef, which has been classified into zones by depth of light penetration. Infested areas are marked in solid black: (b) starfish (dark dots) move around at random; (c) the starfish prefer areas of highest coral cover

organisms with each other and with their environment (cf Chap. 9). There is no need for an overall plan. Global organisation is simply a by-product of local interactions. This idea of emergent behaviour is extremely powerful. It has implications for many areas of activity. Brooks for instance extended the idea to robotics, where he showed that central intelligence was not necessary for coherent behaviour. In computing, it has encouraged the spread of research on multi-agent systems as a paradigm for addressing complex problems. The ant sort mentioned above, for example, is a simple algorithm that is used for organising incomplete information. Computer scientists are beginning to emulate other aspects of behaviour too. For instance, leaving pheromone trails is now seen as a useful way to get search agents to assist each other in their role of simplifying knowledge discovery on the internet. Another form of coordination occurs when groups of animals move through the environment. Some simple kinds of aggregate behaviour emerge because of the way that individuals interact with their environment. Figure 2.11 shows outputs from a simple model of starfish outbreaks on a coral reef. The interesting behaviour in this case arises from the ways in which the starfish, represented as a set of independent agents, interact with the coral, which is represented as a grid. 5 A crucial property is the percentage of coral cover at each location. At each site, the kind of substrate (reef crest, lagoon, etc.) defines the maximum possible cover, and the coral grows and spreads until it reaches that level. Meanwhile the starfish are agents that can move freely around the reef eating coral (i.e. reducing the amount of cover at each location). The crucial behavioural parameter is the rule that determines where the starfish move. Simple changes to the rule result in completely different patterns in the outbreak. If we assume that the starfish move completely at random, then they quickly spread all over the reef. If we assume that they actively prefer areas of high coral cover, then they spread out along the reef crest, but do not move into other areas.  We will discuss this kind of model, which is known as a cellular automaton, in the next chapter.

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Fig. 2.12  Examples of flocking behaviour in a boid simulation of birds

Fig. 2.13  Behaviour of a simulated school of fish. The behaviour of the school arises because of interactions among the fish, with each individual reacting according to simple rules. Here a school spontaneously divides and regroups to avoid a predator (centre). (Based on [21])

The important lesson we can draw from the starfish model is that seemingly minor changes in our assumptions about starfish behaviour can lead to enormous changes in the global behaviour of the system. Another insight the model provides is that interactions, whether agent to agent or agent to environment, are important. In many systems, interactions within aggregates of agents, such as those we have just seen, become the main force driving the system. Bees move in swarms, birds in flocks (e.g. Fig. 2.12). Fish swim in schools, and zebras move in herds. In a series of simulation studies, Craig Reynolds showed that central coordination was not needed to explain many of the group manoeuvres (e.g. avoiding predators) that biologists have observed [21]. For instance, when attacked by a predator, a school of fish will perform a number of kinds of manoeuvres such as splitting and reforming, as the predator passes through their centre (Fig.  2.13). Reynolds’ boids model showed that observed patterns of this kind could be produced by simple rules

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governing the behaviour of individuals and the way they interact with other animals nearby. For instance, basic flocking behaviour arises from the following three rules. 1 . Keep as close as possible to the centre of the group. 2. Avoid crowding too close. 3. Aim in the same direction as the rest of the group. Whether it is a flock of birds, a school of fish, or an ant colony, in each of the above examples we see that the whole emerges from interactions amongst the individuals. In later chapters we will see that many kinds of ecological patterns emerge from interactions of one kind or another. In the next chapter, we will see how patterns in landscapes can emerge this way.

References 1. Anselme P, Güntürkün O (2019) How foraging works: uncertainty magnifies food-seeking motivation. Behav Brain Sci 42(e35):1–59 2. Bartumeus F, Campos D, Ryu WS, Lloret-Cabot R, Méndez V, Catalan J (2016) Foraging success under uncertainty: search tradeoffs and optimal space use. Ecol Lett 19(11):1299–1313 3. Bode NW, Wood AJ, Franks DW (2011) The impact of social networks on animal collective motion. Anim Behav 82(1):29–38 4. Charles-Edwards DA, Doley D, Rimmington GM (1986) Modeling plant growth and development. Academic, North Ryde 5. Chmait N, Dowe DL, Green DG, Li YF (2019) Simulating exploration versus exploitation in agent foraging under different environment uncertainties. Behav Brain Sci 42:e39 6. Chmait N, Dowe DL, Li YF, Green DG, Insa-Cabrera J (2016) Factors of collective intelligence: How smart are agent collectives? Proceedings of the Twenty-second European Conference on Artificial Intelligence IOS Press, Amsterdam, p 542–550 7. Dunbar RIM (1998) Grooming, gossip and the evolution of language. Harvard University Press, Cambridge 8. Dunbar RIM (2013) Primate social systems. Springer, Berlin 9. Fiorucci AS, Fankhauser C (2017) Plant strategies for enhancing access to sunlight. Curr Biol 27(17):931–940 10. Hernández-Orallo J, Dowe DL (2010) Measuring universal intelligence: towards an anytime intelligence test. Artif Intell 174(18):1508–1539 11. Hogeweg P, Hesper B (1983) The ontogeny of the interaction structure in bumblebee colonies: a MIRROR model. Behav Ecol Sociobiol 12(4):271–283 12. Larralde H, Trunfio P, Havlin S, Stanley HE, Weiss GH (1992) Territory covered by N diffusing particles. Nature 355(6359):423–426 13. Lindenmayer A (1968) Mathematical models for cellular interaction in development. J Theor Biol 18(3):280–315 14. Mehlhorn K, Newell BR, Todd PM, Lee MD, Morgan K, Braithwaite VA, Hausmann D, Fiedler K, Gonzalez C (2015) Unpacking the exploration–exploitation tradeoff: a synthesis of human and animal literatures. Decision 2(3):191–237 15. Monk CT, Barbier M, Romanczuk P, Watson JR, Alós J, Nakayama S, Rubenstein DI, Levin SA, Arlinghaus R (2018) How ecology shapes exploitation: a framework to predict the behavioural response of human and animal foragers along exploration–exploitation trade-offs. Ecol Lett 21(6):779–793 16. Papert S (1973) Uses of technology to enhance education. LOGO Memo No 8 MIT Artificial Intelligence Laboratory, Cambridge

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17. Pinter-Wollman N, Hobson EA, Smith JE, Edelman AJ, Shizuka D, de Silva S, Waters JS, Prager SD, Sasaki T, Wittemyer G, Fewell J, McDonald DB (2014) The dynamics of animal social networks: analytical conceptual and theoretical advances. Behav Ecol 25(2):242–255 18. Poskanzer J (1991) Xantfarm – simple ant farm for X11. http://www.acme.com/software/xantfarm/ Accessed 27 Dec 2019 19. Prusinkiewicz P, Lindenmayer A (1990) The algorithmic beauty of plants. Springer, Berlin 20. Quadros AL, Barros F, Blumstein DT, Meira VH, Nunes JAC (2019) Structural complexity but not territory sizes influences flight initiation distance in a damselfish. Mar Biol 166(5):65–71 21. Reynolds CW (1987) Flocks herds and schools: a distributed behavioral model. Comp Grap 21(4):25–34 22. Rimmington GM, Alagic M (2007) From Modeling foliage with L-systems to digital art. In: Bridges Donostia: mathematics, music, art, architecture, culture. Tarquin Publications, pp 269–276 23. Tan Y, Zheng ZY (2013) Research advance in swarm robotics. Def Technol 9(1):18–39 24. Tao Y, Börger L, Hastings A (2016) Dynamic range size analysis of territorial animals: an optimality approach. Am Nat 188(4):460–474 25. Viswanathan GM, Buldyrev SV, Havlin S, Da Luz MGE, Raposo EP, Stanley HE (1999) Optimizing the success of random searches. Nature 401(6756):911–914

Chapter 3

Complexity in Landscapes

Abstract  Complexity often arises in the way things are distributed in a landscape. Sampling is subject to scale and can display properties of fractals. Cellular automata, which represent a landscape as a grid of sites, are often used to model processes in landscapes. These models highlight the phase change that occurs between connected and fragmented landscapes. Keywords  Cellular automata · Connectivity phase · Criticality · Fractal dimension · Game of life · Geographic information · Landscape complexity

Fire in a forest understorey, southern Australia. In this chapter we will see how the spread of fires depends on the connectivity of fuel across the landscape

© Springer Nature Switzerland AG 2020 D. G. Green et al., Complexity in Landscape Ecology, Landscape Series 22, https://doi.org/10.1007/978-3-030-46773-9_3

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In the previous chapter, we saw that interactions among organisms lead to complexity in ecosystems. Interactions between organisms and their environment also contribute. Soil formation, for instance, involves a combination of organic and inorganic processes. Rainforests not only depend on local climate, they also modify it. Flow of water across a landscape affects where plants can grow. But the story does not end there. Landscapes themselves can be complex. Many biological and physical processes lead to interactions between different parts of a landscape. Plants and soils affect runoff and stream patterns. But the flow of groundwater also shapes the landscape itself. Lakes, hills and streams all affect the movement of plants and animals. In this chapter, we look at the nature of the complexity that can exist within landscapes, as well as some of its implications for conservation and management.

3.1  The Eye of the Beholder First things first. Before we can talk sensibly about landscapes and landscape patterns, we must first have some way of representing space, and objects in space. What is more, it is essential to understand the assumptions that underlie ways of modelling landscapes. These assumptions have a bearing on the properties that emerge and on the validity of any conclusions that we might draw from models. Let’s start with location. To indicate any point in space, we use its coordinates. Latitude denotes its deviation (north or south) from the equator; longitude denotes its deviation (east or west) from the Greenwich meridian. Finally, elevation denotes the height of the point above mean sea level. As just indicated, the reference points for the coordinates, where the coordinates take the value of zero, are the equator, the town of Greenwich, England, and mean sea level. So far so good. Now that we can say where things are in space, we have to be able to indicate what they are and when they are. To do this we will adopt the ideas that serve this purpose on computers. The standard way of storing and using data about landscapes is a Geographic Information System (usually called a GIS for short). Many GIS systems are now online (e.g. Chap. 10). Arguably, the best known and most widely used is Google Maps [9].

3.1.1  Geographic Information Systems The way we represent landscapes is reflected by the ways that geographic information systems (GIS) store and transform data about landscapes. A GIS organises data about landscapes into layers. A layer is a set of geographically indexed data with a common theme or type. Examples of themes might include coastlines, roads, topography, towns, and public lands. Geographic layers come in three distinct types: vector layers, raster layers, and digital models.

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Vector layers consist of objects that are points (e.g. towns, sample sites), lines (e.g. roads, rivers) or polygons (e.g. national or state boundaries). Data of this kind are usually stored in database tables, with each record containing attributes about individual objects in space, including their location and point in time. Raster layers consist of data about sites within a region. Examples include satellite images and digitized aerial photographs. The region is divided into a grid of cells (called pixels when displayed as images), each representing an area of the land surface. The layer contains attributes for each cell. For instance, in a satellite image the attributes might be a set of intensity measurements at different light frequencies, or a classification of the land features within the cell (e.g. forest, farmland, water). Digital models are functions that compute values for land attributes at any location. For instance, a digital elevation model would interpolate a value for the elevation, based on values obtained by surveying. In practice, digital models are usually converted to vector or raster layers when they are to be displayed or otherwise used. In the good old days, map layers were literally sheets of clear plastic that had various features drawn on them. Today, they are more likely to be virtual layers that exist inside a computer and become visible only when drawn on a computer screen, or printed out on paper. There is a big difference between a digital map and a GIS layer. In a digital map, the data are simple coordinates for drawing dots, lines and labels. There may be no connection between the different parts of the data. A river, for instance, might be just a set of disconnected wiggly lines. There may be no indication that they all form part of the same object (the river) and there may be intentional gaps in the lines to allow space to draw roads or to print names. The locations recorded for towns may not be the true locations, but instead indications of where the label is to be plotted. The actual site of a town would be given by a separate dot, with no indication of its link to the label. In contrast, in a GIS, the data are organised into objects. So the record for a river would include both its name and the vector data for drawing it. The basic tool in GIS is the overlay. That is, we take two or more data layers and lay them on top of one another. To form a map, a GIS user selects a base map (usually a set of key layers, such as coastlines or roads, at a given point in time) and overlays selected layers on top of it. The process of overlay often involves the construction of a new data layer from existing ones. For instance, if we overlay park boundaries on top of forest distributions, then we can create a new map showing areas of forest within parks. Both vector and raster representations of plant distributions have advantages and drawbacks. Vector models (which represent individual plants as points in space) can require a great deal of computer time and are therefore difficult to apply to problems on large scale or heterogeneous landscapes. Raster models, in which “pixels” represent areas of the landscape, are less precise but more tractable for computation. They also reflect major sources of large-scale data, such as satellite imagery, which consists of arrays of pixels.

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3.1.2  The Game of Life One of the first Alife models, and perhaps the most famous, is the Game of Life, devised by the Cambridge mathematician John Conway [8]. LIFE is a simple 2-D analogue of basic processes in living systems. The game consists in tracing changes through time in the patterns formed by sets of “living” cells arranged in a 2-D grid. The game demonstrates several general features that are common in systems of many agents: simple rules impose order (Fig. 3.1), but at the same time are capable of producing very rich behaviour (Fig. 3.2). The game is a cellular automaton model in which each cell exists in one of two states: “alive” or “dead”. Likewise, the rules of behaviour mimic various life-like properties. The neighbourhood of each cell consists of the eight cells in the 3 × 3 box immediately around it. The model goes through a sequence of generations in which the states of the cell progressively change. The rules governing the transition from one generation to the next are simple: 1 . If a living cell has less than two living neighbours, then it dies from loneliness; 2. If a living cell has more than three living neighbours, then it dies from overcrowding; 3. If a dead cell has exactly three living neighbours, then a birth occurs. The Game of Life proved extremely popular for many years, especially amongst computer programmers, and many versions are widely available. It was the first program to demonstrate what were later to be recognised as common features of cellular automata, and of complex systems in general. The most telling of these features relate to the final state of the system. In a finite universe, any random starting configuration of live cells eventually gives rise to just two kinds of end results (both represented in Fig. 3.2b):

Fig. 3.1  Some of the complex self-replicating forms that can be generated in the Game of Life

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Fig. 3.2  The Game of Life. (a) The chaotic initial state of the system. (b) Its state after many generations. Note the complex, but non-random patterns that have been generated over time

1 . static patterns that never change; 2. cyclic patterns that repeat at a constant interval. These patterns are attractors in the model (see Chap. 5). The early patterns, and most of the interest in the game, centre on the transient behaviour that it undergoes before settling into these attractor states. The above observations about the Game of Life led to research into the computational properties of many kinds of cellular automata, most notably by Stephen Wolfram [43, 44] and later by Christopher Langton [19]. The studies revealed four distinct classes of behaviour: fixed, periodic, chaotic and complex. These behavioural attractors are related to equilibria and limit cycles in dynamic systems. Any finite cellular automaton must ultimately fall into a fixed state or cycle. This is because a cellular automaton consisting of N cells, with a possible S states each, has a total of NS possible configurations of cell states overall. So it is inevitable that the cellular automaton must eventually repeat some earlier configuration after at most NS generations. The behavioural richness of a cellular automaton is related to the number of neighbourhood configurations that force a state change in the cell at the centre of the neighbourhood. In the Game of Life, for instance, there are 512 possible configurations for a cell and its eight neighbours. Of these, the game’s rules force the state of the cell to change state in 216 cases. In general, models that force a change of state for very few configurations tend to freeze into fixed patterns, whereas models that change the cell’s state in most configurations tend to behave in a more active ‘gaseous’ way, that is, fixed patterns never emerge. Systems with intermediate levels of

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activity usually exhibit the most interesting behaviour. For instance, it has been shown that the Game of Life is capable of universal computation. The Game of Life is the prototypic cellular automaton. It has served as the inspiration for the investigation of many other cellular automaton systems and for many of the applications we look at in other chapters, including cellular automaton models of fire spread, epidemics, forest succession. All of these are Alife models.

3.1.3  Cellular Automata Models of Landscapes Now that we have a way of representing things in the landscape, we need ways of using that data to simulate the processes that occur there. Each of the three types of data layers that we described above are sometimes used for modelling. However, models based on raster data are perhaps the simplest and most common. Here, we look at ways in which models of landscapes are built using raster data. In Chap. 10 we shall look at the technology that underlies the modern development and use of geographic information, as well as the modelling tools that make use of them. In a raster-based model, we represent the landscape as a rectangular array of cells (Fig. 3.3). Each cell contains information about the area that it represents. The figure shows an example in which the cells are shaded by landscape category, including fields, trees or roads. In this example, the amount of fuel found at each site is given by a separate table (not shown), which lists the amount of fuel associated with each category. However, in a model of this kind, the cells could contain many other kinds of data, such as vegetation type, altitude, slope, geology.

Fig. 3.3  A raster-based model of fire spread across a landscape. (a) The landscape map, taken from a digitized aerial photograph, is a rectangular grid of cells that are here coloured to indicate their contents. The white lines are roads; the black spots are trees, and the shaded areas represent different kinds of land cover, mostly fields and grasslands. The stippled L-shaped object just North of the road near the Southern part of the South Eastern quadrant, represents a house. (b) A scenario in which a fire starts from a cigarette butt thrown from a car under hot, dry weather and is driven by a strong northerly wind. The house has just been destroyed. The uniform grey area indicates locations that have been burnt out, and the bright area surrounding the grey area shows the fire front. (c) Same scenario but testing the effectiveness of cutting a break across the fire’s path

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Raster models generally use the formalism of cellular automata to simulate plants in landscapes. A cellular automaton is an array of interacting cells, with each cell behaving according to the same program [43]. The essential features of a cellular automaton (“CA” for short) are as follows: • its state is a variable that takes a separate value for each cell; • its neighbourhood function defines the set of nearby cells that each cell interacts with. In a grid, the neighbourhood normally consists of cells physically closest to the cell in question (see Fig. 3.4); • its program is the set of rules that define how each cell’s state changes in response to its current state, and those of its neighbours. Cellular automata models of landscapes consist of fixed arrays in which each cell represents an area of the land surface [11]. The states associated with each cell correspond to environmental features, such as coral cover or topography. This approach is compatible with both pixel-based satellite imagery and with quadrat-based field observations. It also makes it relatively easy to model processes, such as dispersal or fire [16], in heterogeneous landscapes (Fig. 3.3). The most important feature of the CA model is the role of the neighbourhood (Fig. 3.4). Any cell, taken in isolation, behaves in a certain simple way, just like a tree that grows by itself in a glasshouse. However, just as trees in a forest interact with each other to produce a rich variety of growth, form and dynamics, so it is the interactions of each cell with its neighbours that dominate the behaviour of a cellular automaton model. The behaviour of Game of Life is typical of the way in which many cellular automata reproduce features of living systems. That is, regularities in the model tend to produce order. Starting from an arbitrary initial configuration, order usually emerges fairly quickly in the model. This order takes the form of areas with well-­ defined patterns. Ultimately most configurations either disappear entirely or break up into isolated patterns. These patterns are either static or else cycle between several different forms with a fixed period. (See Fig. 3.2)

Fig. 3.4  Examples of neighbourhoods in two different cellular automata (CA). The two diagrams shown represent a typical cell (shaded) in a CA grid. The blue arrows indicate the nearby cells that form part of the shaded cell’s neighbourhood. As shown by the arrows, in the CA at left, each cell has 4 neighbours (known as rook’s rule or NSEW); in the CA at right, each cell has 8 neighbours

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Notice that CA models adopt the raster representation of landscapes, which we introduced in the previous section. The alternative vector representation is embodied in what are known as agent-based models. One important difference between the two kinds of model is a matter of scale. By definition, individual-based models act at the scale of individual plants and animals. In contrast, the states in a CA model summarize the contents of an entire area. Both the size of the area and the detail is variable. Even if they are set up to simulate exactly the same process, cellular automata models and individual (agent based) models of (say) forest dynamics can produce somewhat different results [20]. Ecologists have applied cellular automata models to many problems in landscape ecology. One common application has been to identify the way in which particular kinds of spatial patterns form. For example, a team of scientists used CA models to look at the vegetation patterns that resulted from the combination of tree growth rates and the killing capacity of the wind in the subantarctic forest of Tierra del Fuego [29]. They were able to show that simulated patterns for heterogeneous forests with random age distributions matched the patterns observed in nature. Similarly, CA models have been used to examine the formation of wave patterns in the heights of trees [33]. This study, by Sato and Iwasa, found that both “absolute height (or age) and the height difference between neighbours affect tree mortality”. Another study used CA models to examine interactions and vegetation degradation on a contamination gradient [41]. We shall see many more examples of the application of cellular automaton models to ecological questions, below and in later chapters.

3.2  Sampling and Scale What is the point of sampling? A simple question, and yet it is one that is all too easy to forget. Collecting data and samples from different locations in a landscape is one of the most basic jobs in ecology. However, ecologists often resort to some standard pattern of sampling without considering whether it really will answer the crucial questions. In landscapes, the patterns that we see are reflections of the processes that produced them. Dry hilltops and lush valleys reflect the movement of water and nature of soils. The distributions of plants and animals arise from a multitude of processes that we have to tease apart. Sampling methods stem from these assumptions. Now as we have seen, a lot of traditional theory in ecology is based on reductionist models of cause and effect—plant A grows at site X because of soil type Y—often this assumption is correct, but not always. In keeping with the two main kinds of geographic data layers (vector and raster) introduced in the previous section, there are two ways of collecting spatial data. We can record objects and note their location (vector data), or we can record locations, and note their features (raster data). Both approaches are used in ecology, but the latter is more common in field studies. Ecologists have normally used vector data to

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map large-scale features, such as lakes and streams, or areas of different vegetation type, such as swamp, grassland or forest. They also use vector data to record animal territories. Vector (point) data have sometimes been recorded during intensive studies of spatial pattern, such as mapping every tree within part of a forest. Traditional sampling methods in ecology are designed to provide the maximum amount of information given limited time and resources. The most common methods of sampling landscapes are quadrats and transects. A quadrat is a square within which an ecologist takes samples or records observations. She might, for instance, record every plant within the quadrat, or estimate biomass. The size of quadrats varies according to the degree of spatial variation within the landscape. Metre square quadrats are common in field ecology, but in specialised studies they may vary in size from less than a millimetre to hundreds of kilometres. A transect is a set of quadrats in sequence across a landscape. Transects are often used to sample some quantity that is known to vary systematically across a landscape, such as elevation, water table depth or nutrient levels. Systematic changes like these create environmental gradients, which provide natural experiments that tease out the effects of factors such as elevation. Later we shall see that increasing numbers of studies are examining complexity gradients to understand the effects that increasing complexity has in ecosystems [2, 5, 7]. The usual intent of quadrat sampling is to obtain a sample that is representative of an entire landscape. To allow for random variations, lots of quadrats may be needed. Transects are a way of taking into account systematic variations, such as differences in soil moisture down a slope. In essence, these methods cater for reductionist interpretations. They assume uniformity and attempt to eliminate known sources of non-uniformity. However, it may be that non-uniformity itself is the most telling pattern of all. Ecologists are well aware that individual interactions can be important. A large tree may block the light of a smaller one, or compete for soil moisture. With this in mind, another common sampling method is to consider nearest-neighbours. In the course of quadrat sampling, or monitoring individual trees, it is helpful to know how nearby trees affect the tree concerned. The nearest neighbour is the plant closest to the one under study. The essential problem of landscape complexity is that as often as not, complex processes leave behind complex patterns. To interpret a complex pattern often requires a lot of data. You may have to map features in detail, and on different scales. Sometimes it is simply not feasible to gather detailed information about an entire landscape. Quadrat samples are a compromise between what is desirable and what is possible. They allow field workers to put together a picture of an entire landscape at the cost of reasonable time and effort. Fortunately, new technologies, especially airborne or satellite remote sensing, are making it possible to map entire landscapes, even if some of the fine scale detail may be lost. As we shall see later in this chapter, methods are now available to interpret aspects of the complexity that we see in landscape patterns.

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3.3  Complexity in Spatial Processes Spatial processes are inherently complex. Water percolates through the soil. Wind carries seeds to fresh territory. Chestnut blight spreads from tree to tree. In almost every case, spatial processes involve interactions between objects at different locations in a landscape. The patterns that we see in landscapes are often like frozen memories of the past. Fly over the landscape of eastern Canada, Finland, or Russia, and what strikes you the most are the lakes. There are thousands of them: reminders of the distant Ice Age, they break up the landscape and have immense influence on the ecology of those regions. Likewise, if you fly over the scrublands of Western Australia, you see striations in the vegetation, which are the results of fires burning parallel to the sand dunes. To understand patterns such as these, we can model the processes that lead to them. One important class of processes is percolation. Percolation involves movement of a percolute through a surface or medium. Water seeping through cracks in rocks is one example. Several common landscape processes, both physical and biotic, are essentially percolation. These include the spread of epidemics, wildfire, pestilence, invasion of exotic species, diffusion of soil, water and nutrients, and the spread of new genotypes through a population. Epidemic processes assume that a disturbance spreading across a landscape follows the path of least time from its starting point to any arbitrary location. The cellular automaton representation of landscapes described above readily lends itself to modelling epidemics, and other cases of percolation. Here we treat fire spread as an example of an epidemic-like (percolation) process. In this case, we treat the cells in the landscape as packets of fuel. Within a fuel bed, fire spread is a percolation process, in which ignition of one patch of fuel eventually leads other nearby fuel to ignite. Most models of fire spread tacitly assume that the fuel bed is both continuous and homogeneous: there is plenty of fuel and it is spread evenly everywhere. In such cases, the areas burnt by a fire form an ellipse. In the absence of wind, it would be a circle, with the ignition point at the centre. However, wind blows flames and hot gases in one direction, elongating the area into an ellipse. The eccentricity of the ellipse increases with the wind speed. Real fuel beds, however, are heterogeneous and often discontinuous as well. Moreover, they are heterogeneous at almost any scale relevant to modelling fire spread. Whether they consist of trees, bushes, or tussocks of grass, fuels in the real world are normally discrete, not continuous, at the scale of individual plants. Even where litter provides virtually continuous fuel cover, this cover can display immense variations in depth or concentration. At larger scales, changes in vegetation type lead to similar variations in fuel concentration. Fuels that are both continuous and homogeneous are really one end of a spectrum; the other end being fuels concentrated in patches and separated by bare ground. Models of fire spread show that in patchy fuel beds, the area burnt out by a fire can assume many different shapes: anything from a circle to a straight line. Even in

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a completely flat landscape, a fire can burn out some areas completely while leaving other areas untouched. The most important insight to emerge from fire models is that if the fuel in the landscape is too patchy, then a fire will simply go out of its own accord. It does not just burn more slowly; it simply does not spread at all. We will look at exactly why this happens a little later (see Sect. 3.5). A crucial observation about the above property is its relation to weather conditions. Although the fire burns out without spreading if the fuel is too patchy, patchiness is a property that changes with scale. On a very broad scale, fuel is usually continuous; by large scale here, we mean that the cells in the grid are large in area. If we shrink the scale, so that the cells become smaller, then it takes more and more cells to cover the same area, and more and more of them will be empty of fuel. The role of the weather in this question is that it effectively alters the scale. We stated earlier that a burning piece of fuel affects an elliptical area around it. If the air is hot and dry, then the fuel will be drier to start with, so it will burn faster and hotter and both radiant and convective heat will spread further. So the affected ellipse will be large. If the weather is cool and the fuel is moist, then the size of the ellipse will be small. But if the elliptical area affected by a fire shrinks, then that change is equivalent to shrinking the scale. This means that fuel is effectively much patchier. If the weather conditions are too mild, then the fuel becomes so patchy that it falls below the threshold needed for the fire to spread. This observation means two things. Firstly, fires don’t spread slowly in really mild conditions, they simply go out. Secondly, they go out, not because the cold weather puts them out, but because the fuel becomes too patchy for them to spread. We should point out that models of fire spread have a wide range of practical uses. They can be used to test scenarios for fire control (Fig.  3.3), so that fire-­ fighters are prepared for any contingency. For example, these models allow planners to estimate the risk of fire spreading through an existing landscape, and to design spatial environments to minimise fire risk. However, in general, simulation models of this kind are less useful for real-time fire-fighting. The reason for this is that it is next to impossible to gather the detailed data needed to run the model. A model of the landscape may easily contain 10,000 or more cells in the model grid. Given the potential importance of fuel heterogeneity, it would be desirable to obtain detailed data for the fuel load of each cell. Not only is that impractical in most cases, but the minute-by-minute changes in fuel moisture, as well as shifts in wind direction and speed, mean that the model is probably running with false data anyway. Fortunately, dangerous wildfires that need to be contained normally only happen in extreme weather conditions. Fire fighters often comment that in such conditions they do not need a very detailed simulation model. All they need to know is the forward rate of spread. The main use for a simulation in such conditions would be to identify what might happen in various scenarios (Fig. 3.3), such as whether a fire break will contain the burn, or to determine the effect of a sudden wind shift. Many authors have applied cellular automata models to examine aspects of fire behaviour [6, 10, 21, 34, 37]. An important insight that arises from these models is that many spatial processes that appear to be very different often share deep

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similarities. Fire spread, for instance, belongs to a wider class of epidemic processes. Other examples of epidemics include the spread of disease, expansion of invading species, and the spread of insect pests and dieback [18].

3.4  Complexity in Spatial Patterns How long is the coast-line of Great Britain? At first sight this question, posed by the mathematician Benoit Mandelbrot, may seem trivial. Given a map, one can sit down with a piece of string and a ruler and soon come up with a value for the length. The problem is that repeating the operation with a larger scale map yields a greater estimate of the length (Fig. 3.5). If we went to the coast and measured it directly, then still greater estimates would result. It turns out that as the scale of measurement decreases, the estimated length increases without limit. Thus, if the scale of the (hypothetical) measurements were to be infinitely small, then the estimated length could become infinitely large! When discussing measurement, scale can be characterized in terms of a measuring stick of a particular length: the finer the scale, the shorter the stick. Thus at any particular scale, we can think of a curve as being represented by a sequence of sticks (Fig.  3.5), all of the appropriate length, joined end-to-end. Clearly, any feature shorter than the stick will vanish from a map constructed in this way. Of course, no one actually makes maps by laying sticks on the ground, but the stick analogy reflects the sorts of distortions that are inevitably produced by the limited resolution of aerial photographs, by the length of survey transects, or by the thickness of the lines produced by drafting pens. There is an analogy here, too, with the accuracy or frequency with which any sort of biological measurements are made. The dependence of length (or area) measurements on scale poses serious problems for biologists who need to use the results. For example, lakes that have a very convoluted shoreline are known to offer a larger area of shallows in relation to their

Fig. 3.5  Using sticks of different size S to measure the length S × L of a coastline. Shorter sticks capture finer details, so the measured length is greater. A stick of size 2 needs 3 stick lengths to reach from one end of the coastline to the other, so the measured length is 2 × 3 = 6; but a size of size 0.5, gives the length as 10 (= ½ × 20)

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total surface area, and thus support richer communities of plant and animal life. Attempts to characterize shore-line communities in terms of indices that relate water surface to shoreline length have been frustrated by problems of scale. Mandelbrot proposed the idea of a fractal (short for “fractional dimension”) as a way to cope with problems of scale in the real world. He defined a fractal as any curve or surface that is independent of scale. This property, referred to as self-­ similarity, means that any portion of the curve, if blown up in scale, would appear identical to the whole curve. Thus the transition from one scale to another can be represented as iterations of a scaling process (e.g. see Fig. 3.5). In the coastline example, the implication of Mandelbrot’s definition is that as the scale of your measurements decreases, the total distance that you measure increases. So hypothetically, the answer to the question we asked at the beginning is that the coastline of Britain would be infinite in length! An important difference between fractal curves and the idealised curves that are normally applied to natural processes is that fractals are rough. Although they are continuous - they have no breaks - they are “kinked” everywhere. We can characterize fractals by the way in which the representation of their structure changes with changing scale.

3.4.1  Fractal Dimensions The notion of “fractional dimension” provides a way to measure how rough fractal curves are. We normally consider lines to have a dimension of 1, surfaces a dimension of 2 and solids a dimension of 3. However, a rough curve (say) wanders around on a surface; in the extreme it may be so rough that it effectively fills the surface on which it lies. Very convoluted surfaces, such as a tree’s foliage or the internal surfaces of lungs, may effectively be three-dimensional structures. We can therefore think of roughness as an increase in dimension: a rough curve has a dimension between 1 and 2, and a rough surface has a dimension somewhere between 2 and 3. The dimension of a fractal curve is a number that characterizes the way in which the measured length between given points increases as scale decreases. Whilst the topological dimension of a line is always 1 and that of a surface always 2, the fractal dimension (D) may be any real number between 1 and 2.1 For the shoreline shown in Fig. 3.5, the fractal dimension of the coastline is about 1.5. This is a measure of how crinkly it is. Returning to the L-system representations, described in Chap. 2, for plant like structures, the production rule: A → FA will have a fractal dimension of 1 and as the  The fractal dimension D is defined by the formula D =

log ( L2 / L1 )

where L1, L2 are the mealog ( S1 / S2 ) sured lengths of the curves (in units), and S1, S2 are the sizes of the units (i.e. the scales) used in the measurements. For the shoreline shown in Fig. 3.5, measurements for S1 = 1 and S2 = 1/2 give lengths of L1  =  7 and L2  =  20, respectively. So entering these numbers in the calculation log ( 20 / 7 ) gives D = = 1.51 . log ( 2 / 1) 1

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order of branching increases (e.g. A → F[-FL][+FR]FA and A → F[-FA][+FA]FA and so on) the fractal dimension will approach 2 for 2-dimensional patterns. The idea of fractals is built on the assumption that patterns repeat at different scales, but in the real world, this is not necessarily true. Different processes influence patterns on different spatial scales. In this case, if we repeat the calculation for the transition from S = 1 to S = 2, then we find the smaller estimate of D = 1.22; and the transition from S  =  2 to S  =  3 gives D  ~  1.13. In other words, the coastline becomes smoother at larger scales. It is important to realise that the above way of estimating fractal dimension applies only to certain sorts of data. Suppose that we wish to measure fractal properties of, say, the surface of a coral reef. There are two different measurements that we might make. One measurement would consist of measuring distances between two points on the reef with measuring sticks of varying lengths (as in Fig. 3.5). If instead we moved along the same transect and measured, say, the height of the reef surface above the substrate, then we could not measure fractal index in the same way. No curve or surface in the real world is a true fractal; they are produced by processes that act over a finite range of scales only. Thus estimates of D may vary with scale, as they do in the above example. The variation can serve to characterize the relative importance of different processes at particular scales. Mandelbrot called the breaks between scales dominated by different processes “transition zones”. Fractal geometry has become important in many fields. As we shall see in later chapters, it is related to several important ideas about complexity, such as chaos (see Chap. 4). The repeating nature of fractal patterns is intimately related to basic computation, which consists of repeating operations. It also ties fractals closely to iterative processes in nature, such as cell division. Branching patterns arising during plant growth, for example, are inherently fractal in nature, as we saw in Chap. 2.

3.4.2  Fractals in Nature Many natural structures have fractal properties. In biological patterns the fractal nature arises from the iteration of growth processes such as cell division and branching. Structures with high fractal dimension, such as lungs and branches, are associated with processes that maximise surface to volume ratios. Fractals in nature arise from the action of specific processes. One of the useful insights to be gained from fractals is to help us understand the roughness that we often see in natural things. Landscapes, for instance, are often rough at all spatial scales. This roughness is a result of natural processes, such as climate and weathering, that operate on many different spatial scales. Fractal models capture that roughness. Unlike theoretical models, natural processes operate only over a finite range of scales. For this reason the fractal dimension of many natural structures remains constant only over a limited range. Sometimes there are distinct breaks between

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scales, where one process ceases to become important and another becomes dominant. In coral reefs, for example, coral growth, buttress formation and underlying geomorphology all affect the profile. Each process operates on different scales and the fractal dimension of the reef surface has distinct breaks at the transition from one process to another. Fractal dimension provides a way of measuring the complexity of landscape patterns (e.g. [38]). One application, for instance, has been to examine some of the fractal properties that arise in rainforest gap analysis [22, 23]. For instance, a region with a high fractal index will have a long, convoluted boundary relative to its area. So edge effects are likely to be a major consideration for wildlife conservation in such a region. The increasing use of geographic information systems and satellite imagery to study landscapes and landscape patterns has increased the use of this approach.

3.4.3  Measuring Landscape Complexity Increasing awareness of complexity and its implications has led to a growing body of research into the effects of complexity in landscapes. To undertake such research requires a systematic, usually quantifiable way of representing complexity. How complex is a landscape? A practical challenge facing any study involving the complexity of ecosystems or landscapes is how to measure complexity. In a simulation model, you can follow entire regions and every organism and follow the richness of their interactions. In a real ecosystem, however, gathering such detail is usually prohibitive. For this reason, most studies use features that can stand as proxies for complexity. One proxy feature is diversity: adopt the same method that ecologists do for species and use the diversity of sites as an indicator of landscape complexity. Thus the simplest approach is to count the number of different classes of sites we find in a landscape. However, we first need to understand the data we have about landscapes. Suppose that we have a map of a landscape in the form of a grid, such as a pixel image provided by a satellite, or the array of cells in cellular automata models. In a map of this kind, each cell (or pixel) corresponds to a particular area on the ground. To make an image, we can classify each cell according to its contents, producing a set of classes, such as rainforest, temperate forest, grassland, agriculture. At a large sale, this crude measure would give an idea of differences in the richness of land types. As we saw for species, we can measure the variety of land types found in a region. This uses the Shannon entropy index, which takes into account the relative proportion of different classes of landscape (Fig. 3.6). If all classes are equally represented (Fig. 3.6a), then the entropy measure takes the maximum possible value. For the eight classes shown in the figure, this is log2 8, which equals 3. However, if there is an imbalance, such as a single landscape class occupying a large area, then

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Fig. 3.6  Site diversity in a landscape. Here, cells are coloured to represent scattered distributions of sites that fall into eight classes of land type. (a) Equal numbers of all eight classes. (b) One class (white cells) occupies 50% of the region

the entropy measure is less. In the case shown in Fig. 3.6b, one class occupies ~50% of the area, so the resulting value of the entropy measure is 1.46. One drawback, however, is that considering only proportions does not take into account how different land types are distributed within a region. A refinement of the Shannon index is to consider how different land classes are arranged in relation to one another. This mutual information, or joint entropy,2 measures the variety of combinations of classes found within the landscape [27]. Figure 3.7 shows the differences for two distributions of eight land classes. One distribution has a uniform pattern, whereas the other has cells of each class scattered at random across of the landscape. In both cases, every class occupies the same proportion of cells. The distribution makes no difference to the Shannon entropy measure: which is 3.0 for both distributions. However, the adjacency entropy differs greatly: 3.1 for the uniform distribution versus 6.0 for the scattered distribution. Note that the curves shown in Fig. 3.7 represent the maximum possible value in each case. If the areas covered by different landscape classes are unequal, then the values will be lower in every case. As we have seen, complexity implies a high degree of local interaction, but it is not always clear what those interactions are. A common approach to measuring landscape complexity is to look at structural complexity, especially the richness of habitats or land cover types, as well as their fragmentation, combinations and variations. Metrics of this kind are widely used in studies of complexity gradients. For instance, a study of fish in a Brazilian river found that the diversity of fish was greatest in patches where the structural complexity of habitats was at intermediate levels [5]. If the complexity was low, then predation reduced diversity. If complexity was

 The formula is E joint = ∑∑ p ( i,j ) log2 p ( i,j ) , where i and j are landscape classes.

2

i

j

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Fig. 3.7  Entropy measures of two extremes in the distribution patterns of landscape cover types. (a) & (b): Computer generated examples of two landscape grids, each containing eight land cover classes, which are represented by different colours. (c): The graph shows how the two metrics vary as the number of classes increases in uniform and scattered distributions. In each case, the proportion of all classes are equal. The Shannon entropy measure (solid line) is the same for both uniform and scattered distributions. For a uniform distribution, the joint entropy measure is close to Shannon’s value; however, it is much greater when cells of the different classes are scattered randomly (dotted line)

high, then interstitial spaces between vegetation restricted fish size and oxygen concentration was sometimes low. Many studies assume that edges between different landscape types imply interactions. So the richer the edges, the more complex the landscape. The particular features or edges used depends on the type of issue under study. For instance, for issues concerning interactions between different kinds of land use, borders between pastures and forest or grasslands may be the key source of complexity [2, 7, 36]. Some studies have tried to identify suites of environmental features that are likely to interact in unpredictable ways. Ecosystem restoration, for instance, often focuses on geomorphic complexity, on the assumption that increasing its complexity will lead to heterogeneous habitats, thus achieving greater biodiversity. One

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study of the outcomes of stream restoration projects analysed results of 20 streams. The researchers identified five different “dimensions” of complexity and measured 29 different indicators (e.g. heterogeneity in sediment size, variation in stream profile) that contributed to them [17]. Their results showed that focussing on particular features led to increased biodiversity for different groups of organisms.

3.5  Are Landscapes Connected? 3.5.1  Connectivity in a Grid The issue of landscape connectivity has been a question of increasing concern in conservation. If individuals (or gametes of individuals) cannot come into contact with one another, they cannot breed. If a landscape is fragmented, then barriers to movement between patches may reduce the ability to find enough food. But what does “connected” mean in a landscape? We can define a set of sites in a landscape as connected if there is some process that provides a sequence of links from any one site to any other site in the set. In the CA formalism, connectivity is defined by the neighbourhood function. Two sites are directly connected if one belongs to the neighbourhood of the other. A region in a landscape is connected if we can link any pair of points in the region by some sequence of points (i.e. a path or “stepping stones”) in which each pair of points is directly connected. Before we go any further, let us be clear about what exactly we mean by the word “connected” in ecological terms. Two objects are “connected” if some pattern or process links them. Dingoes, for instance, are linked to kangaroos because they prey on them. The most important sources of connectivity for plants and animals are associated with landscapes. Links within a landscape arise either from static patterns (e.g. landforms, soil distributions, or contiguous forest cover) or from dynamic processes (e.g. dispersal or fire). For instance, a tree growing within a forest can transmit and receive pollen from any other tree that lies within the range of bees or other pollinating vectors. Pollination provides a connection among the trees. We need to make clear several basic aspects of connectivity. First, it is important to realise that a landscape may be connected with respect to one process, but not with respect to another. A river or fence running through the middle of a forest may form a barrier to restrict some processes (e.g. fire or movement of animals) but not others (e.g. movement of bees or wind-borne seeds). Secondly, we need to distinguish between the connectivity of any two objects (e.g. two trees) and the connectivity of an entire system. For instance, a tree at one end of a forest may not be able to pass pollen directly to trees at the other end of the forest, but over a period of generations its genetic information can flow throughout the forest. In a patchy environment, when is a landscape connected and when is it not? To answer this question, we can simulate a patchy landscape by using a grid in which

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each cell represents an area of the land surface [11, 12, 14, 15]. We classify each cell according to (say) the dominant vegetation within the area it represents. Now suppose that there is a random distribution of (say) rainforest within the region represented by the grid. Let us assume that two cells containing rainforest are connected if they are adjacent to one another. In biological terms this assumption means that some process that we are interested in (e.g. dispersal, animal migration) links adjacent cells. Let us define a patch of rainforest as a set of rainforest covered cells in which we can move anywhere within the patch via a sequence of adjacent cells. In this system we find that the connectivity of sites in a landscape falls into three distinct phases: disconnected, critical, and connected (Fig. 3.8). If the density of rainforest sites is sub-critical, then the landscape is broken into many isolated sites and small patches. If the density is critical then a single large region may be connected (shaded cells in Fig.  3.8), but much of the landscape remains as small isolated patches. If the density is super-critical, then almost the entire landscape is connected, with few isolated sites remaining (Fig. 3.8). The relationship of the above results to other kinds of criticality [1] and to percolation theory is well known [35, 42]. As the name implies, percolation is about the flows of percolutes through a surface or medium. As we saw above, the ability of a percolute to spread through a medium depends on the formation of “edges” within a lattice, and is usually determined by density. A phase change occurs when a critical density is reached. It has been shown that all of these criticality phenomena stem from underlying properties of graphs (sets of nodes and edges) [13, 14]. Perhaps the most striking implications of the above phase changes concern epidemic processes, as in the following examples. • Fire spread depends on the density of fuel being greater than a critical level; otherwise the fire quickly dies out [16]. The critical density is a function of environmental factors such as temperature and fuel moisture. This phenomenon has been confirmed in both laboratory experiments [26] and field studies [4]. • Simulation studies of the spread of Crown of Thorns starfish outbreaks on the Great Barrier Reef suggest that reef-to-reef spread relies on inter-reef connectiv-

Fig. 3.8  Density and connectivity of sites in a landscape. As the percentage of covered sites increases, they link to form patches: at high density a single patch encompasses the entire area

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ity, as defined by the dispersal of starfish larvae between reefs being above a critical threshold [3]. • The invasion of a new species in a region depends on there being a critical density of sites available for colonisation [12]. Many weeds and pests are exotic species that perform best in cleared areas. Therefore the connectivity of cleared areas is as important a consideration as the connectivity of undisturbed habitats. These findings have been applied to determine control scenarios in the spread of exotic diseases such as foot and mouth disease [28]. The key result to emerge from studies of connectivity is that landscapes can exist in two different phases: connected and disconnected (Figs. 3.9 and 3.10) [14]. The variability that occurs at the phase change (Fig. 3.10b) means that the size and distribution of landscape patches become highly unpredictable when the density of active regions is at the critical level.

3.5.2  Why Is a Starfish Like an Atomic Bomb? On November fifth, 1983, the Weekend Australian newspaper carried the following startling headline: Reef park a farce in face of starfish threat.

This item, along with other sensationalist reporting, sparked a frenzied debate, [31] followed by intense research, into the problem of starfish infestations along Australia’s Great Barrier Reef (GBR). The controversy erupted because the Crown of Thorns starfish (Acanthaster planci) was appearing in unprecedented numbers on

Fig. 3.9  Phase change in the connectivity of a cellular automaton grid as the proportion of “active” cells increases (after [14]). Light green denotes active cells; white denotes inactive cells. The proportion of active cells increases from left to right. The dark green areas indicate the largest connected patch of active cells

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Fig. 3.10  Critical changes in connectivity of a CA grid (compare Figs. 3.8 and 3.9) as the density of active cells increases. (a) The number of separate, connected regions in the grid; (b) average size of the largest connected subregion (LCS); (c) variation in the size of the LCS. Each point is the result of 10,000 iterations of a simulation in which a given proportion of cells (in a square grid of 10,000 cells) are marked as active. Note that the location of the phase change (here ~0.59) varies according to the way we define connectivity within the model grid (see Fig. 3.4). The results shown here are based on a four-cell neighbourhood

several reefs. In reef after reef, the entire coral population was being literally eaten away while distraught tour operators could only look on in horror. It took four years of research by ecologists at the Australian Institute of Marine Science in Townsville to uncover what was really going on. The team, led by Roger Bradbury, first identified the pattern of outbreaks. The Great Barrier Reef is not a single reef, but a chain consisting of thousands of individual coral reefs. It runs parallel to the North Queensland coast for over 2,000 km. The team began by surveying hundreds of reefs. Destruction was patchy. Some reefs were totally denuded by the starfish; others nearby were completely untouched. A plot of the outbreaks revealed a travelling wave of outbreaks that moved from north to south. This wave pattern was weakest in the north, where outbreaks seemed to be more sporadic, but became progressively stronger as it moved south. The

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systematic pattern implied a causal relationship. Further work revealed that offshore ocean currents carried larvae from one reef to the next. The starfish give birth to their young as larvae on a single night each year. The sea becomes filled with clouds of tiny larvae. These get caught by ocean currents which carry them to other reefs, where they settle in huge numbers. Then the cycle starts again.3 In atomic fission, a neutron hits a nucleus and splits it. In the process, more neutrons are released. These neutrons then split other nuclei and the process continues in a chain reaction. The effect of starfish on the Great Barrier Reef is very similar. Larvae settle on a reef, setting off a local outbreak. When they become adults, they release more larvae, which currents carry to the next reef. In this way, they set off a chain reaction of outbreaks. In a simulation model of this process [39], the ecologists combined data on currents with a map of the entire Great Barrier Reef. This model duplicated the patterns revealed by surveys and confirmed that the “epidemic” model was indeed correct. The only remaining question was how the outbreaks started in the first place. Obviously the process has to be seeded by outbreaks in the waters north of Cape York, but how? Without massive influxes of starfish larvae, why should an outbreak suddenly start? Is it chance? Or is it something more serious? The most contentious theory is that over-fishing reduces the predators that normally feed on starfish larvae and young starfish. The issue is still hotly debated to this day. The Crown of Thorns starfish provides an excellent example of biological connectivity. The starfish larvae move from one reef to another, setting off outbreaks wherever they settle. More generally, we can consider landscape connectivity in terms of the plants and animals that inhabit it. Many animals, for instance, have a home range or territory within which they roam. That territory defines a relationship that connects every site within it. This kind of analysis is most useful where animals inhabit isolated or fragmented habitats. The assumption is that two habitats are connected if an animal can reasonably move between them. In practical terms, most researchers have considered the distance that an animal might normally move to get from A to B. Long distance transport, migration and freak events are not considered. Rodney van der Ree used this approach to study the distribution of squirrel gliders in fragmented forests that consisted mostly of roadside strips bordering farmlands [40]. The question was whether the gliders would reach isolated trees and patches that were separated by open fields from the roadside cover that formed their chief habitat. He found evidence for a threshold of 75 m. That is, the incidence of gliders visiting trees further than this distance in the course of their normal activity was very small. Some birds can fly great distances in the course of a single day. This can lead to landscape connectivity on a grand scale. For instance, David Roshier and his colleagues looked at the connectivity of water bodies in central Australia for the water birds that inhabit them [32]. They did this by assuming that water birds would not

 By 2019, coral bleaching had replaced starfish as the main threat to the GBR.

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normally fly more than about 200 km in the course of a day’s travel while moving from one lake or swamp to another (cf. Chap. 8). On this basis, they found that during wet years the entire continent was effectively connected. It is important to realise that habitat connectivity will vary from species to species. Just because one species finds a habitat connected does not mean that this is the case for all species. For example, Joanne McLure showed that in the Apennine Mountains of central Italy, the environment is well connected for wolves but not for less mobile species such as bears [24]. These differences between species can influence species richness. In Brazil’s Sao Paulo state, Jean Paul Metzger studied the effect of the area and connectivity of forest fragments on the diversity of trees and vertebrates [25, 30]. He found that forest cover was important at local scales, but, in terms of the whole landscape, connectivity between fragments was a better indicator of species richness. The above studies all considered connectivity patterns evident in the environment. However, populations may be fragmented even in the absence of corresponding landscape patterns. Similarly, environmental conditions may change, so populations may have been fragmented in the past. We will look further at some implications in later chapters.

References 1. Bak P, Chen K (1991) Self-organized criticality. Sci Am 264(1):46–53 2. Birkhofer K, Andersson GK, Bengtsson J, Bommarco R, Dänhardt J, Ekbom B, Ekroos J, Hahn T, Hedlund K, Jönsson AM, Lindborg R, Olsson O, Rader R, Rusch A, Stjernman M, Williams A, Smith HG (2018) Relationships between multiple biodiversity components and ecosystem services along a landscape complexity gradient. Biol Conserv 218:247–253 3. Bradbury RH, van der Laan JD, MacDonald B (1990) Modelling the effects of predation and dispersal on the generation of waves of starfish outbreaks. Mathematical & Computer Modeling 13(6):61–67 4. Caldarelli G, Frondoni R, Gabrielli A, Montuori M, Retzlaff R, Ricotta C (2001) Percolation in real wildfires. Europhys Lett 56(4):510–516 5. Cunha ER, Winemiller KO, da Silva JC, Lopes TM, Gomes LC, Thomaz SM, Agostinho AA (2019) α and β diversity of fishes in relation to a gradient of habitat structural complexity supports the role of environmental filtering in community assembly. Aquat Sci 81(2):38 6. Duarte J (1997) Bushfire automata and their phase transitions. Int J Mod Phys C8(2):171–189 7. Gagic V, Tscharntke T, Dormann CF, Gruber B, Wilstermann A (2011) Thies C (2011) food web structure and biocontrol in a four-trophic level system across a landscape complexity gradient. Proc Roy Soc B 278:2946–2953 8. Gardner M (1970) Mathematical games: the fantastic combinations of John Conway’s new solitaire game “life”. Sci Am 223:120–123 9. Google Maps. http://www.google.com.au/maps. Accessed 30 Dec 2019 10. Green DG (1983) Shapes of simulated fires in discrete fuels. Ecol Model 20(1):21–32 11. Green DG (1989) Simulated effects of fire dispersal and spatial pattern on competition within vegetation mosaics. Vegetation 82(2):139–153 12. Green DG (1990) Landscapes, cataclysms and population explosions. Math & Comput Model 13(6):75–82

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13. Green DG (1993) Emergent behaviour in biological systems. In: Green DG, Bossomaier TRJ (eds) Complex systems: from biology to computation. IOS Press, Amsterdam, pp 24–35 14. Green DG (1994a) Connectivity and complexity in ecological systems. Pac Conserv Biol 1(3):194–200 15. Green DG (1994b) Databasing diversity  – a distributed public–domain approach. Taxon 43(1):51–62 16. Green DG, Gill AM, Tridgell A (1990) Interactive simulation of bushfire spread in heterogeneous fuel. Mathematical & Computer Modeling 13(12):57–66 17. Hasselquist E, Polvi L, Kahlert M, Nilsson C, Sandberg L, McKie B (2018) Contrasting responses among aquatic organism groups to changes in geomorphic complexity along a gradient of stream habitat restoration: implications for restoration planning and assessment. Water 10(10):1465–1496 18. Jeltsch F, Wissel C (1994) Modelling dieback phenomena in natural forests. Ecol Model 75:111–121 19. Langton CG (1990) Computation at the edge of chaos: phase transitions and emergent computation. Physica D: Nonlinear phenomena 42(1–3):12–37 20. Lett C, Silber C, Barret N (1999) Comparison of a cellular automata network and an individual-­ based model for the simulation of forest dynamics. Ecol Model 121(2–3):277–293 21. Malamud BD, Turcotte DL (1999) Self-organized criticality applied to natural hazards. Nat Hazards 20(2–3):93–116 22. Manrubia SC, Solé RV (1996) Self-organized criticality in rainforest dynamics. Chaos, Solutions & Fractals 7(4):523–541 23. Manrubia SC, Solé RV (1997) On forest spatial dynamics with gap formation. J Theor Biol 187(2):159–164 24. McLure JT (2006) Conservation through connection: a spatial assessment of landscape connectivity in the central Apennine Mountains, Italy (Doctoral dissertation, University of Reading) 25. Metzger JP (2000) Tree functional group richness and landscape structure in a Brazilian tropical fragmented landscape. Ecol Appl 10(4):1147–1161 26. Nahmias J, Téphany H, Duarte J, Letaconnoux S (2000) Fire spreading experiments on heterogeneous fuel beds. Applications of percolation theory. Can J For Res 30(8):1318–1328 27. Nowosad J, Stepinski TF (2019) Information theory as a consistent framework for quantification and classification of landscape patterns. Landsc Ecol 34(9):2091–2101 28. Pech R, McIlroy JC, Clough MF, Green DG (1992) A microcomputer model for predicting the spread and control of foot and mouth disease in feral pigs. In: Borrecco JE, Marsh RE (eds) Proceedings of the 15th vertebrate Pest conference. University of California, Davis, pp 360–364 29. Puigdefabregas J, Gallart F, Biaciotto O, Allogia M, del Barrio G (1999) Banded vegetation patterning in a subantarctic forest of Tierra del Fuego as an outcome of the interaction between wind and tree growth. Acta Oecol 20(3):135–146 30. Pütz S, Groeneveld J, Alves LF, Metzger JP, Huth A (2011) Fragmentation drives tropical forest fragments to early successional states: a modelling study for Brazilian Atlantic forests. Ecol Model 222(12):1986–1997 31. Raymond R (1986) Starfish wars – coral death and the crown-of-thorns. Macmillan Australia, South Melbourne 32. Roshier DA, Robertson AI, Kingsford RT, Green DG (2001) Continental–scale interactions with temporary resources may explain the paradox of large populations of desert waterbirds in Australia. Landsc Ecol 16(6):547–556 33. Sato K, Iwasa Y (1993) Modelling of wave regeneration in subalpine Abies forests–population dynamics with spatial structure. Ecology 74(5):1538–1550 34. Satulovsky JE (1997) On the synchronizing mechanism of a class of cellular automata. Physica A: Statistical Mechanics and its Applications 237(1–2):52–58 35. Stauffer D (1979) Scaling theory of percolation clusters. Phys Rep 54(1):1–74

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36. Tschumi M, Ekroos J, Hjort C, Smith HG, Birkhofer K (2018) Predation-mediated ecosystem services and disservices in agricultural landscapes. Ecol Appl 28(8):2109–2118 37. Turcotte DL (1999) Applications of statistical mechanics to natural hazards and landforms. Physica A: Statistical Mechanics and its Applications 274(1–2):294–299 38. Turner MG (1989) Landscape ecology: the effect of pattern on process. Annu Rev Ecol & Syst 20(1):171–197 39. Van der Laan JD, Hogeweg P (1992) Waves of crown–of–thorns starfish outbreaks–where do they come from? Coral Reefs 11(4):207–213 40. Van der Ree R (2003) Ecology of arboreal marsupials in a network of remnant linear habitats. PhD Thesis Deakin University, Melbourne 41. Walsworth NA, King DJ (1999) Image modelling of forest changes associated with acid mine drainage. Comput Geosci 25:567–580 42. Wilkinson D, Willemsen JF (1988) Invasion percolation: a new form of percolation theory. J Phys A – Math Gen 16:3365–3376 43. Wolfram S (1984) Cellular automata as models of complexity. Nature 311:419–424 44. Wolfram S (1986) Theory and applications of cellular automata. World Scientific, Singapore

Chapter 4

Oh, What a Tangled Web … Complex Networks in Ecology

Abstract  Networks are inherent in all complex systems. Patterns of interactions influence system behaviour. Many kinds of large scale patterns emerge from local interactions, including critical collapse. Ecosystems are really interconnected networks of many kinds, so changed conditions in one network can affect the entire ecosystem. One example of this was the reintroduction of wolves into Yellowstone National Park, which initiated a trophic cascade that transformed the landscape. Keywords  Clusters · Connectivity avalanche · Emergent properties · Feedback · Modules · Motifs · Networks · Network topology · Self-organisation · Trophic cascade

Complex networks, such as the web of this Spiny Spider (Australacantha minax) from southern Australia, are found everywhere in nature.

© Springer Nature Switzerland AG 2020 D. G. Green et al., Complexity in Landscape Ecology, Landscape Series 22, https://doi.org/10.1007/978-3-030-46773-9_4

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One of the greatest scientific challenges is to understand how the world is put together. How do billions of cells combine to form a living, breathing human being? How does a seed grow into a giant tree? Puzzles such as these have intrigued and frustrated scientists for hundreds of years. The central question is perhaps best captured in the famous saying: “the whole is greater than the sum of its parts.” A tree is more than just a heap of cells. A forest is more than just trees, and an ant colony is more than just thousands of individual ants. All of these systems are organised. The whole emerges out of interactions between many individual parts. But just how does that organisation occur? This is the challenge that complexity theory seeks to answer. In essence, complexity concerns the way the world is put together. An ecosystem is more than just a collection of species. It is made up of plants and animals interacting with each other, and with their abiotic environment. Complexity theory concerns phenomena that arise from interactions within large collections of objects [3, 4]. As such, it cuts across many different phenomena, in many different disciplines. At first sight, some of the properties that we discuss here may seem abstract and far removed from ecology. And yet, as we shall see both here and in later chapters, some of these properties have far-reaching implications for landscape ecology.

4.1  The Roots of Complexity Theory Complexity theory has its roots in several areas of science. One is biology. Beginning in the 1920s, the German biologist Ludwig von Bertalanffy developed what he called General Systems Theory [27]. This theory introduced several important ideas. One concept that von Bertalanffy explored was that identical, common, processes often occur in superficially different systems. A good example is feedback (see Sect. 5.1), which plays a prominent role in engineering control systems, as well as in biology. Feedback occurs when changes in one part of a system cause changes in another part, which in turn ‘feed back’ to affect the source. A thermostat in an oven, for instance, keeps the oven at a constant temperature because any increase in oven temperature heats up the thermostat, causing it to turn off power to the oven, which then cools down again. Similarly, the human body has a thermo-regulatory system that keeps the body’s temperature constant. Feedback can also occur in ecological settings. For instance, if lions eat zebras, then lions reduce the size of the zebra population. But if the zebra population shrinks, there are fewer zebras around for lions to eat, so the lion population shrinks. This in turn allows the zebra population to grow again, thus restoring a (dynamic) balance. As well as stressing common underlying processes, General Systems Theory introduced the idea of self-organisation. Patterns and order in nature arise from two main sources. One source is external constraint. For example, environmental conditions limit the spread of plant and animal populations across a landscape. Processes of this kind are well known and much studied. However, a more subtle source of organisation and order stems from internal interactions and processes. For instance,

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as we saw above, feedback can serve to keep an ecosystem in a more or less constant state. Also, similar species compete for resources and this competition influences where they occur within the landscape. Another key idea of General Systems Theory is the holistic view of nature. This stresses links between a whole system and the parts of which it is made. As well as being separate entities in themselves, objects in nature are also parts of larger wholes [16]. A cell is an object in its own right, but it is also part of the body in which it lives. A tree is an object, distinct from all other trees, but it is also part of the forest where it grows. The traditional reductionist approach in science tries to understand whole systems by breaking them down into simpler parts. Holism stresses the opposite: the need to understand how wholes emerge from their parts. To modify a popular expression, it aims “to see the forest in the trees.” Another, highly practical, source of inspiration for complexity theory was Operations Research. This discipline arose during World War II because of the need to plan complex military operations. In the post-war world, its use soon extended into business management and engineering. In any large system, the need arises to coordinate the activity of large numbers of separate, but inter-dependent elements. For instance, a large manufacturing enterprise, such as making motor cars or aircraft, requires many different organisations to coordinate their activity. When building a car, hundreds, perhaps thousands, of different suppliers and contractors need to provide parts that fit together, that meet high reliability standards, and that arrive at the assembly plant on time. Perhaps the most important lesson of operations research is that large, complex systems tend to be unpredictable. Complex machines tend to break down and unexpected foul-ups tend to plague any large operation. Many of the resulting techniques—including standardization, redundancy, quality control, monitoring and feedback—are aimed at reducing these problems. Operations research also identifies a number of general questions that are associated with the management of any kind of system. One is forecasting—the ability to predict future values of key variables. Another is optimisation, which is finding conditions that produce a maximum or minimum for some value. For instance, a manager wants to minimise costs, and maximise profits. Many environmental problems are examples of optimisation. For instance, allocation of land use might involve minimising loss of biodiversity while maximising commercial use. The period after 1980 saw a rapid growth of research into complexity. The chief stimulus for this surge of interest was the rapid advances of computing and information technology, which made it feasible to address questions that need massive computing power. This, in turn, afforded a huge expansion of research looking at complexity in nature. For instance, instead of merely arguing about how birds form themselves into flocks, it became possible to test the ideas directly, by putting them into simulation models and watching what happens, to see how the birds behave in different conditions. The computer boom also helped to promote computing as a useful analogy for processes in nature. The idea of natural computation—regarding natural processes as forms of computation—plays an important part in this story. If nature is like

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computation, then it follows that discoveries about computing might apply to natural systems as well. Computational models are not new in ecology. For instance, ideas from information theory, such as redundancy (when different parts of a signal say the same thing) and equivocation (when a known message contains uncertainty), are familiar in animal communication. Likewise, the most widely used measure of species diversity is the Shannon-Weaver index, which interprets variety in the species composition of an ecosystem as a form of information (See Chap. 1).

4.2  The Network Model 4.2.1  Interactions and Connectivity Sometimes analogies are useful. A fire spreading through a landscape is like a disease spreading through a population. It is also like an invasion of exotic weeds, an outbreak of starfish, or the spread of a locust plague. They are alike because all of them are examples of epidemic phenomena, in which a process moves from individual to individual (or place to place). But the analogy extends even further. All of these ecological processes bear similarities to well-known physical processes, such as water percolating through porous rock or even a nuclear explosion. In the past, people have been inclined to dismiss such analogies as frivolous curiosities. An important step forward was to identify a rigorous theory to explain them. This theory gives us new insights that apply to a wide range of phenomena, such as wildfires and locust plagues, which are usually treated as completely distinct from one another. If we are going to understand self-organisation, then first we need to understand the impact of interactions and connections between the parts of the system [15]. The first and most important lesson is that interactions do matter. What is more, interactions at a local scale can produce global effects. A good example is the way in which dispersal (interactions between sites in a landscape) can affect the dynamics of whole ecosystems [12].

4.2.2  Networks What is connectivity? In Chap. 3, we looked at landscape connectivity. We can extend the same ideas to connectivity in all kinds of situations. To do this, we need a more general model. There are many ways to represent relationships, but the simplest model is a graph. In this context, a graph is simply a set of objects, which we call nodes or vertices. Pairs of these nodes are joined by edges (Fig. 4.1). Think of the nodes as the objects and the edges as relationships between them. The basic graph model can be refined

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by allowing the edges (now also called arrows) to have direction. This kind of graph is called a directed graph (or digraph for short). A network is a graph in which the nodes and edges have values associated with them. In a food web, for instance, the nodes might represent populations and the arrows might represent the effect of one species on another. Graph theory concerns properties of graphs. It begins with some common features of graphs. Within a given graph, a path is a sequence of nodes such that each consecutive pair of nodes is connected by an edge in the graph (Fig. 4.1b). So if we have nodes A,B,C,D linked by edges A-B, B-C, C-D, then the ABCD is a path. A graph is called connected if there are paths linking each node to every other node; it is called fully connected if every pair of nodes is connected by an edge. A cluster is a fully connected subgraph. The number of edges linked to a node is called the degree of that node. The basic types of graph include the following familiar structures (Fig. 4.1). A cycle is a path in which the two ends are joined by an edge (Fig. 4.1a). A tree is a graph in which there are no cycles (Fig. 4.1c). A rooted tree is a tree in which paths connect every node to one particular node, the latter being the root of the tree.

Fig. 4.1  Some elementary network ideas: (a) a cycle (the nodes shown as open circles), (b) a path, showing the path length, which also is the diameter of the network, (c) a tree, with the root node at the top

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4.2.3  Networks Are Everywhere How do networks relate to the concerns of landscape ecology? As we saw in the previous chapters, spatial processes, such as seed dispersal, make connections between different locations in a landscape. In other words, they create a landscape network. The nodes of this network are sites in the landscape; the edges are processes that link them, such as movement of animals, flow of water, or dispersal of seeds. Note that a single ecosystem really consists of many different networks, depending on which process we are talking about. So there may be a network of duck habitats, a fire fuel network, and so on. Networks turn out to be the key to understanding complexity in all its many forms. The reason for their importance is that networks occur everywhere. This is ensured by the following two properties [11, 13, 14]. 1. the structure of any complex system (i.e. the pattern of interactions) forms a network; 2. the behaviour of any system forms a network. The first of the above two properties shows that networks are present in the structure of virtually all complex systems. In a herd of zebras, the zebras are the nodes and their interactions with each other form the edges of a network. In an ecosystem, we could consider the species that are present as nodes of a network, and which species eats which defines the edges. Likewise, the second property supposes that we can represent the behaviour of any complex system by a computer model. In a game of chess, for instance, the state of a game is the arrangement of pieces on the board at any time. We can see this state as a node, with the moves by the players providing the edges that link these nodes. In an ecosystem, we could view the species composition at any time as a node, with succession and disturbance defining edges that link these nodes. So the second property above shows that we can also regard the behaviour of complex systems as networks. To sum up, the above properties show that both the structure and the behaviour of any complex system can be represented by graphs. This is an important result because it means that any general property of graphs will be felt in all complex systems. A crucial question to ask about any complex system is what features arise from the particular nature of the objects and their interactions? And what features arise from their underlying network structure? It turns out that networks have some attributes that turn up in systems of many kinds. Below, and in later chapters, we shall see some of the ways that these attributes influence ecosystems and landscapes.

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4.2.4  The Connectivity Avalanche Perhaps the most far-reaching property of graphs is the so-called connectivity avalanche. In 1959 two Hungarian mathematicians, Paul Erdõs and Alfred Renyi, discovered a strange property involving the connectivity of graphs [6]. The question they asked was: “what happens if you take a set of nodes and progressively add edges to pairs of nodes chosen at random?” It turns out that at first the set of connected nodes is very small. For the most part, you just get pairs joined together by a single edge. At a certain point in the procedure, a startling change occurs. Suddenly, all the separate pairs and small clusters of nodes start combining together into a large connected network. This process, known as a connectivity avalanche, occurs when the number of edges is approximately half the number of nodes. When the density of edges reaches this critical point, a phase change occurs in the network— from essentially disconnected to almost fully connected (Fig. 4.2). This discovery of Erdõs and Renyi may seem surprising, but to understand why it should be so, consider the idea of saturation. Suppose that in the experiment, we begin with a set of 1000 disconnected nodes (Fig. 4.2). There are no edges at all. Then as we add edges to join pairs of nodes, each edge joins two nodes to form a pair of connected nodes - a clump of two nodes. As we continue to add edges, the network eventually becomes saturated with pairs of connected nodes. With 1000 nodes, we have to add about 500 edges to reach saturation this way. As the network becomes saturated, any new edges that we add can no longer join pairs of single nodes. They begin to join together different pairs, and in doing so they form clumps of size 4 nodes. The process of forming clumps of size 4 continues until the system again becomes saturated. However, because the new clumps are twice as large as before, the process takes only half as long. At that point, the system starts building clumps of size 8. If we continue the process in this fashion, then each time the system becomes saturated, it starts building clumps that are twice the size of the

Fig. 4.2  The connectivity avalanche in a random network of 1000 nodes. (a) Number of new edges needed to saturate the network, starting with clumps of varying size. (b) Changes in the size of the giant component (largest connected set of nodes) as more and more edges are added to the network. This avalanche also underlies the criticality effects shown in Fig. 3.8

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previous clumps, and because they are twice the size, saturation occurs in just half the time. To summarise the above process, we can tabulate the size of clumps at each stage, and the minimum number of edges required to link them in pairs (Fig. 4.2a). Notice that at each step, the clumps become twice as large as before, but it takes only half as many new edges to saturate the graph. In other words, the process of clump formation accelerates. In reality, the process of saturation is not quite as simple as described here. As you add edges at random, many of them will combine existing pairs into larger groups well before full saturation is reached. The point is that the system enters a critical region, in which the formation of larger clumps accelerates rapidly as saturation approaches. This is why more than 2000 edges are needed in Fig. 4.2b before all the nodes are connected. The result of this doubling process is that once it takes hold, the entire graph soon becomes completely connected. Once a clump forms that contains half the number of nodes in the graph, it is known as a unique giant component (UGC) (or Giant Connected Component). Other small clumps may still exist, but the process becomes largely one of adding to the UGC. The way in which the process rapidly accelerates, once it takes hold, leads to a phase change in connectivity. That is, the system abruptly shifts from being essentially disconnected to essentially fully connected (Fig. 4.2b). It is this connectivity avalanche that is responsible for the phase changes in connectivity within landscapes that we saw in Chap. 3. If we represent the landscape as a grid of cells (Fig. 3.8), then we can consider each cell as a node of a graph. Links to neighbouring cells then provide the edges of the graph. So for any given cell A, there is an edge that joins A to another cell B (say) if and only if B falls within the neighbourhood of cell A.

4.2.5  Phase Changes and Criticality Criticality is a property that is associated with change in complex systems. The changes involved are often sudden and dramatic. Critical processes often involve predictable changes in structure or properties. Changes of this kind are called phase transitions. As the name implies, a phase change occurs when a system changes from one state (or ‘phase’) to another. Matter, for example, generally exists in three possible phases: solids, liquids and gases. Criticality and phase transitions abound in nature. Here are some examples: • • • • •

water freezes to form ice or boils to form steam; crystals form when a solution reaches saturation; a sand pile collapses when it grows too large; a laser fires when enough energy is pumped into the system; nuclear chain reactions occur when radioactive fuel reaches a critical mass;

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• below a certain temperature wildfires do not spread; • an aeroplane wing breaks when stresses grow too great. The importance of the connectivity avalanche becomes apparent when we recall that networks underlie the structure and behaviour of all complex systems. In other words, there is the potential for connectivity avalanches to occur in almost any system. One conjecture, not proven, is that the connectivity avalanche is responsible for all critical phase changes. For instance, water turns to ice when small ice crystals grow and link up with one another. In the following sections we shall see how the connectivity avalanche translates into several widespread features of complexity, such as phase transitions and criticality in many kinds of system. Phase transitions normally occur at fixed values of some state variable. For instance, water freezes when the temperature falls to 0 degrees Celsius. Herman Haken called such variables order parameters. A value where a sudden change occurs is called a critical point. So for water freezing, temperature is the order parameter and 0 degrees is the critical point. In the case of a network, the connectivity avalanche occurs when the density of edges reaches a critical value. This critical density turns out to be 0.5. In other words, the critical point occurs when there is one edge for each pair of nodes. In effect, when we identify an order parameter for a critical system, we are making a link between that order parameter and the density of edges in the equivalent graph of relationships. There are order parameters associated with every one of the examples of criticality listed above. Order parameters are important because they make it possible to predict phase changes precisely. They also allow us to draw conclusions about managing the systems concerned. The following examples illustrate this link between order parameters and critical phase changes. 1. Wild fires do not spread if the fuel temperature or fuel density falls below critical thresholds. These two order parameters are related. Lower temperatures mean that fires burn with less intensity. So the area of scorch around (say) a burning bush is smaller on a cool day than it would be on a hot day. This means that the area of fuel ignited around a burning bush (i.e. the neighbourhood for the fire spread) is smaller, so fewer bushes get ignited. This has exactly the same effect as reducing the number of bushes in the fuel bed. 2. Epidemics fail to spread if the infection rate falls below a critical threshold. The two factors that contribute to the infection rate are the disease’s virulence, and the number of exposures. So to halt the spread of a disease we need to cut down the number of exposures. The traditional and most effective way to do this is to isolate the infected. 3. For a plant species to spread across a landscape, the density of suitable habitats needs to exceed a critical threshold. We will come back to this example again in Chaps 7 and 8.

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4.2.6  The Order of Things Another important property of networks is the pattern of connections that link the nodes. In the previous sections we saw that phase changes occur during the formation of randomly connected networks (e.g. Fig. 4.3a). More orderly patterns also have important properties. Regular networks arise when nodes in a network are connected systematically. For instance, rectangular grids (such as shown in Fig. 4.3b) often arise in artificial systems (e.g. city streets). They also arise in environmental contexts as a by-product of human planning or study. A satellite image represents the landscape as a rectangular grid of pixels. Cellular automata models (see Sect. 3.2.2) do the same thing; in a network the nodes would represent the cells in the grid, and the edges would link neighbouring cells. In a cyclic network (Fig. 4.3c), the edges form a closed loop, that is, they form a path that begins and ends at the same node and that includes all of the nodes. For the nodes A, B, C, D, for instance, the sequence of edges A-B, B-C, C-D, D-A forms a cycle that starts and ends at A. Trees are connected networks that contain no cycles (Fig. 4.3d). This means that there are no closed paths. Trees and hierarchies are extremely common, in both natural and artificial systems. Small worlds (Fig. 4.3e) and scale free networks (Fig. 4.3f) appear in the connectivity patterns of many systems. The “small world” phenomenon is the finding that the number of intervening nodes between any two nodes in certain types of networks is astonishingly small [18, 29]. In terms of network theory, small worlds are characterized by relatively small diameter and high clustering. The diameter of a network is the longest path separating a pair of nodes.1 In the cycle (Fig. 4.3c) for instance, the diameter is 6, because you require 6 edges to move from any node to the node on the opposite side of the circle. However, in the small world (Fig. 4.3e), the diameter of the network (the maximum separation between pairs of nodes) has been reduced to 5. Also the average separation between nodes has decreased, from 3.3 down to 2.5. Social networks are the most widely-known example of small world networks. Stanley Milgram, for example, found that although the population of the United States was over 200 million, if he chose two people at random, he could on average trace a path of less than six steps from one to the other via their acquaintances. This finding, which became known as ‘six degrees of separation’ led to statistics such as the ‘Bacon numbers’ for actors, which measures an actor’s distance from Kevin Bacon on the social network of film participation [5, 21]. Since then, many other small world networks have been identified. Networks of collaboration among 1  Here the word “separation” implies the shortest path between two nodes, so the diameter is the longest shortest path between pairs of nodes (taken over all pairs).

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Fig. 4.3  Some common kinds of networks. (a) In a random network there is no pattern to the connections. (b) In a regular network all the nodes are connected to neighbours in identical fashion (except on the border). (c) A cycle (or circuit) is a network in which a single, closed path takes in all the nodes. (d) In a hierarchy (tree), there are no cycles and all connections stem from a singe root node. (e) A small world is obtained from a regular network by replacing local connections with random, long-range ones (in this case we alter the cycle). (f) In a scale free network, the degrees of the nodes follows and inverse power law. This means that some nodes have many connections, but most nodes have very few

scientists, of biochemical reactions within cells, and of links between internet sites, all take a small world structure [7]. As we shall see in Chap. 7, networks of interactions in ecological communities also appear to take a small world structure, and this may be a key factor in understanding how ecosystems persist.

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Small-world networks are particularly notable because their small diameter means that states can be transferred through them astonishingly rapidly. This feature is, in real life, both useful and dangerous. It means that useful information and ideas can spread very rapidly throughout society. It also means that diseases such as avian influenza can do the same. An important feature of small worlds is that they combine local clustering with long-range connections. Thus they fall between regular networks and random networks. One way to form a small world is to take a regular graph and progressively replace the edges with random ones. In the example shown here (Fig. 4.3e), four edges in the cyclic network (Fig.  4.3c) have been replaced. Scale-free networks (Fig. 4.3f) arise in situations where a network forms with preferential attachment. Suppose that you build a network by adding nodes one at a time. Where do you add edges to link the new nodes to existing ones? In many cases, new nodes ‘prefer’ to make links to nodes that already have many connections. In business networks, for instance, people usually prefer to make connections with colleagues who have many business connections. In food webs (see Chap. 7), species new to an ecosystem are more likely to be eaten by species that eat lots of other species, than by those with specialised diets. Preferential attachment leads to a power law in the distribution of connections. This means that most nodes will have few connections (one or two) to other nodes, but a few nodes will have many connections.

4.3  Self-Organisation 4.3.1  Emergent Properties The large-scale behaviour of a system emerges out of the properties and interactions of many individual objects (Fig. 4.4). Temperature, for instance, is an average property of the atoms or molecules that make up a medium. A flock of birds is the result of many individual birds flying together. An ant colony is the product of the activity of many individual ants. Three aspects of aggregates of individuals influence the nature of the system that emerges. First, there is the character of the agents themselves. Bees in a swarm behave differently from birds in a flock. The second aspect is the quality of the interactions between the agents. Individuals in a rioting mob interact very differently from guests at a cocktail party. The third aspect is the “wiring pattern” in the network of interactions between agents. The pathways of influence in a feudal kingdom are different from those in a democracy. An important issue is whether the interactions persist across different scales. In a gas, for instance, the main interactions (molecules colliding and rebounding off one another) are all brief, local and linear. They quickly average out so that the nature of the gas remains the same over a wide range of scales. The temperature of

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Fig. 4.4  A forest emerges out of the interactions between millions of trees and plants, both with each other and with the physical environment

the gas is an emergent property (see Chap. 1) that expresses the average behaviour. In contrast, non-linear systems do not usually scale in such a simple fashion. They are governed by non-linear interactions, which are a common source of complexity. In physics, entropy is a measure of the disorder, or randomness, in a system. For example, a system with hot spots and cool spots has order: there are patterns of heat and cold. A system in which the temperature is uniform throughout has low order, and high entropy. The Second Law of Thermodynamics states that in any closed system, the entropy of the system will increase with time until the system is uniformly disordered. For instance, if you heat the end of a metal bar, the heat will spread until the whole bar becomes hot. An ice cube melts in an oven. In both cases, order decreases. One of the puzzles of physics was to account for living systems, which seem to fly in the face of thermodynamics by accumulating order. Likewise, why do clouds of interstellar gas condense into stars? According to the second law of thermodynamics, this should not happen. But the second law applies explicitly to closed systems, whereas living systems are open. They exchange energy and materials with their environment. Therefore, order can increase locally in gas clouds and in living systems because of the exchange of energy with the outside. The Nobel prize winner Ilya Prigogine developed the idea of dissipative systems to explain how some systems can accumulate order. Dissipative systems are open systems that are maintained in an orderly state by exchanging energy with their environment. That is, they take in energy from their environment, use that energy to generate orderly internal structures, and dissipate it to the environment in a less orderly form.

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In dissipative systems, there is no tendency to smooth out irregularities and for the system to become homogeneous. Instead, irregularities in dissipative systems can grow and spread. Chemical systems provide many examples, such as crystal formation. However, by far the most common dissipative systems are living things. Within intergalactic dust and gas clouds, gravitational interactions between particles lead to the formation of stars and planets. Unlike gas in a bottle, in which molecules rebounding off one another serve to distribute energy evenly throughout the system, gravity is the predominant interaction within interstellar clouds. The global properties of interstellar clouds are therefore very different from bottled gas. Instead of moving towards homogeneity, irregularities in these clouds coalesce into stars, planets and other astronomical objects.

4.3.2  Modules and Motifs Above we saw that pattern and organisation can arise out of the interactions between things. However, complex structures do not magically appear. One of the most important principles at work is modularity. To understand modularity, a good analogy is a carry bag. Think of any bag that you regularly use to carry things, such as a hiking pack, a purse, or a briefcase. The simplest carry bag is just a sack. You throw all the items that you want to carry into the sack and off you go. If there are just one or two, then there is no problem. But people often need to carry dozens of separate items around with them. If you throw them all into a sack, then you could be rummaging around for ages trying to find what you want. Even worse, some items might be incompatible. Your lunch could leak all over your wallet, for instance. So what do we do? We create modules. Hikers, for instance, put clothes in one bag, cooking gear in another bag, and so on. In this way, different classes of items are kept together. This makes it easy to find the cooking gear when you need it, and avoids dirty pots from soiling your clean clothes. Most purses and briefcases provide pockets and compartments to help us store the contents in modular fashion. The same principle holds in many different systems. Large organisations divide their business into sections, branches, divisions and so on. This simplifies the way they operate and enables people to concentrate on specific roles. Engineers use modularity too. Any large system, such as an aircraft or a factory, may consist of thousands or even millions of individual parts. A common source of system failure is undesirable side effects of internal interactions. The problem grows exponentially with the number of parts, and can be virtually impossible to anticipate. The solution is to organise large systems into discrete subsystems (modules) and to limit the potential for interactions between the subsystems. This modularity not only reduces the potential for unplanned interactions, but also simplifies system development and maintenance. The same is true of cities. To support huge concentrations of people, cities have to provide a wide range of services such as housing, transport, distribution of food

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and other commodities, water, sewage, waste disposal, power, and communications. On top of these vital services, there are social infrastructures such as education, police, fire brigade, hospitals, ambulance, and shopping centres. The interactions between so many systems, combined with rapid growth, technological development, and social change, underlie many problems in modern society. The living world around us is also modular. It is full of hierarchies. The body plans of all organisms are highly modular. The human body, for instance, consists of discrete organs, bones, and tissues. Organs form bodies. Each organ is built out of discrete cells. Likewise, the cells are modular too. They each consist of distinct organelles: nucleus, mitochondria, and so on. As we saw in Chap. 2, plants are even more modular than animals. A tree, for instance, consists of thousands of nearly identical branches, leaves, buds, flowers and fruit. At a higher level, species themselves can be seen as a form of genetic modularity. They represent reproductively closed populations. Ecosystems too are modular. They consist of distinct populations, and may also contain niches or habitats that are partially closed subsystems. Likewise, different parts of a landscape may form modules. Interactions within a pond or stream, for instance, are often much richer than with the surrounding woods. Although hierarchies reduce complexity, as described above, they also introduce brittleness into a system. Removing a single node, or cutting a single connection breaks the network into two separate parts. Every node below the break becomes separated from the rest of the system. This brittleness occurs because hierarchies are minimally connected. There is no redundancy in the connections. For instance, the internet is organised as a hierarchy of domains. If a domain name server fails, then computers in that domain may be cut off from the internet. Centralised services, such as city power supplies, suffer from the same problem. Recent studies suggest that instead of hierarchies, many large systems, including metabolic pathways, large societies, and some ecosystems, are organised as scale-­ free networks (Fig. 4.3f). These are networks in which most nodes have few edges linking them to other nodes, but some have large numbers of connections to other nodes. In contrast, in a simple hierarchy, such as a binary tree (Fig.  4.3d), most nodes have three connections per node (one up and two down). Scale-free networks have two important features. One is that they are independent of scale; remove any number of nodes and the overall structure still looks much the same. The other is that they are brittle in face of systematic disturbances. Removing a few critical nodes from a tree breaks the system apart. However, a scale-free network can retain full connectivity even when large numbers of nodes are removed. As we have seen here, complex networks pervade our world and offer a powerful way to look at a wide range of processes. Understanding the structures that occur in networks, and how these influence processes within them, can provide insights into the functioning of complex systems. These structures include trees, cycles, hierarchies, and modules. Some key processes are positive and negative feedback, percolation, phase transitions, criticality and self-organisation. Network theory is closely tied to dynamical systems and chaos theory. In the next chapter, we shall look at these ideas and how they relate to some major questions in ecology.

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Fig. 4.5  Examples of some simple motifs in dynamic networks

An idea closely related to modules is that of motifs. In general, a motif is a meaningful fragment of pattern. For centuries, composers have long known and used motifs, in the form of fragments of tunes, in musical compositions. The idea was introduced into biology in the late twentieth Century. In 1981, Noble and Slatyer adopted a motif-based approach to interpreting the dynamics of ecosystems [20]. In the 1990s, Amos Bairoch introduced the idea of motifs into bioinformatics [1]. In amino acid sequences, particular short segments, and amino acids with particular properties, control the folding patterns in proteins. Identifying such motifs in amino acid sequences provide useful clues to protein structure and functional relationships. The idea became popular in the early twenty-first Century after reviews demonstrated its potential of motifs as tools to interpret complex networks [19, 24]. A network motif is a pattern of nodes and edges (that is a subgraph) that occurs repeatedly, either within a single network, or across different networks (Fig.  4.5). In knowledge discovery, identifying repeated motifs is a way to detect important properties. Locating motifs with known properties in a network provides clues about processes within a network. In Chap. 7, we will see an example of using positive and negative feedback loops as motifs to understand complex ecosystems.

4.3.3  The Shape of Complexity Rainforests and coral reefs are complex ecosystems. Everyone says so. But what does that mean? When is an ecosystem complex? When not? Clearly, complexity is not an all-or-nothing property. Some ecosystems are more complex than others. There are degrees of complexity. So can we measure complexity? In Chap. 1 we saw that ecologists often use the notion of diversity as a surrogate for complexity (see Sect. 1.2). The assumption implicit in this usage is that the more individuals there are, and the more species there are in an ecosystem, the more interactions there will be. However, this is not always the case. A mountain may be host to dozens of plant species, but if they confine themselves to separate bands of altitude, then the interactions between species may be relatively few. The network model provides other ways of looking at ecosystem complexity, especially in terms of direct interactions. Earlier in this chapter, we saw that random

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networks flip from fragmented to connected when the number of edges (interactions) reaches a critical density. This property makes the edge density an important metric in relation to the complexity of networks. Researchers also adopt many other measures, each reflecting particular properties of networks. Earlier we met a network’s diameter, which is the longest path between pairs of nodes. By ‘path’ in this instance, we mean the shortest sequence of edges that joins a given pair of nodes. Some metrics show well-known trends that are related to particular network topologies. For instance, the definition of a scale-free network is that the degrees of the nodes follow an inverse power-law. The exponent of that power law reflects how strongly the links are concentrated. Clustering is the tendency of nodes to form tight groups. A node is the centre of a cluster if its neighbours are also linked to each other by edges. For any node in a network (Fig. 4.6), local clustering is measured by the number of edges that link its neighbours. The clustering coefficient (CC) is the ratio of the number of edges joining pairs of neighbours and the total number of pairs of neighbours (Fig. 4.6). In a landscape, objects, such as trees, form networks by being neighbours of one another. However, to say that two objects are neighbours is a matter of scale. As Fig. 4.7 shows, a small increase in scale can increase the clustering coefficient dramatically. This means that we need to be careful about defining neighbourhood relationships. They are usually defined in terms of some natural process or feature, such as home range of an animal or seed dispersal. The clustering coefficient of an entire network is the clustering ratio, averaged over all nodes in the network. Its value ranges from 0 to 1. In a random network, if the edge density exceeds the critical density, then each node added will increase the degree of clustering, so as the edge density increases, the clustering coefficient will increase too. In a landscape, this means that we need to take care in calculating and interpreting the clustering coefficient. Figure 4.8 provides an example. Here, one of the mapped distributions is clearly clustered, whereas the other is randomly scattered. If we use a small radius to define the neighbourhoods around points, then the clustering coefficient for the real clusters increases much more rapidly than for a random scatter. However, in this example, there are 100 points spread across an area

Fig. 4.6  Local clustering coefficient for a target node (at left) and its four neighbours. The clustering coefficient (CC) for the target is the number of triangles it forms with its neighbours as a fraction of the maximum possible

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Fig. 4.7  Local clustering in a landscape is relative to scale. (a) & (b) The circles show neighbourhoods of different diameter around a central target point (solid circle) and its immediate neighbours (dashed circles). (a) none of the target point’s neighbours are neighbours of each other. (b) However, if the neighbourhoods are large, then they are all neighbours. (c) & (d) The corresponding connectivity graphs implied by the neighbourhood circles in (a) and (b), respectively

Fig. 4.8  Measuring clustering of points in a landscape. At left are examples of clustered (top) and scattered (bottom) point distributions. At right is a plot showing how the value of the clustering coefficient changes as the radius of the measured neighbourhood increases

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of 100 × 100 units. So when the neighbourhood radius exceeds this threshold, the clustering coefficient for the randomly scattered points increases linearly. Assortative mixing is the tendency of nodes in a network to sort themselves by linking to other similar nodes. In a forest, for instance, trees of the same species often grow together in the same area. We can consider assortativity with respect to different properties, such as species or environmental preference. The usual way to measure assortativity is the correlation between properties of similar nodes. For instance, do tall trees tend to grow close to other tall trees? Do plants of the same species tend to grow in soils of similar pH? In networks, node degree is the most common property considered by assortativity. That is, how is the number of edges between nodes distributed? For instance, if nodes with high degree mostly link to nodes of high degree and nodes of low degree mostly link to nodes with low degree, then link assortativity will be positive. But if links of low degree mostly link to nodes of higher degree (in a scale-free network, for example), then assortativity will be negative. In traditional ecology, the limitations of fieldwork make gathering information about (say) the distributions and properties of individual trees a very time-­consuming and difficult task. A complete a census of every tree in a forest is uisually out of the question, so most research relies on sampling. For instance, measuring assortativity for an entire forest was usually impractical, so studies would instead use correlation between nearest-neighbours along a transect.

4.4  Networks of Networks Imagine the ecosystem we might find in a pond. As we have seen, we can regard this ecosystem as a network of interacting populations: fish, plants, insects, and so on. Complex as it might be, this model of the pond is actually a simplification. One simplification is that it treats the organisms in isolation. The nutrients in the pond also form a network, the nutrient cycle. There is also a carbon cycle. Another limitation is that the above model treats the pond as a closed system. In reality, the pond is an open system. There are many connections to the outside. Water, sediment and nutrients flow into the pond from the surrounding catchment basin. Energy flows in from sunlight. The pond is also connected to other ecosystems. Waterbirds, for instance, move from one pond to another, consuming organisms and transferring seeds, eggs and other material. What all the above connections mean is that a pond is not a single network, but the setting for many distinct, but interacting networks: a network of networks. The picture of networks we presented above assumes that all the nodes and edges are of the same kind. In a food web, for instance, the nodes are species and the edges are interactions between species. In reality, however, ecosystems often comprise many different networks. In any ecosystem, there are many kinds of networks. The animals or plants that make up a single population comprise a network of individuals that breed and

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compete with one another. However, those populations, those networks, interact in food webs, which are really networks of networks. Then again, we can regard the landscape itself as a network of sites or locations. Each location is a node and neighbouring sites define the edges. This may seem simple, but in fact, the landscape is actually many different networks all at once. The key is in the word ‘neighbouring’. It implies some link between different sites, but that link can be of many different kinds. Take trees: they produce seeds, which drop on neighbouring sites. In this way, seed dispersal defines one kind of network. However, the distance that seeds disperse varies enormously. Some trees produce heavy seeds that fall more or less at the base of the tree. However, animals may transport them much further, as can storms or floods. Other trees produce light seeds that the wind picks up and blows long distances from the parent tree. For animals, the size of a neighbourhood can vary by orders of magnitude. The distance a worm moves is normally just a few metres. At the other extreme, some birds fly thousands of kilometres while migrating. In between, different species of animals have vastly differing territories or home ranges. Thus the size and nature of a neighbourhood varies enormously from one species to another. One implication of the above variations in neighbourhood is that the same landscape can be seen as many different networks at the same time. We say a network is connected if there is a path from neighbour to neighbour between any pair of nodes in the network. As we saw above, the same landscape may be connected for some species, but fragmented for others. A pond or forest may seem to be a closed ecosystem because its interactions with the outside are slow, rare or have little overall effect (Fig. 4.6). However, if conditions change, then the ecosystem may interact with the outside world in new and significant ways or existing interactions may become stronger. Under many conditions, we can readily regard a pond as a closed system. But when conditions change, those links to the outside may assume great importance. Under normal conditions, say, sediment might flow into a pond at a more or less constant rate, but a severe storm upstream might produce a torrent of water that suddenly dumps large amounts of sediment, changing conditions in the pond. Such changes can have big effects, which can cascade throughout the ecosystem. As we shall see later, they can also introduce new kinds of interactions, for instance positive feedback, that destabilizes the ecosystem (see Sect. 5.1). In terms of conservation, one of the most important kinds of interlinked networks is the way human socio-economic networks impinge on natural ecosystems. Humans interact with ecological networks in many ways, including clearing and fragmenting habitats, introducing exotic species, and polluting the atmosphere and waterways. We can see these problems at work in the deforestation in the Amazon. The loss of forest and species results from interaction between two very different networks: the forest ecosystem and the socio-economic network. As we saw earlier (Chap. 1), the original motivation was the search for farmland. However, deforestation was concentrated at points of maximum human contact, so at first the impact was

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confined to the edges, so the forest remained largely a closed system. The major change was the cutting of roads through the forest: “... deforestation was much higher near roads and rivers than elsewhere in the Amazon; nearly 95% of all deforestation …occurred within 5.5  km of roads or 1  km of rivers. Protected areas near roads and rivers had much lower deforestation (10.9%) than did unprotected areas near roads and rivers (43.6%).” [2]

At the time of writing it remains a problem that private land accounts for 53% of native vegetation in Brazil [26]. After 2004, authorities introduced a number of measures to discourage deforestation [8, 17]. These included imposing fines for clearance and monitoring to detect cases. They also tightened up licensing procedures for logging. The nett result was that annual deforestation rates in the Brazilian Amazon fell by 77% between 2004 and 2011, after 2009 they stabilized at 5000–7000 km2 per annum [10]. but deforestation by fires skyrocketed again from 2018 due to changes in government policy. When connections form between previously isolated networks, outcomes can be unpredictable. Isolated networks function independently. However, a change in the environment can create new pathways by which the networks interact (Fig. 4.9), then in effect they form a single, more complex network. In ecosystems, one of the most serious consequences of interactions between networks is a trophic cascade [28]. A trophic cascade is a series of changes that occur in an ecosystem, with each change triggering more changes. They usually begin in response to some triggering event, such as the loss of a keystone species, the introduction of a new species or some change to the environment. We can see an example of a trophic cascade in the changes wrought by the removal and later reintroduction of wolves in Yellowstone National Park [9, 22, 23]. In the early twentieth Century, wolves were considered a threat, both to wildlife and to livestock in the region. By the mid-1920s, all wolves in Yellowstone National Park had been killed. Over the next seventy years, the virtual absence of wolves led to major changes in the ecology of the park. Without wolves, controlling the elk

Fig. 4.9  Networks of networks. The dashed lines mark out a target network (e.g. a food web). (a) Under “normal” conditions, the target network has few connections to the outside and we can regard it as a ‘closed box’. (b) However, when conditions change, the target network may be subject to strong new influences. These can trigger unpredictable, large-scale changes, such as trophic cascades

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Fig. 4.10  Simplified diagram showing the trophic cascade of changes wrought by the removal and later reintroduction of wolves in Yellowstone National Park. See the text for further explanation. Based on [22, 23]

population was a problem. In addition, the coyote population grew, which severely reduced the population of pronghorn antelopes. In the late 1960s, a ban on hunting elk enabled the population to expand rapidly. This led to widespread environmental changes. For instance, the elk cleared much of the vegetation that bound the soil, which led to erosion, which affected the streams, which affected plants and wildlife that depended on the water (Fig. 4.10). The ecological effects described above began to reverse when authorities reintroduced wolves to the park in the winter of 1995–96. Monitoring revealed a complex cascade of ecological changes that followed. First, the wolves immediately began to hunt the elk. Besides constraining growth of the elk population, hunting forced the elk to change their behaviour. To avoid the wolves, they kept away from areas, such as valleys, where the wolves could trap them. Reduced grazing allowed vegetation to regenerate. Not only did grasses recover, but also forests of aspen, cottonwoods and willows appeared. All of the above changes increased the niches open to birds. The increase in trees provided more nesting sites for birds or all species. Carrion from kills by the wolves provided opportunities for ravens and bald eagles. The wolves also reduced the coyote population, which allowed populations of rabbits and mice to increase. Hawks hunted the rabbits and mice, as did foxes, badgers and weasels. The bear population also increased. Besides feeding on deer calves and carrion left by the wolves, an increase in shrubs provided more berries. The cascading effects extended to the landscape itself. With less erosion, and with forests stabilizing riverbanks, channels became narrower, rivers meandered

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less and more pools formed. Beavers took advantage of all these changes, felling trees and creating dams. In this way, they created habitats for many plants and animals, including ducks, frogs, muskrats and otters. The story of wolves in Yellowstone is just one of many examples of cascading effects through an ecosystem. In Botswana, for example, numbers of elephants and ungulates were greatly reduced by excessive hunting for more than a hundred years. The ecological effects of hunting are not documented. However restrictions on hunting have allowed those populations to increase rapidly. The flow-on effects of the changes are on a similar scale to the Yellowstone story [25]. Cascading effects occur in many other ecosystems, especially where humans have activity is involved. Human movement has created pathways between many previously isolated ecological communities. Seeds and spores are carried on shoes and vehicles; early explorers transported rodents and colonists brought domestic live-stock. Today, reef-dwelling organisms such as sea stars, barnacles, seaweeds and anemones hitch rides in bilge water. And each new arrival carries its own community of symbiotic microbes. Newly introduced organisms often cannot survive, being less adapted to local conditions than the established biota. However, those that do survive usually succeed by exploiting local resources more effectively than any native species. In this way they pose a two-fold danger to natives, both out-competing species who overlap their ecological niche, and over-exploiting populations they use as resources. Cats arrived in mainland Australia shortly after European settlement. Their ecological niche as predators of small vertebrates was previously occupied in Australia by marsupial carnivores such as quolls and other dasyurids. The dasyurids have relatively small brains compared with felines, and are much less efficient hunters. Australian small vertebrates had adapted to this relatively weak predation pressure. Thus the arrival of cats resulted in rapid, wholesale slaughter of native birds, reptiles and small mammals. Cat populations exploded, and in most places, prey became too scarce for quolls to survive. Many native bird and mammal species became extinct or endangered due to predation, and quolls are now rare and confined to isolated regions where cat populations remain low. The loss of these vertebrate species had many flow-on effects on distributions of native plants and invertebrates, transforming the Australian landscape as a whole.

References 1. Bairoch A (1991) Prosite: a dictionary of sites and patterns in proteins. Nucleic Acids Res 19(S):2241 2. Barber CP, Cochrane MA, Souza CM Jr, Laurance WF (2014) Roads, deforestation, and the mitigating effect of protected areas in the Amazon. Biol Conserv 177:203–209 3. Bossomaier TRJ, Green DG (1998) Patterns in the sand. Allen & Unwin, Sydney 4. Bossomaier TRJ, Green DG (2000) Complex systems. Cambridge Univ Press, New York 5. Collins JJ, Chow CC (1998) It’s a small world. Nature 393(6684):409–410 6. Erdos P, Renyi A (1960) On the evolution of random graphs. Publ Math Inst Hung Acad Sci 5:17–60

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7. Farkas I, Derenyi I, Jeong H, Neda Z, Oltvai ZN, Ravasz E, Schubert A, Barabasi A-L, Vicsek T (2002) Networks in life: scaling properties and eigenvalue spectra. Physica A 314(1–4):25–34 8. Fearnside PM (2016) Brazil’s Amazonian forest carbon: the key to southern Amazonia’s significance for global climate. Reg Environ Chang 18(1):1–15 9. Fortin D, Beyer HL, Boyce MS, Smith DW, Duchesne T, Mao JS (2005) Wolves influence elk movements: behavior shapes a trophic cascade in Yellowstone National Park. Ecology 86(5):1320–1330 10. Godar J, Gardner TA, Tizado EJ, Pacheco P (2014) Actor-specific contributions to the deforestation slowdown in the Brazilian Amazon. PNAS 111(43):15591–15596 11. Green DG (1993) Emergent behaviour in biological systems. In: Green DG, TRJ B (eds) Complex systems: from biology to computation. IOS Press, Amsterdam, pp 24–35 12. Green DG (1994a) Connectivity and complexity in ecological systems. Pac Conserv Biol 1(3):194–200 13. Green DG (1994b) Databasing diversity  – a distributed public–domain approach. Taxon 43(1):51–62 14. Green DG (1994c) Connectivity and the evolution of biological systems. J Biol Syst 2(1):91–103 15. Green DG (2000) Self-organization in complex systems. In: Bossomaier TR, Green DG (eds) Complex systems. Cambridge University Press, New York, pp 7–41 16. Koestler A (1967) The ghost in the machine. Hutchinson, London 17. Lapola DM, Martinelli LA, Peres CA, Ometto JP, Ferreira ME, Nobre CA, Joly CA (2014) Pervasive transition of the Brazilian land-use system. Nat Clim Chang 4(1):27–35 18. Milgram S (1967) The small-world problem. Psychol Today 2(1):60–67 19. Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii D, Alon U (2002) Network motifs: simple building blocks of complex networks. Science 298(5594):824–827 20. Noble IR, Slatyer RO (1981) Concepts and models of succession in vascular plant communities subject to recurrent fire. In: Gill AM, Groves RH, Noble IR (eds) Conference on fire and the Australian biota. Australian Academy of Science, Canberra. 9 Oct 1978, pp 311–335 21. Reynolds P (2019) Oracle of bacon. https://oracleofbaconorg/. Accessed 15 Dec 2019 22. Ripple WJ, Beschta RL (2004) Wolves and the ecology of fear: can predation risk structure ecosystems? AIBS Bull 54(8):755–766 23. Ripple WJ, Beschta RL (2012) Trophic cascades in Yellowstone: the first 15 years after wolf reintroduction. Biol Conserv 145(1):205–213 24. Shen-Orr SS, Milo R, Mangan S, Alon U (2002) Network motifs in the transcriptional regulation network of Escherichia coli. Nat Genet 31(1):64–68 25. Skarpe C, Aarrestad PA, Andreassen HP, Dhillion SS, Dimakatso T, du Toit JT, Halley DJ, Hytteborn H, Makhabu S, Mari M, Marokane W, Masunga G, Modise D, Moe SR, Mojaphoko R, Mosugelo D, Motsumi S, Neo-Mahupeleng G, Ramotadima M, Rutina L, Sechele L, Sejoe TB, Stokke S, Swenson JE, Taolo C, Vandewalle M, Wegge P (2004) The return of the giants: ecological effects of an increasing elephant population. Ambio 33(6):276–282 26. Soares-Filho B, Rajão R, Macedo M, Carneiro A, Costa W, Coe M, Rodrigues H, Alencar A (2014) Cracking Brazil’s forest code. Science 344(6182):363–364 27. von Bertalanffy L (1968) General systems theory: foundations, development, applications, vol 4. Braziller, New York 28. Walsh JR, Carpenter SR, Van der Zanden MJ (2016) Invasive species triggers a massive loss of ecosystem services through a trophic cascade. PNAS 113(15):4081–4085 29. Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small–world’ networks. Nature 393:440–442

Chapter 5

The Imbalance of Nature … Feedback and Stability in Ecosystems

Abstract  Ecosystems are dynamic and apparent stability may be an illusion of scale. Some ecosystems are subject to chronic disturbance. In dynamic systems, equilibrium is difficult to achieve and maintain. Systems often exhibit sensitivity to initial conditions and chaotic behaviour. Negative feedback promotes stability. Positive feedback is destabilizing, but also promotes the emergence of large-scale order in complex systems. Keywords  Attractor · Chaos · Equilibrium · Feedback · Logistic growth · Self-­ organisation · Sensitivity to initial conditions · Stability · Stigmergy · Succession

A peaceful lakeside scene such as this may seem unchanging, but over long periods the forests can undergo massive change. Shown here is Everitt Lake, Nova Scotia, which is discussed in Chap. 6.

© Springer Nature Switzerland AG 2020 D. G. Green et al., Complexity in Landscape Ecology, Landscape Series 22, https://doi.org/10.1007/978-3-030-46773-9_5

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It is a quiet spring evening by a lake in eastern Canada. The sun is disappearing behind the distant beech trees. The sound of loons singing drifts across the water. There is a perfect scene of peace and tranquillity. With everything so still, it is easy to believe it will stay this way forever. The reality is very different. Every day, the trees and plants you see must compete with one another for light, for water and nutrients, and for space. The birds you hear struggle to avoid being eaten, to find mates and to feed their young. One moment, frozen in time, hides the struggle for survival that goes on day after day, season after season, generation after generation. Likewise, in popular debate about conservation you will often hear people talking about the ‘balance of nature’. This expression reflects a widely-held assumption that balance, or equilibrium, is the natural state of any ecosystem. At best, this assumption is only partly correct. The illusion of equilibrium arises from the long time-scales associated with many ecological phenomena. Just as the lake at sunset looks still and unchanging, it is easy to be fooled into thinking that ecosystems are unchanging because we cannot see them change within our familiar time frame. Studying the ecology of a region is a bit like trying to hit a moving target. The problem is that we do not realise that it is moving. The time scale (and occasionally the spatial scale as well) on which things happen is sometimes so large that to a human observer an ecosystem looks still and unchanging. Often a single tree will live for many human lifetimes. The Coastal Redwoods (Sequoia sempervirens) of California, for instance, predate the arrival of Columbus1 and in many parts of Australia, there remain stands of Eucalyptus  regnans  still growing that predate European colonisation in 1788. Simple succession—the transition from one community to another—can take thousands of years to complete [13]. And yet those populations are competing and changing all the time. For instance, studies of long-­ term forest history, using pollen and other evidence continuously preserved in lakes, reveal that the forests of eastern North America have undergone vast transformations over the past 10,000 years (see Chaps. 6 and 7). A single fire can wipe out a forest that has persisted unchanged for hundreds or even thousands of years. The ecological processes that we see enacted in the ecosystems around us are just skirmishes in the never-ending struggles for survival that confront all species. Traditionally, researchers have focused on small parts of a landscape over relatively short time-frames (at most a human lifetime). Studies on these ‘small’ scales can lead to the false impression that the landscape as a whole is in a stable equilibrium. Ecologists now know that the appearance of permanence is often an artefact of the scales at which people operate. In this chapter, we begin by looking at what it means to speak of balance, stability and equilibrium. We will examine why these concepts are rarely if ever applicable to populations, and how they can nonetheless be useful for understanding wider ecological questions. We will conclude the chapter by considering how the idea of equilibrium has influenced ecology. However, we begin by considering how a single population changes over time.

 Some trees are more than 2000 years old.

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5.1  Feedback At the simplest level, connections in an ecosystem form chains. Wedgetail Eagles eat foxes, foxes eat rabbits; rabbits eat grass, so these connections constitute a chain, which creates an indirect effect of eagles on grass, or grass on eagles. More generally, suppose that species A eats species B, that species B eats species C, and that species C eats species D. These species form what is known as a food chain, where each species exerts an influence on other species further down the chain.2 Often these chains of interaction are circular. The number of foxes influences the number of rabbits, but the number of rabbits also influences the number of foxes. These circular interactions, known as feedback loops (Fig.  5.1), play a key role in ecosystems.

5.1.1  Negative Feedback Promotes Stability Negative feedback occurs when activation of one part of a system reduces the activity of the component that activated it. Negative feedback loops result in convergent behaviour. For example, increasing numbers of rabbits might lead to increasing numbers of foxes, which in turn would lead to reducing the numbers of rabbits again. In controlling any kind of system, negative feedback is desirable because it tends to return a system to its previous state following any perturbation. In theory, negative feedback would stabilise ecological systems. The eagle, fox and rabbit populations converge on numbers that are in balance. Any small, random change in the number of rabbits would be compensated instantly by an equivalent change in the

Fig. 5.1  Feedback loops are formed by the interactions between forests and various human activities (based on parts of the Limits to Growth model, see Chap. 9). The arrows indicate causal relationships: a plus sign means that the interaction is positive (e.g. forests promote ecotourism); a minus sign means the interaction is negative (e.g. forestry decreases the area of forests). Feedback loops are closed sequences of arrows, linked head to tail. Any loop with an odd number of negative arrows is a negative feedback loop; otherwise, the loop forms positive feedback. So Forestry and Forests form a negative feedback loop, but Forests and Ecotourism form a positive loop. The remaining three variables form a positive feedback loop, as does the loop that includes all five variables  We will explore food webs further in Chap. 7.

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number of foxes and then eagles (and vice versa), so the numbers of rabbits, foxes and eagles would remain roughly constant. In practice, the response of one population to another often lags, because it takes time to reproduce. This can lead to oscillations in population size, and to a wide variety of non-linear and chaotic behaviour [19]. Chaotic changes in population size occur when the time for response of reproduction by prey to changes in predation is much shorter than the response of predators to prey population.

5.1.2  Positive Feedback Promotes Self-Organization Sometimes, growth rate of a system component increases as its size increases (Fig. 5.1). This is positive feedback. It results in divergent behaviour. Rolling snowballs downhill, compound interest, growth of a plant seedling and nuclear explosions are all examples of positive feedback. In ecology, we see positive feedback loops wherever species compete with one another. An increase in one species leads to a decrease in its competitor, leading to accelerating increase in the first species population and decline in the competitor population. From the perspective of control, positive feedback is undesirable because it destabilizes a system. In a system where positive feedback operates, any small deviation leads to further deviations, and the deviations grow larger and larger until the entire system collapses. Paradoxically, this process, which is so destructive when trying to control and maintain the status quo in a complex system, can also act as a constructive process that promotes self-organization and enables order to emerge out of random chaos. This is possible because positive feedback can make a small, local variation grow to become a large, global property. One example of positive feedback contributing to the emergence of order occurs in the connectivity avalanche, which we saw in Chap. 4. As more edges are added to a graph, connected clumps grow larger. As they grow larger, it becomes increasingly likely that they will become linked to other large clumps, so the rate of growth accelerates. We can see another example of positive feedback creating order in the way stigmergy [34] creates order within ant colonies (Fig. 5.2). Stigmergy is a process in which actions of individuals collaborate with each other indirectly through their environment. In an ant colony, ants carry items (eggs, food, rubbish etc.) around more or less at random. However, they follow a simple rule that like goes with like. If an ant is carrying an egg and it comes upon another egg, then it will drop the egg it is carrying. The next time and ant comes along carrying an egg, it adds to the pair of eggs left by the first ant. In this way, it cooperates indirectly with the first ant. Positive feedback comes into play because the larger a clump of eggs grows, the more likely it is that other ants will deposit eggs there too.

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Fig. 5.2  A simple example of stigmergy. An initially random collection of objects (a) is rapidly sorted into concentrated piles (b) by ants without any central scheme. The ants sort food and other resources within their nest by following two simple rules: (1) if you find an item, pick it up; (2) if you find a similar item, drop the one you are carrying. The process relies on positive feedback, which operates while the ants move items between piles. The rate of loss of items from a pile depends on the ratio of perimeter to area. This ratio decreases as piles grow larger, so large piles grow at the expense of smaller piles Fig. 5.3 Exponential growth in the world’s human population. This curve shows the historical trend in the growth of the world’s human population (in billions) over the last 2000 years

5.2  The Big Get Bigger In ideal conditions, any population will grow at a rate that is proportional to its present size. Just like compound interest acting on an investment, or a snowball rolling down a hill, the bigger it gets, the faster it grows. It is the result of a positive feedback loop (see Chap. 2). In mathematics, this kind of behaviour is known as exponential growth.3 Fig. 5.3 shows that human population has grown exponentially over the last 2000 years. A graph of mould spreading on a loaf of bread would look quite similar. Both show exponential growth.

3  In the exponential model the rate of increase in population size at any time t is proportional to its size Nt. This means that Nt can be calculated from the initial size of the population No, the intervening time t and the rate of growth r, using the formula: Nt = No ert.

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In theory, all populations will grow exponentially when there are unlimited resources. In practice, resources are never unlimited. When there is not enough food, mortality increases or reproduction declines as animals die of starvation or use their energy reserves to survive instead of to breed. In the case of plants, they grow until the overall cost of respiration exceeds the output of photosynthesis, so the net growth rate falls to zero. Populations may also be limited by less obvious resources: nesting sites, hiding places, minerals, sunlight and water. In the case of bread mould, the mould runs out of bread. In the case of humans, we have yet to meet our limiting resource. Indeed, mainstream economic theory is based naïvely on the idea of eternal, exponential growth. The effect of a limiting resource on exponential population growth is a pattern known as logistic or sigmoidal growth (Fig. 5.4a). The logistic growth model represents exponential reproduction, but with increasing mortality as the population increases.4 The population increases exponentially at first, while the influence of mortality is very small. As the population increases, however, mortality becomes significant. At a certain population size, growth is balanced by mortality. This maximum population size is termed the carrying capacity. If the reproduction rate is low, then logistic growth produces the smoothly rising curve of population size that levels off when it reaches the carrying capacity (Fig. 5.4a). In reality, however, the above condition is very often violated. What happens if the reproduction rate is rapid? Or if the food supply varies from year to year? Instead of remaining more or less constant as population size approaches the carrying capacity, a population may continue growing until its demand exceeds the available resources. At this point it crashes due to elevated mortality and population falls. It then recovers and crashes in a cyclical fashion. In this latter case, the population oscillates at regular intervals. This population, represented in Fig. 5.4b, may be a herbivore grazing on plants that take time to recover, causing the herbivore population to crash and when the plants recover, the growth of the herbivore population may lag due to delays in reproduction. These lags and recovery times are responsible for the oscillations. Note that oscillations in population size can arise in other ways. For instance, the limiting resource might take time to disappear, or its disappearance might be disproportional to the size of the population. In many cases, the availability of this resource is not solely determined by the population size, but is influenced by outside factors as well. Food supply, for example, is affected by many factors, including soil moisture, soil nutrition, seasonal variation of sunlight and temperature, rainfall, runoff and evapotranspiration. Additional factors can include diseases or invasive species. The combination of all these factors and population levels can result in an irregular  The equation for logistic growth is:

4



X ( t + 1) = rX ( t ) (1 − X ( t ) / K )



where X(t) is the population size at time t, r is the rate of population growth, and K is the carrying capacity.

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Fig. 5.4  Logistic growth for populations that reproduce continually (a) and seasonally (b). The cycles occur because the population size repeatedly jumps above the carrying capacity, which initiates a population crash

oscillation or chaotic variation. Often, the resource in question is itself a population, which may drop more quickly than linearly in response to increasing use. In any of the above scenarios, the population will fluctuate. As the population grows, resources become severely and patchily depleted. Many individuals die, but not before they have further depleted resources that the survivors require. Thus, the population can drop rapidly and suddenly (‘crash’) for no obvious reason. In economics, fluctuations of this kind are known as boom and bust cycles. One of the biggest factors that loosen the connection between population growth and resource availability is seasonal variation. This can cause the fluctuations depicted in Fig. 5.4b. If breeding occurs in spring when resources are abundant, but food becomes scarce in winter, then populations can grow unsustainably in spring, only to suffer winter famine. This, incidentally, is a major reason why populations in the tropics are more stable and less vulnerable to extinction than populations at higher latitudes, as we mentioned in Chap. 1. It also results in migration for some species. A severe crash can drive a small population to extinction. For this reason, ecologists and mathematicians are very interested in the causes of fluctuating population sizes. Unfortunately, one of their major findings is that both the timing of population booms and crashes, and the size of these fluctuations, are usually unpredictable. There are two reasons for this unpredictability of population booms and crashes. First, as mentioned above, there may be large, effectively random factors that alter the carrying capacity or the growth rate from year to year, such as climate variation and interactions with other species. Secondly, if the population growth rate is high, then the logistic equation falls into a state in which even the smallest random disturbance can change its entire future behaviour. This state is called deterministic chaos. Later in this chapter we will look at deterministic chaos in detail: what it is, why it occurs, and how it affects predictions about populations. Also, we have not considered the additional complexity generated by migration within and between

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populations. We will look at that issue in Chap. 6. First, however, we need to recast these ideas in more general terms by looking at populations as dynamic systems.

5.3  Who Eats Whom? How do we describe the changes that go on in a system? Often we can express important properties, such as population size or annual profit, in terms of numbers. So changes within the system are reflected by the changes in the values of the variables that represent these properties. Changes through time are known as the dynamics of a system. A set of interrelated variables (for example, population size) whose values change through time is called a dynamic system. Besides the variables used to represent values of particular quantities, dynamic systems may also include variables to represent rates of change through time. For example, in a predator-prey system, such as foxes eating rabbits, we would have variables for the size and reproduction rate of each population (Fig. 5.5). So the dynamic system would have four variables: • • • •

size of the rabbit population; size of the fox population; rate of change for the rabbit population; rate of change for the fox population.

Besides system variables, a dynamic system also includes parameters. These are fixed values that determine particular properties and relationships. For instance, in the case of the foxes and rabbits system, we have four parameters: • • • •

birth rate (per season) for rabbits; birth rate (per season) for foxes; mortality rate for rabbits; mortality rate for foxes.

To fully define this dynamic system we also need to specify the initial conditions, that is the values of all the variables (in this case, the sizes of the populations) at the start. We can represent the behaviour of a dynamic system in several ways. The most obvious is to plot values of the variables (vertical axis) versus time (horizontal axis), Fig. 5.5 Population changes in a simple predator-prey system

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Fig. 5.6  Population changes in a model of foxes preying on rabbits. (a) A plot of population size for foxes (solid line) and rabbits (dashed line) through time. (b) The trajectory of the system through time. In this case the system is unstable, so the trace spirals outwards until the fox population crashes to zero (not shown)

as shown in Fig. 5.6a. Where there is more than one variable involved, an alternative way to view change is to plot one variable against another to show the trajectory that the system follows. For instance, for the foxes and rabbits system (Fig. 5.6b) each point on the graph defines a state of the system. So the area on the graph shows all possible states of the system. This is called the state space of the system. The trajectory shown in Fig. 5.6b is diverging from a point, so the system is moving away from equilibrium. Note that the notion of predator-prey relations or even food chains is an oversimplification. In reality, many prey species and predators, herbivores, producers and decomposers form complex food webs (see Chap. 7) that can exhibit high levels of redundancy or high levels of compensatory feedback behaviour. Finally, it is important to realise that dynamic systems are not confined to systems that take numerical values, such as logistic growth or the foxes and rabbit model. Any system that passes from one state to another is a dynamic system. That includes all computers as well as (say) animal behaviour or the working of the brain. So all of these systems potentially exhibit properties similar to those we are discussing here.

5.3.1  Equilibrium and Stability Equilibrium is a state of balance. A system in equilibrium shows no tendency to change of its own accord. A variable is said to be an equilibrium point if there is no tendency for its value to change through time. That is, the forces that act to change it are balanced out. Place an orange at the bottom of a fruit bowl and it will stay there. It is in equilibrium. If you roll it up the side of the bowl and release it, then it will roll back

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Fig. 5.7  Three kinds of equilibria. In each case, a ball starts at rest on a surface, as shown, and does not move unless pushed. Left: if moved, the ball rolls back down the slope to its initial position. Centre: if moved, then the ball continues to roll away down the slope. Right: if moved, the ball will remain in its new position

down to the bottom again. So the equilibrium is stable. In general, any equilibrium point is said to be stable if the system tends to return to the equilibrium when displaced from it (Fig. 5.7). This condition means that the rate of displacement away from the equilibrium point must be opposite in sign from the direction of the displacement. If the system continues to move away from an equilibrium point when a displacement occurs, it is called unstable. If a displacement leads to no further movement, it is termed neutral. One of the simplest ways to visualise the above ideas is to imagine the system as a ball rolling across a landscape (Fig. 5.7). When the ball stops rolling, we say it is in equilibrium: it has no tendency to move of its own accord. Suppose the ball has rolled down into a hollow in the grass. Outside forces, such as the wind, might push the ball up one side of the hollow a little, but it quickly rolls back down into the deepest part of the hollow. In this case, the ball has found a stable equilibrium. The walls of the bowl provide negative feedback. Note that for most systems, a very strong perturbation can push the system away from an equilibrium point, just as a really strong gust of wind might push the ball out of the hollow and back into the wider landscape. There are limits to stability. That is to say, stable equilibria are usually relatively, but not absolutely stable. Now suppose that, instead of falling into a hollow, the ball has been placed at the summit of a steep hill. Even a slight nudge will tend to send it rolling down the hill and far away. Sitting poised at the top of the hill, the ball is at an unstable equilibrium point. The sides of the hill provide positive feedback. Lastly, imagine not a ball, but a heavy cube sitting on a flat plane. It will move in response to an outside force, but halt as soon as the force is removed. It is in neutral equilibrium. Neutral equilibrium is not a likely scenario in any ecological context; it is included here merely for completeness. Formally analysing the stability of an ecological system can lead to useful insights. In many parts of the world, dryland ecosystems are turning into deserts. Because drylands include a high proportion of farms, this loss is devastating for farming communities and contributes to famines and poverty in the developing world. For many years, the reason for this widespread pattern remained unclear. The geographer Jonathon Phillips analysed the stability of a well-known systems model of dryland ecosystems [22]. He was able to show that drylands are likely to be inherently unstable because of positive feedback between key factors such as

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vegetation cover, light absorption, rainfall, soil moisture and erosion. Perturbations to dryland ecosystems, Phillips showed, would not average out, but would persist and grow. The dryland is like a ball poised at the top of a hill: even a small disturbance can magnify to push it far from its original state. Thus, chance disturbances can easily cause a dryland to become desert, but will rarely if ever cause a desert to revert to dryland. Sustainable management of drylands therefore requires close monitoring and quick correction of small disturbances. Stability analysis has also been used to understand the dynamics of food webs and ecological communities. We will look at these systems in Chap. 7.

5.3.2  Transients and Attractors As we saw above, a dynamic system is stable if its behaviour converges over time to a fixed point. However, not all systems behave in such a simple way. In many systems the behaviour converges to a cycle, instead of a single equilibrium point. Cycles of this kind are called limit cycles. Equilibria and limit cycles are both cases of a class of behaviour known as attractors. In general, an attractor is any set of states to which a system is ‘attracted’. In an equilibrium that set consists of a single point. In a limit cycle the system passes through the set in sequence, then repeats. Later we shall see that chaotic systems also have attractors, but with more complicated structure. The term ‘attractor’ is used because the system tends to be attracted towards those states. In like fashion, there are sometimes states - repellers - that repel the system. A characteristic of an attractor is that once a system falls into an attractor, it stays there unless subject to some external disturbance. Thus stable equilibria are attractors and unstable equilibria are repellers. In the desert-dryland system mentioned above, deserts are attractors, while drylands are repellers. Starting from an arbitrary initial state, a dynamic system will normally pass through a series of transient states until it falls into an attractor (Fig. 5.8). A system may contain more than one attractor. If so, which attractor it ends up in depends on where it starts. In general, any attractor is surrounded by a region known as its basin of attraction.

Fig. 5.8  How dynamic systems work. In each case the label T denotes transient behaviour and the label A indicates an attractor. (a) An equilibrium is a point attractor. (b) Limit cycles plotted against time. (c) Limit cycles in the foxes and rabbits system (plotted in state space as foxes versus rabbits)

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We can imagine a dynamic system as a ball dropped into a rugged landscape and pushed at random by the wind. The basin of attraction behaves like the slope of a valley: once the ball has passed the rim, it tends to roll downwards, and becomes increasingly unlikely to escape the attractor as it falls. Which valley the ball finally lands in may be impossible to predict, because it depends on its precise initial position, as well as on the magnitude and direction of random perturbation by the wind.

5.3.3  Sensitivity to Initial Conditions So far we have discussed dynamic systems that are either in equilibrium or have fallen into a limit cycle. A common feature of non-linear systems is that when they are far from equilibrium, their behaviour often diverges, rather than converging. This property, which makes them sensitive to small changes in their initial conditions, was discovered independently by several researchers. Edward O. Lorenz was trying to simulate weather systems on an early computer. He was puzzled to see that when he restarted a simulation, using results he had printed previously, the output rapidly diverged from his earlier findings. After just a handful of iterations, the model behaved in a completely different way. At first, Lorenz thought he must have accidentally entered an incorrect digit; but he had not. Intrigued, Lorenz repeated his experiment – and once again got completely different results (Fig. 5.9). There could be only one explanation. The difference was caused by tiny changes introduced when numbers in the simulation were rounded off for printing. Around the same time, mathematical biologist Robert May was experimenting with discrete logistic models of rapid population growth. Varying the initial population size by just a tiny fraction, May found that in models that were completely identical, apart from small differences in initial conditions, the magnitude and Fig. 5.9  The Lorenz attractor. This (three-­ dimensional) figure shows the trajectory followed by Lorenz’s weather model. Within the attractor, the position is unpredictable, but if a system starts outside, it soon falls into the attractor. The attractor itself is densely braided. If you restart the system at any two points close together, then they soon drift apart

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Fig. 5.10  Sensitivity to initial conditions in discrete logistic growth. The two populations are identical, except that the population represented by the dashed curve started with one extra individual. Notice that after just a few cycles the dotted line deviates from the solid line

timing of fluctuations in population size rapidly diverged. Even the most miniscule variation would effectively make population size impossible to predict within a few tens of generations (Fig. 5.10). The sensitivity of non-equilibrium systems to changes in their initial conditions has many implications. Perhaps the most important is that the detailed behaviour of such systems is unpredictable. For example, we can make reasonably accurate predictions of the weather up to a few days ahead. But beyond that limit, real weather patterns become increasingly different from our predictions. Similarly, if we have an excellent understanding of the ecology of a species and it lives in a predictable environment, we may be able to make reasonably accurate predictions of population size across a generation or two, but as time progresses, our confidence rapidly declines and predictions become meaningless. Economic projections are likewise limited. The essential reason for this unpredictability is that we can never measure every aspect of weather or populations or economic activity with total accuracy. The sensitivity of the system means that these errors rapidly become magnified. Likewise, small random fluctuations add tiny errors that are also magnified by the system. This unpredictable behaviour is the domain of chaos theory.

5.3.4  The Onset of Chaos Chaos theory deals with systems that are deterministic, but whose long-term behaviour is unpredictable nevertheless. As we saw in the previous section, this unpredictability is a result of the sensitivity to initial conditions. By deterministic, we mean that no outside perturbation is necessary to generate unpredictable behaviour: it emerges from our finite knowledge of initial conditions. Chaotic behaviour may be unpredictable, but it is constrained. To understand how it is constrained, first consider constraints in ordered conditions. As mentioned above, ordered systems contain either equilibrium points or limit cycles. Both of these may be considered as attractors, in the sense that if the system starts in some arbitrary state, or else is disturbed, then it is attracted steadily closer to the equilibrium or to the limit cycle.

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5  The Imbalance of Nature … Feedback and Stability in Ecosystems

Fig. 5.11  The onset of chaos in a model of logistic growth. (a) When the growth rate is low, the population rises smoothly to an equilibrium level. (b) At higher rates the system settles into cycles of period 2. (c) As the growth rate increases still further the cycling period doubles. (d) Eventually this period doubling leads to chaotic behaviour. These changes reveal the system’s sensitivity to growth rate

The equivalent of an equilibrium point or a limit cycle for a chaotic system is a strange attractor. The difference is that a strange attractor is a far more complex structure (Fig. 5.9). Whereas a limit cycle is a closed loop, a strange attractor never crosses itself. That is, the system is never in exactly the same state twice. It may exhibit quasi-cyclic behaviour at times, but the cycles are never closed and separate strands of the strange attractor may be braided together with infinite density. The extreme sensitivity of chaotic systems to initial conditions is a consequence of this braiding: changing the initial state even slightly moves a chaotic system onto a different braid. It is sometimes called the butterfly effect: in theory, even the minute air disturbance caused by a butterfly beating its wings could tip large-scale weather patterns in unpredictable ways. When first discovered, chaos seemed to be a mysterious and puzzling anomaly in otherwise orderly systems. However, it is now known that the transition of a dynamic system from equilibrium to chaos is itself a very well-defined process. Typically, as we change the value of a particular number, called the order parameter L, the onset of chaos is associated with a transition of behaviour from stable equilibria, through limit cycles of increasing period, to chaotic variation associated with a strange attractor. This transition occurs via a process known as period doubling. That is, at certain critical values of L, the period of the limit cycle doubles (Fig. 5.11). Furthermore,

5.3  Who Eats Whom?

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Fig. 5.12 Period-doubling for logistic growth. The horizontal axis shows changes in the value of the growth parameter r. The vertical axis shows the size x of the population relative to the ‘carrying capacity’. The points on the graph show the values of x that lie on the attractor of the process for each value of r

after the period starts doubling, the increase in L required to produce the next doubling steadily decreases.5 We can represent the transition to chaos using a bifurcation diagram (Fig. 5.12). This diagram plots the attractors of the system against L. In the example shown the system settles into equilibrium when L is small. This value changes as L increases, so it appears as a single line in the diagram (Fig. 5.12). However as L increases the system switches to limit cycles of period 2. So the line bifurcates into two lines - one for each point in the limit cycle. As L increases again, the system bifurcates again and again, so that two lines become four, then eight, etc. (Fig. 5.12). Notice that at any bifurcation (Fig. 5.12), the pattern from that point onwards resembles the overall pattern, except that it is squashed. This resemblance is no coincidence: if the two axes are rescaled by appropriate amounts, then the pattern is identical to the overall pattern. This decrease occurs in a fixed way with the changes in L between successive sets of doublings falling in the fixed ratio of 4.66920 [10]. Although the above theory suggests that chaotic behaviour should be a widespread property of ecological systems, demonstrating chaos in practice has proven difficult. One reason for this difficulty is that it is difficult to control populations sufficiently to rule out the influence of external factors causing their unpredictable dynamics. There has been no simple way to test whether a natural system is chaotic or merely random [24]. Another possibility is that chaos may tend to be damped by natural selection, as populations with chaotic dynamics are vulnerable to extinction because of their large and frequent fluctuations. This seems improbable, however, because natural selection acting on individuals is often likely to favour reproductive strategies that lead to chaotic population dynamics [11]. One of the clearest laboratory experiments on ecological chaos concerns flour beetles (Tribolium sp.). By manipulating adult mortality and recruitment, researchers were able to demonstrate that flour beetle population dynamics changed from 5  In the case of discrete logistic growth x’ = Lx(1 − x), for example, the period is 1 (i.e. an equilibrium point) for 2