Modelling Insect Populations in Agricultural Landscapes (Entomology in Focus, 8) 3031430972, 9783031430978

Table of contents :
Foreword
Preface
Acknowledgments
Contents
About the Editors
1 Introduction
References
2 Introducing Different Modelling Scenarios to Entomologists
2.1 Introduction
2.2 Types of Models
2.2.1 Mechanistic and Statistical Modelling
2.2.2 Experimental Dynamic Models: Combining Data and Population Theory at Different Observation Scales
2.2.2.1 Order Diptera
2.2.2.2 Order Coleoptera
2.2.2.3 Order Lepidoptera
2.2.2.4 Order Hymenoptera
2.2.2.5 Order Blattaria (Isoptera)
2.2.2.6 Order Hemiptera
2.2.2.7 Order Orthoptera
2.2.3 Outbreak Analysis with Ecological Theory
2.2.4 Models and the Agricultural Landscape
References
3 Monte Carlo Simulations to Model the Behaviour of Agricultural Pests and Their Natural Enemies
3.1 Introduction
3.2 An `Appetiser'
3.3 Random Number Generators
3.4 Parameters and Their Statistical Distribution
3.5 Analysing the Obtained Results
3.6 Looking for Optimised Values
3.7 A More Advanced Example
3.8 Conclusion
References
4 Representing Insect Movement in Agricultural Areas using Spatially Explicit Models
4.1 Introduction
4.2 Modelling Insect Movement Step by Step
4.3 Case Studies
4.3.1 Modelling Movement of Hosts and Their Respective Natural Enemies in a Crop Area (Weber et al., 2022)
4.3.1.1 Host Dispersal Function
4.3.1.2 Parasitoid Foraging Function
4.3.2 Modelling Movement of S. frugiperda in an Heterogeneous Landscape Considering Evolution of Resistance (Tomé et al., 2022)
4.3.2.1 Crop Scenario
4.3.2.2 Biological Parameters and Genetic Component
4.3.2.3 Insect Dynamics
4.3.2.4 Genetic Component
4.4 Final Considerations
References
5 Transition Models Applied to Interactions Involving Agricultural Pests
5.1 Introduction
5.2 Motivational Study
5.2.1 Host Plants and Insect Colony
5.2.2 Experimental Design
5.3 Statistical Methodology
5.3.1 Categorical Data
5.3.2 Generalized Logit Model
5.3.3 Markov Chains
5.3.4 Generalized Logit Transition Model
5.3.4.1 Tests to Assess Stationarity
5.4 Analysis of the Motivational Study
5.5 Computational Recourses
5.6 Final Considerations
References
6 Agent-Based Modelling with Rules Inspired by Game Theory: Case Studies in Insect Resistance Management
6.1 Fundamental Concepts
6.1.1 Components, Classification, and Examples of Games
6.2 Simulation Studies with a Spatial Game Model
6.2.1 Example 1: Competition Involving Two Lepidopteran Species in Cotton Fields
6.2.2 Example 2: Evolution of Resistance Driven by Competition
6.3 Practical Insights from the Mathematical Model and Future Directions
References
7 Pest Biocontrol and Allee Effects Acting on the Control Agent Population: Insights from Predator-Prey Models
7.1 Introduction
7.2 A Predator-Prey Model with Allee Effect in the Pest and Agent Populations
7.3 A Tritrophic Food Chain Model with Allee Effect in the Pest and Agent Populations
7.4 A Tritrophic Omnivory Food Web Model with Allee Effect in the Pest and Agent Populations
7.5 Discussion
References
8 On Matrix Stability and Ecological Models
8.1 Introduction
8.2 Notation
8.3 Lyapunov Stability
8.3.1 Equilibria and Asymptotics
8.3.2 Lyapunov Functions and Linear Matrix Inequalities
8.3.3 Application: Nonlinear Matrix Models
8.4 Diagonal and D-Stability
8.4.1 Continuous-Time Case
8.4.1.1 Application: Lotka–Volterra Systems
8.4.2 Discrete-Time Case
8.4.2.1 Application: Population Projection Matrices
8.5 Quiescent Stability
8.5.1 Continuous-Time Case
8.5.1.1 Application: Predator–Prey Dynamics
8.5.2 Discrete-Time Case
8.5.3 Application: Host–Parasitoid Dynamics
8.6 Diffusive and Connective Stability
8.6.1 Proper Cones and Diffusively Coupled Systems
8.6.2 Heterogeneous Dispersal
8.6.3 Application: Invasion Dynamics
8.6.3.1 Background
8.6.3.2 Model
8.6.3.3 Parametrisation
8.6.3.4 Stability
8.7 Conclusion
References
9 Machine Vision Applied to Entomology
9.1 Introduction
9.2 Machine Vision Pipeline
9.3 Insect Classification Using Deep Learning Methods
9.4 Insect Localisation with Deep Learning Methods
9.5 Platforms of Computing
9.6 Final Considerations
References
10 Bayesian N-Mixture Models Applied to Estimating Insect Abundance
10.1 Introduction
10.2 N-Mixture Model for a Closed Population
10.3 Model Extensions
10.3.1 N-Mixture Models for Multiple Species
10.3.2 N-Mixture Models for an Open Population
10.3.3 N-Mixture Models for Zero-Inflated Data
10.4 Case Study: Bee Abundance
10.5 Discussion
10.6 Code
10.6.1 Bayesian N-Mixture Models for Closed Populations in JAGS
10.6.2 Bayesian N-Mixture Models for Open Populations in JAGS
10.6.3 Covariance Diagnostic
References
11 Tools for Assessing Goodness of Fit of GLMs: Case Studies in Entomology
11.1 Introduction
11.2 Generalized Linear Models
11.2.1 The Normal Model
11.2.2 The Gamma Model
11.2.3 The Inverse Gaussian Model
11.2.4 The Poisson Model
11.2.5 The Negative Binomial Model
11.2.6 The Binomial Model
11.3 Residuals
11.3.1 Raw Residuals
11.3.2 Pearson Residuals
11.3.3 Deviance Residuals
11.4 Influence Measures
11.4.1 DFBETA
11.4.2 DFFIT
11.4.3 Cook's Distance
11.5 Half-Normal Plots with a Simulated Envelope
11.6 Examples
11.6.1 Biological Control of Ticks
11.6.2 Sustainable Management of Parasitic Nematodes Using Bioagents: The `Plant Height' Data
11.7 Discussion
References
Index

Citation preview

Entomology in Focus  8

Rafael A. Moral Wesley A. C. Godoy   Editors

Modelling Insect Populations in Agricultural Landscapes

Entomology in Focus Volume 8

Insects are fundamentally important in the ecology of terrestrial habitats. What is more, they affect diverse human activities, notably agriculture, as well as human health and wellbeing. Meanwhile, much of modern biology has been developed using insects as subjects of study. To reflect this, our aim with Entomology in Focus is to offer a range of titles that either capture different aspects of the diverse biology of insects or their management, or that offer updates and reviews of particular species or taxonomic groups that are important for agriculture, the environment or public health. The series results from an agreement between Springer and the Entomological Society of Brazil (SEB) and as such may lean towards tropical entomology. The aim throughout is to provide reference texts that are simple in their conception and organization but that offer up-to-date syntheses of the respective areas, offer suggestions of future directions for research (and for management where relevant) and that don’t shy away from offering considered opinions. Editorial Committee Series Editor Sam Elliot is Associate Professor in Entomology at the Universidade Federal de Viçosa (Brazil), also coordinates the Postgraduate Programme in Entomology currently rated maximally by the relevant authority in Brazil (CAPES) and is Associate Editor at Ecology and Evolution. He works on diverse aspects of insect-microbe interactions, with emphases on leafcutter ants, noctuid caterpillars, triatomine bugs, entomopathogenic fungi and microbial control of pests. Adam Hart is Professor of Science Communication at the University of Gloucestershire (UK). His particular interest is in social insects but he has written and broadcasted on a broad range of biological subjects. He presents documentaries for BBC Radio 4, BBC4 and BBC2, as well as the weekly BBC radio programme Science in Action. Eugenio Oliveira is Assistant Professor in Entomology at the Universidade Federal de Viçosa (Brazil), and scholar researcher of the Brazilian National Council of Scientific and Technologic Development (CNPq). He has also working as Associate Editor at the journals Neotropical Entomology and Invertebrate Neuroscience. He works principally on insect neurophysiology, applying this in particular to entomological/agricultural questions.

Rafael A. Moral • Wesley A. C. Godoy Editors

Modelling Insect Populations in Agricultural Landscapes

Editors Rafael A. Moral Department of Mathematics and Statistics Maynooth University Maynooth, Ireland

Wesley A. C. Godoy Departamento de Entomologia e Acarologia Universidade de São Paulo Piracicaba, São Paulo, Brazil

ISSN 2405-853X ISSN 2405-8548 (electronic) Entomology in Focus ISBN 978-3-031-43097-8 ISBN 978-3-031-43098-5 (eBook) https://doi.org/10.1007/978-3-031-43098-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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 Paper in this product is recyclable.

To our colleagues and respective students, whose enthusiasm for disseminating the best of their efforts working on highly relevant research projects combining theory and experimentation in entomology resulted in this book. With this book, we hope to reach the broadest possible audience interested in insects and models.

Foreword

Agricultural pests are among the most challenging problems that ancient and modern agriculture have faced anywhere in the planet. For centuries, several techniques and methodological approaches have been created and improved with the specific goal of controlling these pests. Particularly for insect pests, interesting technologies other than the use of insecticides have emerged with promising results within the applied and theoretical scope, such as plant resistance and the use of biological control agents. Although the use of mathematical and computational skills have admittedly huge implications for the better understanding of many biological and ecological processes, how to transfer these important tools to the applied scientific field is challenging. For example, mathematical and computational models can significantly help to predict pest outbreaks, to understand their movement, specific behaviours, how their abundances and damage vary in time and space, and how trophic interactions may occur under field conditions. These are powerful insights that must be developed to ultimately become useful tools for scientists and agricultural practitioners. The primary goal of Modelling Insect Populations in Agricultural Landscapes is to provide readers with modelling approaches related to agricultural pests and their interactions with other species (e.g., with their host plant and with their natural enemies) and to construct an updated theoretical framework with several applications in the field of entomology. In summary, the chapters of this book include: • A presentation and discussion of mathematical and computational tools used to model the dynamics and the behaviour of agricultural insect pests and their interactions with their host plants and their natural enemies in space and time • A clear illustration of how fundamental ecological theory, represented by mathematical models, can be used for modelling pest dispersion and biocontrol • An introduction to different classes of statistical models that can be useful for understanding pest interactions, how insect resistance is developed in agricultural systems, and for estimating insect abundance

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Foreword

• A clear presentation of how theoretical frameworks can be used in entomological studies, in its most different aspects The intention of the book is to show that modelling is a powerful instrument, whether by creating new ways of thinking about a problem or developing new skills to solve it. I would still venture to say that modelling is more than that: it is a necessity if we really want to create innovative strategies to solve new and old problems that impact our planet. The editors and collaborators did a splendid job showing modern and relevant approaches in modelling insect populations. In doing so, they have addressed with rare ability how to connect theoretical knowledge with applied entomology. Federal University of São Paulo, São Paulo, Brazil

Marcelo Nogueira Rossi

Preface

The term agricultural landscape has become increasingly present in scientific texts. Its use makes possible a more comprehensive view of the aspects involved in crops and refuge areas, facilitating a deeper understanding of the biological processes inherent to the system. The concept of an agricultural landscape involves interactions between their primary elements: agriculture, environment, and natural resources. A brief reflection on this topic indicates that agricultural landscapes are highly complex systems, demanding specific tools to retain their main aspects, thus facilitating a quick interpretation of their nature and respective problems. The concept of a mathematical model meets this demand since models provide a simplification of systems, and because of this, they can be seen as “caricatures” of complex systems in which the main features correspond to the essence of the system. Entomology provides systems rich in interactions between organisms and plants distributed in several different agricultural mosaics, which offer intriguing interactive scenarios (especially in tropical areas), but also pose challenging demands in terms of experimental design and mathematical and computational modelling. Although the models have played a significant role in understanding population patterns and predicting relevant scenarios for pest ecology, management, and conservation of natural enemies, laboratory and field studies can provide validation or a basis for theoretical developments, and have led to numerous advances in entomology over many years. The connection between models and data experimentally obtained has contributed to the establishment of a highly relevant database for studying agricultural landscapes. This book also reflects several decades of research development and teaching to undergraduate and graduate students at South American and European universities. Each chapter brings the authors’ perspectives on what is essential when carrying out research at the interface between entomology and mathematical/statistical/computational modelling. Maynooth, Ireland Piracicaba, São Paulo, Brazil

Rafael A. Moral Wesley A. C. Godoy

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Acknowledgments

The São Paulo Research Foundation (FAPESP-SPARCBIO 2018/02317-5) and National Council for Scientific and Technological Development (CNPQ 305367/2020-0) partially supported the research in this book.

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Contents

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wesley A. C. Godoy and Rafael A. Moral

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Introducing Different Modelling Scenarios to Entomologists . . . . . . . . . Wesley A. C. Godoy and Rafael A. Moral

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3

Monte Carlo Simulations to Model the Behaviour of Agricultural Pests and Their Natural Enemies . . . . . . . . . . . . . . . . . . . . . . Eric Wajnberg

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Representing Insect Movement in Agricultural Areas using Spatially Explicit Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adriano Gomes Garcia, Igor Daniel Weber, and Maysa Pereira Tomé

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Transition Models Applied to Interactions Involving Agricultural Pests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Idemauro Antonio Rodrigues de Lara, Carolina Reigada, and Cesar Augusto Taconeli Agent-Based Modelling with Rules Inspired by Game Theory: Case Studies in Insect Resistance Management. . . . . . . . . . . . . . . José Bruno Malaquias and Cláudia Pio Ferreira

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Pest Biocontrol and Allee Effects Acting on the Control Agent Population: Insights from Predator-Prey Models . . . . . . . . . . . . . . . 101 Lucas dos Anjos and Michel Iskin da S. Costa

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On Matrix Stability and Ecological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Blake McGrane-Corrigan and Oliver Mason

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Machine Vision Applied to Entomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Gabriel R. Palma, Conor P. Hackett, and Charles Markham

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Bayesian N-Mixture Models Applied to Estimating Insect Abundance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Niamh Mimnagh, Andrew Parnell and Estevão Prado

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Tools for Assessing Goodness of Fit of GLMs: Case Studies in Entomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Darshana Jayakumari, John Hinde, Jochen Einbeck, and Rafael A. Moral

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

About the Editors

Rafael A. Moral is an Associate Professor of Statistics at Maynooth University. With a background in Biology and a PhD in Statistics from the University of São Paulo, he has a deep passion for teaching and conducting research in statistical modelling applied to Ecology, Wildlife Management, Agriculture, and Environmental Science. As director of the Theoretical and Statistical Ecology Group, Rafael brings together a community of researchers who use mathematical and statistical tools to better understand the natural world. As an alternative teaching strategy, he produces music videos to promote Statistics in social media and in the classroom. Wesley A. C. Godoy is an Associate Professor of Entomology at the University of São Paulo. With a passion for insect ecology and population dynamics, he leads the Laboratory of Ecology and Forest Entomology. His research encompasses a wide range of applications in ecology, including trophic interactions, and the development and application of mathematical, computational, and statistical modelling. Wesley’s interests include the practical implementation of ecological modelling to advance the fields of entomology and pest science.

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Chapter 1

Introduction Wesley A. C. Godoy

and Rafael A. Moral

Agriculture is vital in feeding the world’s population by providing essential food and raw materials. However, insects of agricultural importance can pose a significant challenge to farmers, causing damage to cultivated plants and productivity (Ferreira & Godoy, 2014). On the other hand, certain groups of insects are of extreme relevance for providing ecosystem services such as pollination, nutrient recycling, and biological control (Noriega et al., 2018). Mathematical and computational modelling emerges as a valuable tool to better understand the dynamics of these insects in agricultural landscapes and to develop effective strategies for integrated pest management and conservation of species relevant to ecosystem services (Ferreira & Godoy, 2014). These tools are capable of describing ecological fluctuation patterns that determine relevant scenarios, such as outbreaks, economic damage, or even the prediction of spatiotemporal trends in which pests and natural enemies occur (Sokame et al., 2021). Mathematical and computational models are simplified representations of reality based on principles and assumptions described by mathematical equations and computational rules resulting in algorithms, which allow simulating and predicting the behavior of complex systems (MacLeod, 2021). In the context of insects of agricultural importance in agricultural landscapes, these models can provide valuable insights into interactions between insects, host plants, environmental factors, and agricultural practices (Alexandridis et al., 2021). These models can address various issues related to insects of agricultural importance. For example, they can help predict insect population dynamics over time and space, considering

W. A. C. Godoy () Departamento de Entomologia e Acarologia, Universidade de São Paulo, Piracicaba, São Paulo, Brazil e-mail: [email protected] R. A. Moral Department of Mathematics and Statistics, Maynooth University, Maynooth, Ireland © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. A. Moral, W. A. C. Godoy (eds.), Modelling Insect Populations in Agricultural Landscapes, Entomology in Focus 8, https://doi.org/10.1007/978-3-031-43098-5_1

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factors such as insect ecology, climatic conditions, host plant distribution, and the effects of agricultural practices, such as pesticide use (Tonnang et al., 2017). In addition, the models can also be used to evaluate the effectiveness of different integrated pest management strategies, such as using pheromones to monitor and control insect populations, adopting crop rotation, and establishing refuge areas for natural enemies (Tang & Cheke, 2008). The application of mathematical and computational models in agricultural landscapes allows an evidence-based descriptive and predictive approach for the sustainable management of insects of agricultural importance (Benítez et al., 2022). By combining empirical data with computer simulations, researchers and farmers can make more informed decisions about management practices, thereby maximizing efficiency and minimizing negative impacts on the environment (Ferreira & Godoy, 2014; Tonnang et al., 2017). However, we must emphasize that the models are subject to uncertainties. Therefore, their application must be accompanied by field data, experimental validation, and continuous adaptation to guarantee the relevance and reliability of their (inferential) results (Garcia et al., 2021). Therefore, mathematical and computational models play a crucial role in the study and management of agriculturally relevant insect species in agricultural landscapes (Fernandes et al., 2022). These tools offer an integrated and predictive approach, helping to develop more efficient and sustainable pest management strategies, thus contributing to food security and agricultural sustainability (Mellaku & Sebsibe, 2022). Modern approaches and technologies have revolutionized the field of agricultural entomology, offering new avenues to gain a comprehensive understanding of the challenges faced in this domain (Høye et al., 2021). One powerful approach that has emerged is the integration of mathematical and computational models, which enable researchers to delve deeper into complex problems and unravel intricate dynamics (Ferreira & Godoy, 2014). In agricultural entomology, mathematical models provide a framework to describe and analyze various processes related to insect populations, such as growth, dispersal, and environmental interactions (Garcia et al., 2021). These models are built on existing knowledge, empirical data, and biological principles. Researchers can simulate and explore different scenarios in a controlled environment by incorporating variables such as insect life cycles, behavior, and ecological factors. Computational models, on the other hand, leverage advanced computing technologies to enhance the accuracy and complexity of mathematical models. With the exponential growth in computing power, researchers can now develop sophisticated simulations that mirror real-world agricultural systems (Maino & Kearney, 2015). These models can account for multiple variables simultaneously, allowing for a more comprehensive understanding of insect population dynamics and their impact on crop health. Furthermore, modern technologies such as remote sensing, satellite imagery, and geographical information systems (GIS) play a crucial role in entomological modelling (Acharya & Thapa, 2015). These tools provide a wealth of spatial and temporal data that can be integrated into mathematical models. Additionally, emerging techniques like machine learning and data mining have the potential to revolutionize agricultural entomology modelling (Gherman et al., 2023).

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These approaches can be used to analyze large volumes of data and identify complex patterns that may not be apparent through traditional statistical methods. By using artificial intelligence, researchers can uncover hidden relationships, optimize pest management strategies, and even predict future insect outbreaks with higher precision. Integrating modern approaches and technologies into mathematical and computational models has opened up new horizons in agricultural entomology (Høye et al., 2021; Palma et al., 2023). These tools give researchers a better comprehension of the intricate problems and dynamics involved, enabling more effective pest management strategies and ultimately contributing to sustainable agricultural practices (Gherman et al., 2023). Over the next chapters, this book brings together recent approaches developed by researchers from different continents, with contributions focused on the field of entomology in different scenarios of the agricultural landscape. Below we make a brief description of the content of each of the chapters that will be found sequentially. Chapter 2 brings important examples of modelling applications for different scenarios in entomology applied to different insect orders, and with this, it is expected to provide the reader with an expansive vision of the approaches currently employed in mathematical and computation modelling. Chapter 3 brings a comprehensive description of methods involved in simulation methods for complex models, which can be mathematically intractable. Although the large number of parameters in a model allows it to capture intricate patterns and relationships within the data, model complexity can induce loss of computational efficiency or mathematical tractability. Monte Carlo methods involve sampling random numbers to estimate complex systems or solve analytically intractable problems (Harrison, 2009). With this, method simulations rely on generating a large number of random samples to approximate the behavior of a complex system (Kitou et al., 2008). Optimization techniques can help improve the efficiency of these simulations by optimizing the sampling scheme or reducing the variance of the estimated quantities, leading to more accurate results (Forrest, 1996). In Chap. 4, the authors provide an opportunity to discuss the movement of insects from different perspectives and use interesting examples of application to individual-based models. Understanding the factors that drive the movement of insects in an agricultural landscape is crucial for effective pest management and natural enemy conservation efforts (Garcia et al., 2020). Several critical events influenced by ecological theory can trigger insect movement and shape their spatial distribution within agricultural ecosystems (Mazzi & Dorn, 2012). Insects are strongly influenced by the availability of food, shelter, and breeding sites. Ecological theory suggests that when resources become scarce or limited in a particular area, arthropods may exhibit movement (De Meester & Bonte, 2010). The theory of habitat fragmentation highlights the importance of landscape structure in influencing insect movement. When agricultural landscapes are fragmented into smaller patches with increased spatial isolation, insects may face challenges finding appropriate resources and mates. In response, they may move between patches to locate resources and establish new populations (Wimberly et al., 2018).

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Insect movement can also be influenced by interactions with other species, such as predators, competitors, or parasitoids (Clark et al., 2019). The ecological theory emphasizes that the presence or absence of these interactions can impact insect dispersal patterns. Insects respond to various environmental cues, such as temperature, humidity, wind patterns, and chemical signals (Jaworski & Hilszczan´ ski, 2014; Fleischer & Krieger, 2018). These cues play a crucial role in guiding insect movement across agricultural landscapes. For instance, favorable weather conditions, such as warm temperatures and favorable winds, can facilitate the longdistance dispersal of certain insects (Prinster et al., 2020). In contrast, adverse weather events or unsuitable environmental conditions may trigger movement to more favorable microhabitats or regions within the landscape (Prinster et al., 2020). Understanding the ecological theory behind critical events that influence insect movement in agricultural landscapes allows for better predicting and managing pest outbreaks, conservation of beneficial insects, and sustainable agricultural practices. Chapter 5 essentially introduces the reader interested in predictions of categorical responses in insect populations using transition models in discrete time. Transition models, which combine stochastic processes and generalized linear models (GLMs), are powerful tools in entomology (Lara et al., 2020). These models can provide a framework to study and understand the dynamics of insect populations and the factors that influence their behavior (Lara et al., 2020). By incorporating both stochasticity and the effects of covariates, transition models offer valuable insights into insect ecology and aid in developing effective pest management strategies (Stirzaker, 2005). By combining this stochastic component with GLMs, entomologists can examine how covariates, such as habitat characteristics or climatic variables, influence the probabilities with which insects transition between different behaviors (Bod’ová et al., 2018). Chapter 6 focuses explicitly on insecticide usage and the emergence of resistance in regions where transgenic technology has been adopted, by employing game theoretical approaches. Introducing game theory to the field of entomology opens up exciting possibilities for understanding the intricate dynamics of insect behavior, interactions, and population dynamics (Leimar & McNamara, 2023). Game theory, as a branch of mathematics and economics, provides a framework for studying strategic decision-making in situations where the outcomes of individuals’ actions depend not only on their own choices but also on the choices of others (Leimar & McNamara, 2023). By applying game theory concepts to entomology, researchers can gain insights into insect interactions, evolutionary strategies, and the dynamics of insect communities (Brown & Staˇnková, 2017). In Chap. 7, the authors analyze how trophic interactions and the Allee effect may be connected with each other, and especially how the Allee effect can play a significant role when pest control is implemented. This phenomenon refers to the positive correlation between population size and individual fitness in certain species (Kramer et al., 2018). In other words, individuals within a population benefit from high population sizes, resulting in improved reproductive success and overall survival rates (Kramer et al., 2018). One key aspect of the Allee effect is its impact on population growth. When a pest population experiences the Allee effect, small

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populations struggle to grow due to limited mate finding, reduced cooperative behaviors, or decreased resource utilization (Dennis, 2002). Consequently, pest populations exhibiting the Allee effect can be more vulnerable to control measures, as they are less likely to rebound quickly from population reductions. The Allee effect also affects the spread of pest populations, such that the probability of successfully invading new habitats or dispersing to new areas decreases as population size diminishes (Taylor & Hastings, 2005; Boukal & Berec, 2009). This knowledge can guide the development of targeted containment strategies, focusing on reducing population numbers to critical thresholds where the Allee effect weakens the pests’ ability to expand their range. Moreover, the Allee effect highlights the importance of early detection and rapid response in pest management. Identifying and addressing pest populations in the early stages, when they are still relatively small, can be more effective due to their increased vulnerability to control measures. Pest control programs should prioritize proactive monitoring and early intervention to prevent population growth and potential outbreaks (Courchamp & Gascoigne, 2008). In Chap. 8, the authors provide a helpful review about matrix stability analysis aiming to give a theoretical background for possible application in entomological systems, particularly incorporating spatiotemporal structures. Matrix models are commonly used in mathematical frameworks to describe the population dynamics of insect species within ecosystems (Erguler et al., 2022). Matrix stability analysis plays a crucial role in assessing the long-term behavior and resilience of entomological systems (Mwalusepo et al., 2014), particularly when incorporating spatiotemporal structures. In entomological models, matrix stability takes into account the spatial distribution and temporal dynamics of insect populations across different habitats or geographical regions (Vinatier et al., 2011). This approach acknowledges that insect communities exhibit spatial heterogeneity, with variations in environmental conditions, resource availability, and species interactions among different locations. By incorporating spatiotemporal structures, it becomes possible to assess stability across multiple locations simultaneously (Vinatier et al., 2011). This enables researchers to investigate how spatial heterogeneity, dispersal patterns, and environmental gradients influence the persistence and stability of insect populations. Understanding stability in entomological systems is essential for conservation and management efforts. It helps identify vulnerable insect populations and areas at high risk of population decline or extinction (Kehoe et al., 2021). This information guides the allocation of conservation resources and assists in the development of effective management strategies. Additionally, it facilitates the evaluation of how environmental changes, such as habitat fragmentation or climate change, impact the stability and persistence of insect communities (Death and Winterbourn, 1994). Chapter 9 provides an essential introduction, bringing an interesting vision of the interface between computer vision and machine learning (i.e., “machine vision”) tools applied to entomology (Høye et al., 2021). Machine vision is a rapidly advancing field that has found practical applications in various domains (Chen et al., 2002). In recent years, it has emerged as a powerful tool in entomology, offering

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new possibilities for studying and understanding insects (Høye et al., 2021). By leveraging sophisticated algorithms and image processing techniques, it enables researchers to extract meaningful information from images or videos of insects, facilitating a range of entomological studies and applications (Asefpour Vakilian & Massah, 2013). When applied to entomology, machine vision involves using computer-based systems to analyze visual data related to insects automatically (De Cesaro & Rieder, 2020). These visual data can include images of insects captured through cameras, microscopes, or other imaging devices. The primary objective is to extract valuable information about the insects’ morphology, behavior, abundance, or species identification (De Cesaro & Rieder, 2020; Høye et al., 2021). One significant application of machine vision in entomology is the automated identification and classification of insect species. Traditionally, insect identification has relied on expert taxonomists who visually examine physical characteristics under microscopes. However, this process is time-consuming and requires specialized knowledge. Machine vision systems offer an alternative by automating the identification process using image analysis algorithms (Kasinathan & Uyyala, 2021). These algorithms can learn and recognize patterns, allowing for the rapid and accurate identification of insects based on their visual features. Furthermore, machine vision can assist in studying insect behavior (Høye et al., 2021). By analyzing videos or image sequences, researchers can track insects’ movement patterns, activity levels, or interactions in their natural habitats or controlled laboratory settings (Gerovichev et al., 2021). This capability provides valuable insights into insect ecology, population dynamics, and the impact of environmental factors on their behavior (Gerovichev et al., 2021; Høye et al., 2021). Additionally, machine vision applied to entomology plays a crucial role in pest management and agricultural applications (Clark, 2020). Insects can cause significant damage to crops, and early detection is vital for implementing timely control measures (Abd El-Ghany et al., 2020). By using machine vision systems, detecting and monitoring pests in real time is possible, enabling the implementation of targeted and efficient pest control strategies (Bjerge et al., 2022). In Chap. 10, the potential of Bayesian N-mixture models is presented as an alternative approach to be used in problems of demography and population dynamics of insects. Bayesian N-mixture models have emerged as a powerful statistical tool for predicting animal abundance data in ecological studies (Joseph et al., 2009). This modelling approach has significant applications in entomology, providing valuable insights into insect population dynamics, estimating abundance, and addressing important questions related to insect ecology, epidemiology, management, and conservation (Manica et al., 2019). In entomology, Bayesian N-mixture models are particularly useful when studying insect populations that are challenging to monitor directly or when traditional monitoring methods are impractical or costly (Gomez et al., 2018). These models employ Bayesian statistics principles to estimate the true abundance of insect populations based on imperfect detection data obtained from surveys or monitoring efforts. Bayesian inference allows for integrating prior knowledge or beliefs about the population parameters, such as abundance or detection probabilities, with the

1 Introduction

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observed data (Conn et al., 2018). This integration provides a more robust and flexible approach to population estimation compared to traditional frequentist methods. Bayesian N-mixture models can handle complex data structures commonly encountered in entomological studies. For instance, these models can account for temporal variation in population abundance, spatial heterogeneity, and the influence of environmental covariates on insect populations. This flexibility enables researchers to investigate the drivers of population dynamics and understand how environmental factors impact insect abundance (Van Klink et al., 2022). Another advantage of Bayesian N-mixture models is their ability to handle missing or incomplete data, a common issue in entomological studies. Incomplete detection or survey data can arise due to observer bias, incomplete sampling, or imperfect monitoring techniques. By explicitly modelling the detection process and estimating the detection probability, these models can overcome these limitations and provide more accurate estimates of population abundance. Bayesian N-mixture models have proven helpful in various entomological applications. They have been employed to estimate the abundance of rare or elusive insect species, assess the impact of habitat quality or management practices on insect populations, and evaluate the effectiveness of conservation interventions or pest control strategies. Additionally, these models can account for uncertainties in the estimation process, providing researchers with credible intervals or posterior distributions that quantify the uncertainty around population abundance estimates. Chapter 11 provides an important review on goodness-of-fit approaches for GLMs, aiming to give to reader a general vision about potentialities and limitations of diagnostic tools. Goodness-of-fit assessment is highly relevant in entomology as it helps to evaluate the appropriateness and reliability of statistical models used to understand insect populations, their dynamics, and the factors influencing their behavior (Moral et al., 2020). Goodness-of-fit evaluation ensures that the chosen statistical model provides accurate and reliable estimates of insect population parameters, such as abundance, density, or occupancy. In summary, this book brings together a combination of state-of-the-art methodologies and theory allied with practical examples and data in agricultural entomology. We hope it is helpful to the interested reader when applying current methods to their own research, as well as when developing and/or extending methods applied to modelling insect populations in agricultural landscapes.

References Abd El-Ghany, N. M., Abd El-Aziz, S. E., & Marei, S. S. (2020). A review: Application of remote sensing as a promising strategy for insect pests and diseases management. Environmental Science and Pollution Research, 27, 33503–33515. https://doi.org/10.1007/s11356-020-095172 Acharya, M., & Thapa, R. (2015). Remote sensing and its application in agricultural pest management. Journal of Agricultural and Environmental, 16, 43–61. https://doi.org/10.3126/ aej.v16i0.19839

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Alexandridis, N., Marion, G., Chaplin-Kramer, R., Dainese, M., Ekroos, J., Grab, H., Jonsson, M., Karp, D. S., Meyer, C., O’Rourke, M. E., Pontarp, M., Poveda, K., Seppelt, R., Smith, H. G., Martin, E. A., & Clough, Y. (2021). Models of natural pest control: Towards predictions across agricultural landscapes. Biological Control, 163, 104761. https://doi.org/10.1016/ j.biocontrol.2021.104761 Asefpour Vakilian, K., & Massah, J. (2013). Performance evaluation of a machine vision system for insect pests identification of field crops using artificial neural networks. Archives of Phytopathology and Plant Protection, 46, 1262–1269. https://doi.org/10.1080/ 03235408.2013.763620 Benítez, M., Rosell, J. A., & Perfecto, I. (2022). Editorial: Mathematical modeling and complex systems in agroecology. Frontiers in Sustainable Food Systems, 6, 1–3. https://doi.org/10.3389/ fsufs.2022.829551 Bjerge, K., Mann, H. M. R., & Høye, T. T. (2022). Real-time insect tracking and monitoring with computer vision and deep learning. Remote Sensing in Ecology and Conservation, 8, 315–327. https://doi.org/10.1002/rse2.245 Bod’ová, K., Mitchell, G. J., Harpaz, R., Schneidman, E., & Tkaˇcik, G. (2018). Probabilistic models of individual and collective animal behavior. PLoS One, 13, 1–30. https://doi.org/ 10.1371/journal.pone.0193049 Boukal, D. S., & Berec, L. (2009). Modelling mate-finding Allee effects and populations dynamics, with applications in pest control. Population Ecology, 51, 445–458. https://doi.org/10.1007/ s10144-009-0154-4 Brown, J. S., & Staˇnková, K. (2017). Game theory as a conceptual framework for managing insect pests. Current Opinion in Insect Science, 21, 26–32. https://doi.org/10.1016/j.cois.2017.05.007 Chen, Y. R., Chao, K., & Kim, M. S. (2002). Machine vision technology for agricultural applications. Computers and Electronics in Agriculture, 36, 173–191. https://doi.org/10.1016/ S0168-1699(02)00100-X Clark, R. D. (2020). Putting deep learning in perspective for pest management scientists. Pest Management Science, 76, 2267–2275. https://doi.org/10.1002/ps.5820 Clark, R. E., Basu, S., Lee, B. W., & Crowder, D. W. (2019). Tri-trophic interactions mediate the spread of a vector-borne plant pathogen. Ecology, 100, 1–8. https://doi.org/10.1002/ecy.2879 Conn, P. B., Johnson, D. S., Williams, P. J., Melin, S. R., & Hooten, M. B. (2018). A guide to Bayesian model checking for ecologists. Ecological Monographs, 88, 526–542. https://doi.org/ 10.1002/ecm.1314 Courchamp, F., & Gascoigne, J. (2008). 5. Conservation and management. Oxford University Press. De Cesaro, J. T., & Rieder, R. (2020). Automatic identification of insects from digital images: A survey. Computers and Electronics in Agriculture, 178, 105784. https://doi.org/10.1016/ j.compag.2020.105784 De Meester, N., & Bonte, D. (2010). Information use and density-dependent emigration in an agrobiont spider. Behavioral Ecology, 21, 992–998. https://doi.org/10.1093/beheco/arq088 Death and Winterbourn. (1994). Environmental stability and community persistence : A multivariate perspective author ( s ): Russell G . Death and Michael J . Winterbourn Source. Journal of the North American Benthological Society, 13. , Published by: The Univ. 13, 125–139. Dennis, B. (2002). Allee effects in stochastic populations. Oikos, 96, 389–401. https://doi.org/ 10.1034/j.1600-0706.2002.960301.x Erguler, K., Mendel, J., Petri´c, D. V., Petri´c, M., Kavran, M., Demirok, M. C., Gunay, F., Georgiades, P., Alten, B., & Lelieveld, J. (2022). A dynamically structured matrix population model for insect life histories observed under variable environmental conditions. Scientific Reports, 12, 1–15. https://doi.org/10.1038/s41598-022-15806-2 Fernandes, L. D., Mata, A. S., Godoy, W. A. C., & Reigada, C. (2022). Refuge distributions and landscape connectivity affect host-parasitoid dynamics: Motivations for biological control in agroecosystems. PLoS One, 17, 1–17. https://doi.org/10.1371/journal.pone.0267037 Ferreira, C. P., & Godoy, W. A. C. (2014). Entomology in Focus 1 Ecological modelling applied to entomology. Springer.

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Fleischer, J., & Krieger, J. (2018). Insect pheromone receptors – Key elements in sensing intraspecific chemical signals. Frontiers in Cellular Neuroscience, 12, 1–14. https://doi.org/ 10.3389/fncel.2018.00425 Forrest, S. (1996). Genetic algorithms. ACM Computing Surveys, 28, 77–80. https://doi.org/ 10.1145/234313.234350 Garcia, A. G., Godoy, W. A. C., Cônsoli, F. L., & Ferreira, C. P. (2020). Modelling movement and stage-specific habitat preferences of a polyphagous insect pest. Movement Ecology, 8, 1–11. https://doi.org/10.1186/s40462-020-00198-7 Garcia, A. G., Malaquias, J. B., Ferreira, C. P., Tomé, M. P., Weber, I. D., & Godoy, W. A. C. (2021). Ecological modelling of insect movement in cropping systems. Neotropical Entomology, 50, 321–334. https://doi.org/10.1007/s13744-021-00869-z Gerovichev, A., Sadeh, A., Winter, V., Bar-Massada, A., Keasar, T., & Keasar, C. (2021). High throughput data acquisition and deep learning for insect ecoinformatics. Frontiers in Ecology and Evolution, 9, 1–11. https://doi.org/10.3389/fevo.2021.600931 Gherman, I. M., Abdallah, Z. S., Pang, W., Gorochowski, T. E., Grierson, C. S., & Marucci, L. (2023). Bridging the gap between mechanistic biological models and machine learning surrogates. PLoS Computational Biology, 19, e1010988. https://doi.org/10.1371/ journal.pcbi.1010988 Gomez, J. P., Robinson, S. K., Blackburn, J. K., & Ponciano, J. M. (2018). An efficient extension of N-mixture models for multi-species abundance estimation. Methods in Ecology and Evolution, 9, 340–353. https://doi.org/10.1111/2041-210X.12856 Harrison, R. L. (2009). Introduction to Monte Carlo simulation. AIP Conference Proceedings, 1204, 17–21. https://doi.org/10.1063/1.3295638 Høye, T. T., Ärje, J., Bjerge, K., Hansen, O. L. P., Iosifidis, A., Leese, F., Mann, H. M. R., Meissner, K., Melvad, C., & Raitoharju, J. (2021). Deep learning and computer vision will transform entomology. Proceedings of the National Academy of Sciences, 118, e2002545117. Jaworski, T., & Hilszcza´nski, J. (2014). The effect of temperature and humidity changes on insects development their impact on forest ecosystems in the expected climate change. Forest Research Papers, 74, 345–355. https://doi.org/10.2478/frp-2013-0033 Joseph, L. N., Elkin, C., Martin, T. G., & Possingham, H. P. (2009). Modeling abundance using N-mixture models: The importance of considering ecological mechanisms. Ecological Applications, 19, 631–642. https://doi.org/10.1890/07-2107.1 Kasinathan, T., & Uyyala, S. R. (2021). Machine learning ensemble with image processing for pest identification and classification in field crops. Neural Computing and Applications, 33, 7491–7504. https://doi.org/10.1007/s00521-020-05497-z Kehoe, R., Frago, E., & Sanders, D. (2021). Cascading extinctions as a hidden driver of insect decline. Ecological Entomology, 46, 743–756. https://doi.org/10.1111/een.12985 Kitou, S., Hanai, K., & Obata, Y. (2008). Monte carlo simulation CT. Proc 15th EGS users’ meet Japan 3. pp. 41–46. Kramer, A. M., Berec, L., & Drake, J. M. (2018). Editorial: Allee effects in ecology and evolution. The Journal of Animal Ecology, 87, 7–10. https://doi.org/10.1111/1365-2656.12777 Lara, I. A. R., Moral, R. A., Taconeli, C. A., Reigada, C., & Hinde, J. (2020). A generalized transition model for grouped longitudinal categorical data. Biometrical Journal, 62, 1837– 1858. https://doi.org/10.1002/bimj.201900394 Leimar, O., & McNamara, J. M. (2023). Game theory in biology: 50 years and onwards. Philosophical Transactions of the Royal Society B, 378, 20210509. MacLeod, M. (2021). The applicability of mathematics in computational systems biology and its experimental relations. European Journal for Philosophy of Science, 11, 1–21. https://doi.org/ 10.1007/s13194-021-00403-3 Maino, J. L., & Kearney, M. R. (2015). Testing mechanistic models of growth in insects. Proceedings of the Royal Society B: Biological Sciences, 282. https://doi.org/10.1098/rspb.2015.1973 Manica, M., Caputo, B., Screti, A., Filipponi, F., Rosà, R., Solimini, A., della Torre, A., & Blangiardo, M. (2019). Applying the N-mixture model approach to estimate mosquito

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population absolute abundance from monitoring data. Journal of Applied Ecology, 56, 2225– 2235. https://doi.org/10.1111/1365-2664.13454 Mazzi, D., & Dorn, S. (2012). Movement of insect pests in agricultural landscapes. The Annals of Applied Biology, 160, 97–113. https://doi.org/10.1111/j.1744-7348.2012.00533.x Mellaku, M. T., & Sebsibe, A. S. (2022). Potential of mathematical model-based decision making to promote sustainable performance of agriculture in developing countries: A review article. Heliyon, 8, e08968. https://doi.org/10.1016/j.heliyon.2022.e08968 Moral, R. A., Hinde, J., Ortega, E. M. M., Demétrio, C. G. B., & Godoy, W. A. C. (2020). Location-scale mixed models and goodness-of-fit assessment applied to insect ecology. Journal of Applied Statistics, 47, 1776–1793. https://doi.org/10.1080/02664763.2019.1693522 Mwalusepo, S., Tonnang, H. E. Z., Massawe, E. S., Johansson, T., & Le Ru, B. P. (2014). Stability analysis of competing insect species for a single resource. Journal of Applied Mathematics, 2014. https://doi.org/10.1155/2014/285350 Noriega, J. A., Hortal, J., Azcárate, F. M., Berg, M. P., Bonada, N., Briones, M. J. I., Del Toro, I., Goulson, D., Ibanez, S., Landis, D. A., Moretti, M., Potts, S. G., Slade, E. M., Stout, J. C., Ulyshen, M. D., Wackers, F. L., Woodcock, B. A., & Santos, A. M. C. (2018). Research trends in ecosystem services provided by insects. Basic and Applied Ecology, 26, 8–23. https:/ /doi.org/10.1016/j.baae.2017.09.006 Palma, G. R., Godoy, W. A. C., Engel, E., Lau, D., Galvan, E., Mason, O. Markham, C., Moral, R. A. (2023). Pattern-based prediction of population outbreaks. https://doi.org/10.1016/ j.ecoinf.2023.102220. Prinster, A. J., Resasco, J., & Nufio, C. R. (2020). Weather variation affects the dispersal of grasshoppers beyond their elevational ranges. Ecology and Evolution, 10, 14411–14422. https:/ /doi.org/10.1002/ece3.7045 Sokame, B. M., Tonnang, H. E. Z., Subramanian, S., Bruce, A. Y., Dubois, T., Ekesi, S., & Calatayud, P. A. (2021). A system dynamics model for pests and natural enemies interactions. Scientific Reports, 11, 1–14. https://doi.org/10.1038/s41598-020-79553-y Stirzaker. (2005). Stochastic process and models. Oxford University Press. Tang, S., & Cheke, R. A. (2008). Models for integrated pest control and their biological implications. Mathematical Biosciences, 215, 115–125. https://doi.org/10.1016/j.mbs.2008.06.008 Taylor, C. M., & Hastings, A. (2005). Allee effects in biological invasions. Ecology Letters, 8, 895–908. https://doi.org/10.1111/j.1461-0248.2005.00787.x Tonnang, H. E. Z., Hervé, B. D. B., Biber-Freudenberger, L., Salifu, D., Subramanian, S., Ngowi, V. B., Guimapi, R. Y. A., Anani, B., Kakmeni, F. M. M., Affognon, H., Ndjomatchoua, F. T., Pedro, S. A., Nana, P., Johansson, T., Nedorezov, L. V., Tanga, C. M., Nana, P., Fiaboe, K. M., Mohamed, S. F., Maniania, N. K., Ekesi, S., & Borgemeister, C. (2017). Advances in crop insect modelling methods—Towards a whole system approach. Ecological Modelling, 354, 88–103. https://doi.org/10.1016/j.ecolmodel.2017.03.015 Van Klink, R., Bowler, D. E., Gongalsky, K. B., & Chase, J. M. (2022). Long-term abundance trends of insect taxa are only weakly correlated. Biology Letters, 18, 1–6. https://doi.org/ 10.1098/rsbl.2021.0554 Vinatier, F., Tixier, P., Duyck, P. F., & Lescourret, F. (2011). Factors and mechanisms explaining spatial heterogeneity: A review of methods for insect populations. Methods in Ecology and Evolution, 2, 11–22. https://doi.org/10.1111/j.2041-210X.2010.00059.x Wimberly, M. C., Narem, D. M., Bauman, P. J., Carlson, B. T., & Ahlering, M. A. (2018). Grassland connectivity in fragmented agricultural landscapes of the north-central United States. Biological Conservation, 217, 121–130. https://doi.org/10.1016/j.biocon.2017.10.031

Chapter 2

Introducing Different Modelling Scenarios to Entomologists Wesley A. C. Godoy

and Rafael A. Moral

Abstract Spatiotemporal modelling has received increased attention from entomologists as it meets various demands, from understanding ecological patterns to predicting pest outbreaks. Recent challenges from modern agriculture require efficient analytical methods to analyze population dynamics and insect movement patterns. Also, the efficacy of control actions and outbreak predictions depends on well-trained professionals properly employing models to guarantee successful management programs. This chapter aims to provide the reader with an overview of the diversity of analytical tools available for application in various scenarios, accompanied by examples of applications in entomology taxonomically classified by orders of insects. Keywords Insects · Mathematical modelling · Population dynamics

2.1 Introduction The history of mathematical and computational modelling reveals that models were developed to address the need for understanding how biological processes influence dynamic systems (Brodland, 2015). These systems can be formalized using mathematical and algorithmic languages that integrate real data obtained from laboratory or field observations with mathematical functions or computational rules. Such models are proposed to analyze important ecological patterns and explain spatiotemporal trends (Pointer et al., 2021). In entomology, various models have been employed to examine population dynamics, relationships between insects and their environment, and interactions between species. These models are particularly

W. A. C. Godoy () Departamento de Entomologia e Acarologia, Universidade de São Paulo, Piracicaba, São Paulo, Brazil e-mail: [email protected] R. A. Moral Department of Mathematics and Statistics, Maynooth University, Maynooth, Ireland © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. A. Moral, W. A. C. Godoy (eds.), Modelling Insect Populations in Agricultural Landscapes, Entomology in Focus 8, https://doi.org/10.1007/978-3-031-43098-5_2

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valuable in understanding trophic relationships and are crucial in integrated pest management programs (Ferreira & Godoy, 2014). There has also been an increasing demand for models capable of forecasting trends in time and space, thus qualifying them as predictive models (Tonnang et al., 2017). The interest of entomologists in mathematical modelling has been significantly influenced by publications focusing on theoretical ecology, which emerged as a discipline in the 1920s. This emergence was inspired by Verhulst’s (1838) proposition of the logistic growth model, the introduction of models for interspecific interactions by Lotka (1925) and Volterra (1926), and the development of models for predator-prey or host-parasitoid relationships by Nicholson and Bailey (1935). Although the modern approach to modelling in entomology involves investigating more complex issues, particularly considering different scales in space and time, the fundamental framework still traces back to theoretical ecology principles inspired by the pioneering work of the abovementioned authors. The current methodology incorporates approaches that were not available decades ago, such as technological advancements enabling extending analyses spatially. This has facilitated the inclusion of individual movement in population growth models (Tilman & Kareiva, 1997; Law et al., 2003; Lima et al., 2009; van Klink et al., 2022). In recent years, models incorporating artificial intelligence concepts have gained attention from researchers interested in deep learning to address various aspects of entomology, including frameworks for implementing machine learning solutions (Høye et al., 2021; Gerovichev et al., 2021; Palma et al., 2023). In this chapter, we aim to present a mini-review highlighting examples of approaches and demands in modelling applied to entomology, attracting growing interest from entomologists studying ecological patterns across different taxonomic groups and populations. Additionally, there is a focus on projecting relevant trends for insect biodiversity conservation, including natural enemies and integrated pest management. Through various scenarios, we illustrate the use of models in emphasizing the dynamics of economically significant insects in space and time.

2.2 Types of Models 2.2.1 Mechanistic and Statistical Modelling Modelling generally describes the relationship between process parameters, quality, and performance attributes. Mechanistic models utilize mathematical expressions that effectively describe the physical or biological processes, as they are based on natural laws and employed for process optimization (Gherman et al., 2023). All parameters in the mechanistic model have biological definitions and, as a result, can be independently measured regardless of the dataset. On the other hand, statistical models utilize mathematical expressions to better describe the data while making probabilistic assumptions involving parameters of interest. In other words, their

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purpose is to separate signal from noise to identify patterns within existing data, while describing variability (Tredennick et al., 2021).

2.2.2 Experimental Dynamic Models: Combining Data and Population Theory at Different Observation Scales In this topic, we have selected several examples of models that utilize data obtained from laboratory or field studies, focusing on species representing various insect orders. The volume of published articles emphasizing mathematical, statistical, or computational modelling in this field is extensive. Here we have chosen to present species that effectively represent different taxonomic groups and illustrate various scenarios explored using diverse models.

2.2.2.1

Order Diptera

The order Diptera exhibits a notable impact on agriculture, particularly concerning fruit cultivation, where two families, Drosophilidae and Tephritidae, with species commonly known as fruit flies, play a significant role. However, even species not directly associated with agriculture have proven excellent study models for proposing and testing analytical frameworks applicable to economically important species. Experimental laboratory studies have long been crucial in supporting advancements in insect ecology (Desharnais, 2005; Pointer et al., 2021). Comprehending ecological aspects involves utilizing theoretical models and an interface between modelling and laboratory experimentation. This combination allows for relatively quick results by establishing experimental insect colonies and minimizing challenges related to survival, replication, and estimation of demographic parameters (Pointer et al., 2021). Furthermore, the possibility of numerically generating long time series in species that exhibit a short life cycle is highly relevant to test hypotheses about population regulation mechanisms (de Godoy et al., 2023). Although we are constantly faced with recognizing the complexity and unpredictability of nature, observable at different spatial scales, the development and use of models based on theoretical ecology and with parameters obtained through laboratory experimentation allow the construction and understanding of high-quality ecological phenomena. Therefore, models are relevant for understanding natural systems’ complexity (Desharnais, 2005; Pointer et al., 2021; Westwick & Rittschof, 2021). Among the many examples of insect species created in the laboratory for research that emphasizes modelling in microcosms, species of the order Diptera undoubtedly stand out, with initial emphasis on studies with populations of Drosophila melanogaster. Pioneering studies on population growth theory demonstrated the density dependence in experimental populations of D. melanogaster, employing

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analyses based on finite difference equations to investigate population stability in relation to allelic selection at the individual level (Mueller & Ayala, 1981). Since then, numerous studies have been conducted at the interface of laboratory experiments and model development. Notably, the study conducted by Prout and McChesney (1985) deserves mention, as they analyzed survival and fertility data obtained in the laboratory and combined it with a population growth model based on Ricker theory. Estimates obtained in the laboratory were analyzed by functions dependent on intraspecific density in D. melanogaster to assess population stability through numerical simulations. The mathematical formalism of Prout and McChesney (1985) paved the way for several studies with experimental populations of blowflies, with the implementation of the spatial dimension and exploration of the parametric space of different demographic parameters, both in the deterministic and stochastic scope (Castanho et al., 2006; Serra et al., 2007, 2011), in addition to providing a theoretical basis for the mathematical analysis of biological invasions by blowflies in the Americas (Coutinho et al., 2012). Other drosophilids, Drosophila suzukii and Zaprionus indianus, of high relevance for fruit growing, were also investigated in the laboratory, aiming to evaluate processes of intra and interspecific competition with statistical and dynamic models of life stages structured by the Leslie Matrix (de Paiva et al., 2023). Drosophila suzukii has been studied with models following different approaches, such as a mathematical model with insertion of dispersion was based on results of laboratory experiments to investigate the persistence dynamics of the insect in a scenario of exchange of host plants (de Godoy et al., 2023). Drosophila suzukii was also investigated considering different temperature profiles, by combining laboratory data and mechanistic models, with results mapping the potential distribution in areas capable of supporting high insect population densities (Langille et al., 2016).

2.2.2.2

Order Coleoptera

The order Coleoptera has a wide diversity of species, and several of them are directly or indirectly associated with agricultural and forestry systems, with high economic relevance, either by the harmful action on crops by species considered pests or by species widely recognized as natural enemies. There are many population models proposed for this order involving laboratory experiments, with emphasis on the beetles called flour beetles, belonging to the genus Tribolium (Tenebrionidae), which, in addition to their importance as pests of stored products, constitute an excellent model for studies for laboratory experiments (Cushing et al., 2003; Desharnais, 2005). The flour beetle, Tribolium castaneum, has a long history of combining laboratory experiments and population modelling because of its biological characteristics and relevance to agricultural entomology. The species’ biology is already well known, with a high degree of detail, and its life cycle has ideal complexity for

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exploring rich scenarios in terms of stability and population dynamics (Cushing et al., 2003). In addition, there is a process of cannibalism between life stages, powerfully and quickly triggered in the species, which exerts a significant influence on dynamic behaviors observable in laboratory experiments and also in simulations based on mathematical models structured in life stages (Cushing et al., 2003; Costantino et al., 2005). This complex scenario led to the creation of a mathematical formalism taking into account different life stages of T. castaneum, resulting in the proposition of the LPA model, also formulated for the probabilistic scenario in order to consider possible deviations from the predictions generated in the deterministic scope (Desharnais 2005). There is a group of beetle species known as bark beetles, which have high relevance for coniferous forests, especially when environmental conditions favor the occurrence of outbreaks (Müller et al., 2022). Climate change has triggered recurrent problems in different areas of the planet, and indeed, factors such as water scarcity, among others, can make trees more susceptible to attacks by these species (Kˇrivan et al., 2016; Müller et al., 2022). Associated with the characteristics mentioned above, the dynamics involving beetles and trees have a strongly nonlinear dimension due to the inherent behavioral complexity of the aggregation of beetles attacking trees (Kˇrivan et al., 2016). In this sense, mathematical models can help elucidate the variation in tree resistance as a function of beetle aggregation. There has been a growing interest in the subject, which is why several dynamic models were based on the thermal variation that affects insects and the distribution of host trees as crucial factors for the development, population dynamics, and damage of beetles to trees (Powell & Bentz, 2014). These propositions substantially supported the proposition of models that go beyond obtaining data in the laboratory (Kˇrivan et al., 2016). Diabrotica speciosa (Chrysomelidae) is another important beetle species from the agricultural perspective, mainly because it is a polyphagous species capable of colonizing different plantations, especially when configured intercropping. Models exploring the agricultural landscape with different crops were proposed using the formalism of cellular automata to investigate the mobility of the species in neighborhood scenarios, combining computational rules and mathematical functions with data obtained from the literature, also estimated in the laboratory (Ferreira & Godoy, 2014; Garcia et al., 2014, 2020). Greater detail on the approach used in the articles written by Ferreira & Godoy (2014) and Garcia et al. (2014, 2020) can be found in Chap. 4 of this book.

2.2.2.3

Order Lepidoptera

The order Lepidoptera has several insect species with significant economic importance in agricultural or forestry entomology. We highlight some examples of computational modelling applied to Spodoptera frugiperda (Lepidoptera, Noctuidae). This species has polyphagous habits, estimated to be able to exploit food

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resources of dozens of different crops, with emphasis on soybeans, corn, and cotton, host plants that are the target of intense movement and damage by insects (Garcia et al., 2021). Spodoptera frugiperda differentiates into two strains, originating in the tropics of the Americas and widely occurring throughout the world, thus exhibiting a high ability to invade new areas, as observed in recent years in large areas of Africa, as well as Asia and Australia (Kenis et al., 2022). Cellular automata models have been used to analyze distribution patterns of S. frugiperda, especially contrasting transgenic plantations with different percentages and refuge options, aiming to analyze the evolution of resistance of populations in Bt plantations (Garcia et al., 2021; Tomé et al., 2022). Details on the research carried out with S. frugiperda can be found in Chap. 4 of this book. Outbreaks in herbivore insect populations, including several species of the order Lepidoptera, are a constant concern in plantations. The delay associated with plant defenses in reaction to insect attacks and the spatial diffusion of insects has a high potential to generate significant outbreaks in plantations. Reaction-diffusion models with a space-time dimension bring relevant approaches to the spatial ecology scenario of the interaction between herbivorous insects and plants. In particular, some studies have already shown that the relationship between the time delay in plant-insect interactions and the probability of an outbreak occurring is compatible with population peaks of defoliator geometrid species (Sun et al., 2014). It is known that plant genotypes can influence individual insect performance, in addition to community composition structure (Barker et al., 2019). Therefore, investigating the effect of genotypes on insect population dynamics is equally important. A relevant mathematical model approach investigated how the genetic composition of a given forest can influence the dynamics of pest outbreaks, examining how the plant genotype and the rate of insect dispersion between genotypic patches can influence the amplitude and duration of outbreaks (Moran et al., 2013). The intense use of chemical products for pest control has brought many problems to agriculture, and among them, indeed, the emergence of resistance by insect populations is a well-known fact. Integrated pest management lacks strategies capable of including strategic plans considering space and time as essential dimensions to meet demands such as those mentioned above. A model that comparatively considers the two dimensions, space and time, was proposed to describe the dynamics of Lobesia botrana (Lepidoptera: Tortricidae), considering the resistance to insecticides. The model provides an understanding of the main factors that can lead the population of moths to persistence or extinction (Bedreddine et al., 2023).

2.2.2.4

Order Hymenoptera

Leaf-cutting ants constitute an important taxonomic group for studies involving mathematical or computational models, mainly when foraging optimization approaches are employed (Burd & Howard, 2005; Farji-Brener et al., 2015). The importance is due to their social complexity and relevance as a polyphagous species

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in agroforestry systems (Roces, 2002). A relevant aspect of subsistence and foraging strategies in leaf-cutting ants is the food load carried by foragers and the possible adverse effects capable of interfering with foraging success (Calheiros et al., 2019). A model was proposed to investigate the foraging of workers in load transport, considering the effect of wind speed on the size of the selected load to ensure an adequate food intake rate, which may affect the colony’s fitness depending on their geographical location (Alma et al., 2017). Among the insects known as providers of ecosystem services, bees stand out for their recognized function as pollinators in addition to honey production. Therefore, it is worth remembering the growing concern with the colony collapse disorder (CCD) phenomenon, which has alarmed those interested in conservation, beekeeping, and pollination, which is why a mathematical model of CCD was proposed to assist in the investigation of possible causes (Atanasov et al., 2021). The complexity of possible factors involved in the phenomenon is notorious, from the destruction of habitats to the indiscriminate use of pesticides or parasites capable of negatively impacting the persistence of bee colonies (Desai & Currie, 2016). A study modelling the dynamics of bee colonies, emphasizing their development from the food stock and populations in different life stages, in addition to internal and external factors as fundamentals for the parameterization of the model, was proposed by Hong et al. (2022). Results were discussed in the context of multifactorial influence, with effects analyzed using sensitivity to perturbation and Monte Carlo simulations. Discussion of the results also included the action of factors on the development of colonies considering the equilibrium level of the system (Hong et al., 2022). Parasitoid wasps constitute a vast and relevant taxonomic group used in biological pest control programs through releases in areas affected by agricultural pests (Zhi et al., 2019). Although the classic model of Nicholson and Bailey (1935) has traditionally brought the structural foundation for the construction and understanding of predator-prey models, emphasizing the host-parasitoid relationship, several other formulations derived from Nicholson and Bailey’s theory have been proposed (Briggs & Hoopes, 2004; Singh, 2021; Molter et al., 2023). The formulations cover different dimensions, including mathematical, computational, and statistical modelling, emphasizing biological pest control. The models proposed to investigate the host-parasitoid relationship have clarified several aspects: the impact of parasitoid actions on specific host life stages (Molter et al., 2023), the effects of population connectivity in both agricultural and natural landscapes, and the implications of natural habitat distribution among different crops (Fernandes et al., 2022), Furthermore, some studies propose strategies for investigating parasitoid release in plantation areas (Weber et al., 2022) and approaches to assess host depletion and model behavior in response to parameter variations. These variations have implications for different forms of functional response and statistical distributions, especially concerning the number of eggs per host (Bruzzone et al., 2023).

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Order Blattaria (Isoptera)

Among the best-known groups of termites, subterranean termites stand out, which have the habit of building tunnels that serve as galleries for transporting food obtained to maintain colonies (Cornelius & Osbrink, 2010). Sim and Lee (2022) have been developing statistical tools to study termite movement in underground galleries (Sim et al., 2015; Sim & Lee, 2022). Markov models have been used to investigate the behavior of termites moving in tunnels with different curvatures (Sim et al., 2015). Models of this nature can explore differences in displacement patterns between different termite species and allow interpretations of the relationship between sinusoidal tunnel construction and energy expenditure (Sim et al., 2015). Sim and Lee (2022) also evaluated how spatial statistics can be employed using fractal dimension, local density, and junction count statistics to estimate termite population size in underground galleries. Applying agent-based models based on experimental data is a usual way of obtaining the above estimates. Also, the training of neighborhood algorithms on gallery pattern images has been carried out aiming at population size estimates, with an estimated accuracy between 60% and 97% (Sim & Lee, 2022).

2.2.2.6

Order Hemiptera

The complexity of life cycles in insect populations can often be explained by the combination of factors, such as environmental ones, with processes triggered by density dependence (Solbreck et al., 2022). The interaction between density dependence and abiotic factors, particularly temperature, was investigated in Largus californicus (Hemiptera, Largidae) in order to understand how the interaction between factors can influence insect dynamics, combining abundance data obtained in the field with a structured mathematical model across life stages (Johnson et al., 2016). The Asian psyllid Diaphorina citri has received increasing attention from entomologists due to its importance in the transmission of pathogens responsible for the dissemination of Huanglongbing, a disease considered of high relevance for areas with citrus orchards (Bassanezi et al., 2020). The most used methodology for psyllid control has been using chemicals (Li et al., 2021), which can lead to physiological and behavioral resistance (Chen & Stelinski, 2017). The mathematical models used to explore this subject have been more restricted to investigating physiological resistance, not considering important aspects of behavioral resistance. Gao et al. (2021) proposed a model to understand the mechanisms of physiological and behavioral resistance, investigating the optimization of the use of insecticides based on product exchange strategies to reduce resistance, especially evaluating the impact of the frequency of application of chemical products. A computational model was also applied to investigate the spatial structure of the landscape formed by commercial and non-commercial areas (Garcia et al., 2022). Different effects were considered in the model. The effect of chemical control on

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the Asian psyllid was simulated by considering commercial areas, the effect of external management involving biological control with the psyllid parasitoid, and the elimination of infected trees. The combination of effects significantly reduced the percentage of infected trees. A relevant taxonomic group with fascinating bioecological aspects for population dynamics is the cicadas, which exhibit spatially synchronized periodic cycles. When they become adults, some species emerge synchronously every 13 or 17 years (Sheppard et al., 2020). Among the hypotheses capable of explaining possible mechanisms behind the unusual synchrony, the predatory action of birds considered as participants in the control to regulate the high densities of cicadas could be listed. In addition, competition between litters may also be associated with the phenomenon. Mathematical models based on the Leslie Matrix were proposed to analyze the hypotheses formulated to investigate synchrony and periodicity in cicadas. The models have a nonlinear dimension and include the Allee effects based on competition, predation, reproduction, and survival. Through numerical and analytical techniques, it was possible to demonstrate that the presence of a litter depends on the relationship between the interactions and the aforementioned demographic parameters with predation and competition (Machta et al., 2019).

2.2.2.7

Order Orthoptera

Among the species of this order, migratory locusts have received increasing attention due to recurrent swarms in certain areas of the planet, with serious damage to agriculture in general (Le Gall et al., 2019). There are not many recent studies aimed at modelling collective migrations, mainly of locusts. One of them provides a comprehensive history, with comparative analysis of models developed to investigate the collective movement of locusts. Modelling and simulations are presented with approaches centered on the use of agent-based models and also models employing integral-differential equations for the analysis of grasshopper movement (Ariel & Ayali, 2015). Foraging modelling was also employed to investigate how food availability can influence swarm formation (Georgiou et al., 2021).

2.2.3 Outbreak Analysis with Ecological Theory The occurrence of outbreaks in insect populations is scientifically intriguing and extremely important (Berryman, 2002), as highlighted in the previous paragraphs when mentioning the orders Coleoptera, Lepidoptera, and Orthoptera. Outbreaks, in general, have been investigated with statistical tools, taking advantage of the verification of cycles through autocorrelation analysis or peaks through spectral analysis, especially using time series analysis, either by using Fourier transform

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or Wavelets (Damos, 2016; Nenzén et al., 2017). In a recent review, Ong and Vandermeer (2023) demonstrated how past and recent approaches using theoretical ecology and dynamics of complex systems are capable of producing a deep understanding of a general and intuitive nature for pest recurrence problems through a predator-prey model, aiming at highlighting the relevance and complexity of interactions, considering pests and natural enemies in alternative stable states revealing significant results on resurgence dynamics.

2.2.4 Models and the Agricultural Landscape Although in the previous pages we have presented some examples of modelling applications in different taxonomic groups of insects, we would like to indicate review papers that can also succinctly present the use of modelling under different aspects in the scale of agricultural landscapes (Haan et al., 2020). In addition to using models in different insect species, knowledge about using different models to answer specific questions is relevant. Different issues may be aligned with different model perspectives, whether mechanistic, deterministic, or stochastic, emphasizing predictions or also aimed at describing important ecological patterns for applications in pest control programs (Kogan & Jepson, 2007). Among the current demands, we highlight approaches that consider aspects related to the sustainability of landscapes, such as agroecological models, which also have the potential to provide forecasts emphasizing the natural control of pests in agricultural landscapes (Garcia et al., 2014; Alexandridis et al., 2021). In the review made by Alexandridis et al. (2021), mechanistic models for natural pest control are described in a comparative way with correlative models, proposed for the exploration of patterns and theories that relate natural control to land use in the vicinity at different scales, including landscape scale or even larger (Alexandridis et al., 2021). Correlative models can be better used when initial estimates of the potential distribution of species are needed, mainly when little is known about the species or even when one wants to obtain information about the factors associated with the limitation of the distribution of the organism focus of the study (Robertson et al., 2003). It is also necessary to consider the dimensions and proportions of the areas intended for the planting of specific crops concerning the natural areas, where the expectation is generally that they function as refuges for the maintenance of natural enemies capable of naturally controlling pests and increasing the chance of ecosystem services (Segre et al., 2020). Although, there is an expectation that the growth of areas destined for monocultures will accelerate the negative impact of pests on the system, mechanistic mathematical models suggest the eventual possibility of positive effects in addition to the negative ones (Rosenheim et al., 2022). Rosenheim et al. (2022) reported area size effects on 14 species in different crops and found no evidence that more extensive areas of plantation cause consistently

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worse impacts than smaller areas. The study highlighted the perception that the inherent complexity of agricultural systems deserves attention from different aspects. It is relevant to revisit concepts that involve the dichotomous concept of monoculture/diversity since the topic may need to be analyzed in greater depth to investigate the mechanistic roles played by habitat components, especially considering different scales in the sustainable management of pests (Rosenheim et al., 2022). The reduction in monocultures can benefit biodiversity conservation and pollination. However, its effect on pest management lacks research combining ecological theory with experimentation. As noted in other scientific areas, entomology has also received new technological tools, including those used for monitoring insects, species interaction networks, quantifying ecosystem services, tracking species movements, occurrences from local to continental scales, and energy and biomass fluxes within and across habitats (van Klink et al., 2022). However, many challenges have motivated the development and application of new insect technologies; whether from the perspective of biodiversity conservation or integrated pest management, there is still much to be done. There are significant challenges that modern society faces, including climate change, the fragmentation of habitats that are important for maintaining biodiversity, the improper use of soil, the inappropriate use of chemical producers that cause severe problems for the environment and human health, in addition to resistance developed by insects to pesticides and also to transgenic crops (Lima et al., 2009; Carrière et al., 2004; Oliver et al., 2016; Camacho et al., 2021; Halsch et al., 2021). Added to this is perhaps the biggest challenge of all, food security, which essentially depends on the success of food production under different circumstances and, of course, on the successful management of agricultural systems (Bruce, 2010). Mathematical, statistical, and computational tools are increasingly playing an important role in pest management and species conservation programs, in addition to other applications with regard to public health (Garcia et al., 2021). A variety of applications with mathematical and computational models have been gaining the attention of entomologists with an emphasis on ecology, among them the modelling of dispersion, interaction between species, resistance to insecticides, biological control, and climate change (Lima et al., 2009; Hackett & Bonsall, 2019; Garcia et al., 2021). Particularly, the spatial modelling of insects in agricultural landscapes has gained prominence (Fernandes et al., 2022; Tomé et al., 2022). These models consider the spatial distribution of insects, as well as the influence of landscape characteristics, such as crop heterogeneity and the presence of natural habitats. Spatial modelling can help to understand the dynamics of insect populations on a regional scale and to optimize management strategies (Fernandes et al., 2022). These approaches provide powerful tools for understanding the dynamics of pest populations, predicting their behavior, and evaluating control strategies. Optimizing control strategies is among the most relevant aspects for modelling applied to pest management at the agricultural landscape level (Parry, 2022). Optimization algorithms can be applied to determine the best pest control strategies in an agricultural landscape. These algorithms consider multiple variables, such as control

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costs, the effectiveness of control methods, environmental impact, and other relevant factors (Parry, 2022). It is possible to find solutions that maximize control efficiency and minimize associated costs through computational and statistical modelling. Another relevant aspect is the analysis of the results arising from pest monitoring. Statistical analysis of historical data can help identify seasonal patterns, trends, and risk factors for pest emergence in an agricultural landscape (Donatelli et al., 2017). Based on this analysis, predictive models can be developed to anticipate the emergence of pests and guide preventive measures. Furthermore, continuous monitoring of pest populations can be combined with different technologies, such as remote sensing, which allows early detection of outbreaks and implementation of timely corrective actions (Hall et al., 2016). Furthermore, risks and environmental impacts are also assessed. Mathematical and statistical models help assess risks associated with pesticide use and other control strategies. They can predict the effects of these products on the pest population, beneficial organisms, and the ecosystem as a whole. These assessments allow informed decisions about using different control methods, taking into account environmental impacts and long-term sustainability (Forbes et al., 2016). Acknowledgements We thank the financial support from the São Paulo Research Foundation (FAPESP), Brazil (Grants Number: 2014/16609-7 and 2018/02317-5) and the National Council for Scientific and Technological Development (CNPq).

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Sim, S. W., & Lee, S. H. (2022). Estimating termite population size using spatial statistics for termite tunnel patterns. Ecological Complexity, 52, 101025. https://doi.org/10.1016/ j.ecocom.2022.101025 Sim, S. W., Kang, S. H., & Lee, S. H. (2015). Using hidden Markov models to characterize termite traveling behavior in tunnels with different curvatures. Behavioural Processes, 111, 101–108. https://doi.org/10.1016/j.beproc.2014.12.013 Singh, A. (2021). Stochasticity in host-parasitoid models informs mechanisms regulating population dynamics. Scientific Reports, 11. https://doi.org/10.1038/s41598-021-96212-y Solbreck, C., Knape, J., & Förare, J. (2022). Role of weather and other factors in the dynamics of a low-density insect population. Ecology and Evolution, 12, 1–11. https://doi.org/10.1002/ ece3.9261 Sun, G. Q., Chakraborty, A., Liu, Q. X., Jin, Z., Anderson, K. E., & Li, B. L. (2014). Influence of time delay and nonlinear diffusion on herbivore outbreak. Communications in Nonlinear Science and Numerical Simulation, 19, 1507–1518. https://doi.org/10.1016/j.cnsns.2013.09.016 Tilman, & Kareiva. (1997). Spatial ecology: The role of space in population dynamics and interspecific interactions. Princeton University Press. Tomé, M. P., Weber, I. D., Garcia, A. G., Jamielniak, J. A., Wajnberg, E., Hay-Roe, M. M., & Godoy, W. A. C. (2022). Modeling fall armyworm resistance in Bt-maize areas during crop and off-seasons. Journal of Pest Science, (2004). https://doi.org/10.1007/s10340-022-01531-2 Tonnang, H. E. Z., Hervé, B. D. B., Biber-Freudenberger, L., Salifu, D., Subramanian, S., Ngowi, V. B., Guimapi, R. Y. A., Anani, B., Kakmeni, F. M. M., Affognon, H., Ndjomatchoua, F. T., Pedro, S. A., Nana, P., Johansson, T., Nedorezov, L. V., Tanga, C. M., Nana, P., Fiaboe, K. M., Mohamed, S. F., Maniania, N. K., Ekesi, S., & Borgemeister, C. (2017). Advances in crop insect modelling methods—Towards a whole system approach. Ecological Modelling, 354, 88–103. https://doi.org/10.1016/j.ecolmodel.2017.03.015 Tredennick, A. T., Hooker, G., Ellner, S. P., & Adler, P. B. (2021). A practical guide to selecting models for exploration, inference, and prediction in ecology. Ecology, 102. https://doi.org/ 10.1002/ecy.3336 van Klink, R., August, T., Bas, Y., Bodesheim, P., Bonn, A., Fossøy, F., Høye, T. T., Jongejans, E., Menz, M. H. M., Miraldo, A., Roslin, T., Roy, H. E., Ruczy´nski, I., Schigel, D., Schäffler, L., Sheard, J. K., Svenningsen, C., Tschan, G. F., Wäldchen, J., Zizka, V. M. A., Åström, J., & Bowler, D. E. (2022). Emerging technologies revolutionise insect ecology and monitoring. Trends in Ecology & Evolution, 37, 872–885. https://doi.org/10.1016/j.tree.2022.06.001 Verhulst. (1838). Correspon- dance mathématique et physique. Correspon- Danc mathématique Phys, 10, 113–121. Volterra. (1926). Fluctuations and abundance of a species considered mathematically. Nature, 118, 558–560. Weber, I. D., Garcia, A. G., de Freitas, B. A., de Oliveira, R. C., & Godoy, W. A. C. (2022). Release strategies of Telenomus podisi for control of Euschistus heros: A computational modeling approach. Pest Management Science, 78, 4544–4556. https://doi.org/10.1002/ps.7074 Westwick, R. R., & Rittschof, C. C. (2021). Insects provide unique systems to investigate how early-life experience alters the brain and behavior. Frontiers in Behavioral Neuroscience, 15, 1–13. https://doi.org/10.3389/fnbeh.2021.660464 Zhi, W. Z., Quan, L. Y., Shi, M., Hua, H. J., & Xin, C. X. (2019). Parasitoid wasps as effective biological control agents. Journal of Integrative Agriculture, 18, 705–715. https://doi.org/ 10.1016/S2095-3119(18)62078-7

Chapter 3

Monte Carlo Simulations to Model the Behaviour of Agricultural Pests and Their Natural Enemies Eric Wajnberg

Abstract Over the last decades, diverse modelling approaches have been used to understand insect behaviour and population dynamics in agricultural landscapes and to improve our ability to manage crop pests. When there are too many parameters used to define insect behavioural reproductive strategies and environmental characteristics, standard modelling tools become mathematically intractable, and so-called Monte Carlo computer-assisted simulation methods can be developed instead. Most of the time, simulations are done in spatially defined environments; hence, these simulations are usually said to be ‘spatially explicit’. Such approaches can be coupled with numerical tools to find the parameters that optimise some pre-defined objective criteria, such as fitness output or pest control efficacy. Through examples, this chapter will present these methods and how they can help us to understand insect behaviour and their populations, and thus to potentially optimise pest control strategies. Keywords Monte Carlo · Simulation · Pest control · Natural enemies · Stochasticity · Insect behaviour · Biological control

3.1 Introduction Theoretical models developed to understand animal, and especially insect behaviour over the last decades, have followed a variety of approaches. Mostly these models are based either on the aim to understand the demographic trajectories – in both time and space – of the species studied or on trying to find optimal reproductive strategies adopted by individuals in different environmental situations (Godfray,

E. Wajnberg () INRAE, Sophia Antipolis Cedex, France INRIA – Projet Hephaistos, Sophia Antipolis Cedex, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. A. Moral, W. A. C. Godoy (eds.), Modelling Insect Populations in Agricultural Landscapes, Entomology in Focus 8, https://doi.org/10.1007/978-3-031-43098-5_3

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1994; Wajnberg et al., 2008). In this last case, the modelling techniques used were borrowed from optimisation theory (Houston & McNamara, 1999). Most models are sufficiently simple and tractable to be solved using standard optimisation tools. However, in an increasing number of situations, the problems that need to be addressed are of an increasing complexity and are difficult to be solved analytically (Hoffmeister & Wajnberg, 2008). For example, these models can now take into account changes in the state of the animal during its resource foraging time using so-called stochastic dynamic programming models (SDPs; Clark & Mangel, 2000). In addition, interactions with competitors that are also trying to maximise their reproductive strategy can now be considered using so-called game theoretical models (Maynard-Smith, 1982). For an increasing number of questions, problems are becoming far too complicated to be solved with these modelling techniques, and other, computer-assisted tools must be used instead. These models are based on so-called Monte Carlo simulation approaches. These are currently the only powerful tool available to tackle problems in which we need to consider situations defined by many different parameters (and/or their interactions), and especially if stochasticity (e.g. variation in the environment in which the simulated animals forage for resources) needs to be considered. The aim of this chapter is to present these simulation methods, to see how they can be implemented in computers, how their results can be analysed, and how optimised solutions can be identified. Monte Carlo methods were initially developed in the 1940s, when the first fast computers became available. They are based on repeated random sampling to collect numerical results. Such simulation techniques are frequently used in a variety of fields, including physics (e.g. to study interacting particle systems), chemistry (e.g. to study molecule interactions), engineering (e.g. to study fluid dynamics or understanding variation in microelectronic circuits), and climate change dynamics. More recently, such methods started to be used in the field of ecology (Giró et al., 1985, 1986). The name ‘Monte Carlo’ was coined by the physicist Nicholas Metropolis to refer to the Monte Carlo casino in Monaco in which a significant amount of numbers is continuously drawn randomly (Metropolis, 1987). In such models, the trajectory of each individual is followed in time and/or in space, so these models are usually called individual-based models (i.e. ‘IBM’) and, in the usual terminology, individuals are sometimes called ‘agents’, so these models are also sometimes called ‘multi-agent models’ or ‘multi-agent-based simulation’ (i.e. ‘MABS’). Despite being usually conceptually and algorithmically simple, the computational cost associated with Monte Carlo simulations can be staggeringly high since getting accurate numerical outputs usually requires many replicates to be run. Hence, these methods became progressively more popular with the increasing availability of powerful computers and especially with the capability nowadays to access to local or even worldwide computer grids enabling the running of several replicates on different computers at the same time (see below). Of course, like any other modelling tool, the process being simulated with these Monte Carlo methods is a simplification of the real situation we are endeavouring to understand. However, since many rules can be added to the simulation framework,

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the obtained results are generally closer to reality than those produced by other modelling approaches.

3.2 An ‘Appetiser’ In this section, I present a simple didactic example to explain the general framework used to build a Monte Carlo simulation model in the field of ecology. The idea is to simulate the behaviour of a single parasitoid female foraging for hosts distributed in patches in a 2D space. The space is a 500 × 500 cell grid and the location of the hosts over the grid is drawn randomly using two steps. The locations of host patches are first randomly drawn all over the grid. It was arbitrarily decided that the number of patches represents 4% of the total number of hosts present in the grid. Each patch contains the same number of hosts, whose location is then drawn using a Normal distribution centred on the location on the patch each host belongs to and with a standard deviation (SD) of 30 cells. This produces reasonable levels of host aggregation (see Fig. 3.1 for an example). A single parasitoid female is ‘released’ in the centre of the grid and moves following a discrete time process. At each time step, a linear speed is drawn from a Normal distribution with a mean of 5.0 and a SD of 2.0 cells, and an angular speed (direction) is also drawn from a Normal distribution with a mean equal to the angular speed used in the previous time step and a SD of 0.9424 radian (i.e. 54.0 degrees). For the first time step, the mean direction is drawn randomly between 0.0 and 2π . During its walking process, the female can perceive the nearest unattacked host from a distance of 80 cells (i.e. reactive distance; Roitberg, 1985; Bruins et al., 1994) and, if a host is perceived, the angular speed in the next time step is in the Fig. 3.1 An example of the simulated walking behaviour of a single parasitoid female foraging for hosts having an aggregated distribution in a 2D space; 300 hosts are available. The female is released in the centre of the grid and the simulation stops when the female reaches the border of the grid. White and black circles represent unattacked and attacked hosts, respectively

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direction of this nearest host. Following this process, if the female gets closer than 5 cells to this targeted host, its location becomes this host location and the host is considered to be attacked. Finally, the simulation stops when the female reaches the border of the grid. Figure 3.2 gives the flowchart of the entire simulation process and Fig. 3.1 provides an example of the corresponding walking pattern obtained when 300 hosts are available.

Fig. 3.2 Flowchart of a Monte Carlo simulation model used as an example to simulate the walking behaviour of a single parasitoid female foraging for hosts exhibiting an aggregated distribution in a 2D space

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After running each simulation, we can compute – as an estimation of the ability of the foraging parasitoid female to produce progeny (i.e. its fitness) – the number of hosts discovered and attacked per time unit. The model specifically depends on two important parameters defining the walking strategy to the simulated individual: (1) the mean linear speed defining the Normal distribution in which the distance travelled at each time step is randomly drawn, and (2) the SD of the angular speed defining the Normal distribution in which the direction of the animal is also drawn at each time step. The smaller this SD, the more the animal walks in a straight line. This model was run with different mean linear speeds, ranging from 5.0 to 20.0, with a step of 1.0 cell, and different SD of the angular speed, from 0.0 to 0.6, with a step of 0.05 radian. In each case, 200 replicates were run, and the average values obtained are shown in Fig. 3.3. As we can see, there are some intermediate values of the mean walking speed that maximise the number of hosts a foraging female discovers and attacks per time unit. This makes sense since females walking slowly will take more time to find hosts to exploit, while females walking more rapidly will miss hosts, losing foraging time. Also, for the conditions used in the simulation, smaller SD for the angular speeds (i.e. walking more in a straighter line) leads the females to increase their rate of host attack efficiency. As is shown in Fig. 3.3, there

Fig. 3.3 Average fitness (i.e. number of hosts discovered and attacked per time unit), obtained by running the simulation model presented in Figs. 3.1 and 3.2, 200 times each for different values of the mean linear speed (expressed in cell unit) and the SD of the angular speed (expressed in radian). The grey surface is a fitted local regression. The arrow shows the value of the parameters (i.e. linear speed = 12.29 cells; SD of angular speed = 0.13 radian) that maximises the overall fitness output of the simulated females, as computed with a genetic algorithm (see the explanation in the text below)

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is a way to find the value of these two parameters that maximises the number of attacks per time unit. This will be presented later in this chapter. As explained above, this is a simple example to present how a Monte Carlo simulation can be conceived and how the results can be presented and discussed. More generally, the simulation framework usually depends on several (usually more than two) parameters that are called ‘state parameters’ (here the mean linear speed and the SD of the angular speed). These parameters are then combined into the simulation framework to produce a so-called objective function that produces an output criterion to be optimised (here the number of hosts discovered and attacked per time unit). Depending on the scientific problem addressed, this can be the number of progeny produced, the pest control ability of a biological control agent, the economic productivity/profitability of a crop, etc. (see, e.g. Plouvier & Wajnberg, 2018). Figure 3.4 gives a diagrammatic representation of a possible general framework of such a process.

Fig. 3.4 Diagrammatic representation of the development of a Monte Carlo simulation process. In this example, the simulation model is based on three parameters that have different distributions (from left to right: Normal, Poisson, and Exponential). Randomly drawn values of these three parameters are used in the objective function simulating the situation studied, leading to produce an output criterion, and the process is replicated multiple times to collect average results, with their variance, to produce graphs, etc., and thus to understand the predictions of the model

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3.3 Random Number Generators Monte Carlo simulations are similar to ‘random experimentations’. To run each simulation, as this is the case in the example described above, we need to draw a large amount of random numbers. In the first simulation approaches that were developed, these random numbers were generated using manual techniques, such as flipping a coin and spinning a roulette (Rubinstein & Kroese, 2017). These methods were rapidly abandoned for at least three reasons: (1) the manual methods were far too slow for running advanced simulation models; (2) the generated sequences could not be reproduced; and (3) the generated sequences obtained were not always truly random. Nowadays, sequences of random numbers are generated using simple deterministic algorithms that can be easily implemented in computers. Hence, the corresponding generators are rather called ‘pseudo-random’ (Rubinstein & Kroese, 2017), and they are all built to produce sequences of numbers that are supposed to be uniformly distributed between 0.0 and 1.0. Such pseudo-random algorithms must satisfy a certain number of properties for them to be ‘random enough’. They must pass a series of statistical tests demonstrating that they actually produce random numbers that are uniformly distributed in the [0, 1] interval. Also, the random numbers generated must be independent, that is, a value in the generated sequence must not be related to the previous one(s). Since the corresponding computation is based on a sequence of numbers that use finite precision arithmetic, the sequence will repeat itself with a finite period, but this period must be as long as possible, and it must be much longer than the amount of random numbers needed for the simulation. Finally, as we will see below, these pseudo-random algorithms are computed from a starting seed, but both the randomness and the period must not depend on the initial seed used. Nowadays, most programming languages provide a build-in pseudo-random number generator, but not all of them accurately conform to the properties listed above, and they thus cannot all be used for scientific applications. Just to make this a bit more concrete, the simplest methods that are used to generate pseudo-random sequences of numbers are called linear congruential generators, and were initially proposed by Lehmer (1951). Succinctly, they are based on the following recursive formula: Xt+1 = aX t + c (mod m) .

.

The initial value, X0 , is called the seed, and a, c, and m are positive, integer constants. The expression ‘mod m’ means that, at each step t, the expression aXt + c is divided by m and the remainder is the generated value used for the next step. Careful choices for a, c, and m produce sequences of pseudo-random numbers that pass most of the statistical tests for uniformity, randomness, and independence mentioned above. A good example is the pseudo-random generator described by Lewis et al. (1969) with a = 74 , c = 0, and m = 231 − 1. However, such linear

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congruential generators are not frequently used anymore since they usually no longer totally meet the requirements of Monte Carlo simulation applications (see L’Ecuyer & Simard, 2007). Other methods are used, but their general frameworks are still based on linear recurrences like in the methods described above. The most successful and widespread one is called MRG32k3a and has been proposed by L’Ecuyer (1999). This is the pseudo-random generator currently implemented in most scientific computing languages, for example, in the R statistical and programming language (R Core Team, 2020). An implementation of this pseudorandom generator in C can be easily found online. The tricky point regarding the use of these pseudo-random generators is to find a seed to start a Monte Carlo simulation. In very rare cases, the seed should remain the same between runs, and the generator will then produce the same sequence of random numbers. For example, this will be useful when there is a need to correct mistakes in a computer programming code being developed. However, in the vast majority of the cases, the seed must be different for each run of the simulation in order to generate independent replicates. In many applications, the system time of the computer in which the simulations are launched is used as a possible seed. For example, on computers running under the Linux operating system, such system time is expressed as the number of milliseconds that have elapsed since the 1st of January 1970 at 00h00m00s. Hence, since different runs of the same simulation will usually not start during the same millisecond, the seeds used will be different. Such a procedure, however, poses a problem since Monte Carlo simulations are now becoming more and more based on intensive computations that require long computation times. In such cases, the simulation runs are now frequently distributed on local or even worldwide computer grids. These are widely distributed, interconnected computers used to reach a common goal. In this case, there is a higher chance that several replicates will start at exactly the same time, and the corresponding replicates consequently will not produce different and independent results. Another method must thus be used to define the seed in each run of a simulation. On all computers running under the Linux environment, there is a special file that can be used for this. Its name is ‘/dev/urandom’ and it gathers the environmental noise, for example, coming from device drivers. This noise can be read directly to generate possible seeds to launch the pseudo-random generator in each specific simulation run. Several works propose more advanced methods to set up the seeds in this case (see, e.g. Maigne et al., 2004).

3.4 Parameters and Their Statistical Distribution Upon designing a Monte Carlo simulation model, two sets of parameters are usually considered. As seen above, state parameters are those on which simulations are based, and are the ones we vary when the outputs of the model are analysed (see the next section). The other parameters are sometimes called ‘forcing parameters’ and are used to define external influences acting upon the simulation process being

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addressed. In the example presented above, the state parameters of interest are the linear speed and the angular speed of the walking simulated females foraging for hosts to attack, while the forcing parameters are the total number of hosts and their spatial distribution in the environment. State parameters have their own statistical distribution from which their values must be drawn, since Monte Carlo simulations are based on stochastic processes that are repeated multiple times. Such simulation processes are thus based on a statistical framework, and these methods are sometimes used to statistically estimate parameters. This is what is done, for example, when bootstrap techniques are used to estimate the robustness of statistically estimated phylogenetic trees (e.g. Augusta de Moraes & Selvatti 2018). In this section, we will see how pseudo-random generators (that are producing random numbers uniformly distributed in the [0, 1] interval, as we have seen above) can be used to draw sequences of random numbers from any kind of statistical distribution. There are different methods that can be used for such a purpose. The simplest one is called the inverse-transform method. Let us say we want to draw a value x from a statistical distribution having a cumulative distribution function F(x). By definition, F(x) is the probability that a variable X takes a value less than or equal to x. Such cumulative distributions always represent monotonically increasing functions from 0.0 to 1.0, as seen in two examples shown Fig. 3.5. These functions can be derived based on theoretically known distributions or they can be purely empirical. Once we have such a cumulative distribution function F(x), the inversetransform method consists of drawing a random value U uniformly from the [0, 1] interval with a pseudo-random generator, and to deliver the value X that corresponds to the inverse function F−1 (U), written as: F −1 (U ) = inf {X : F (x) ≥ U } .

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This equation means that we are looking for the minimum value of x that satifies the equation F(x) ≥ U. Figure 3.5 shows two examples of this procedure, one with a semi-quantative trait and another with a quantitative, continuously distributed trait. A simple application of the inverse-transform method can be seen when we want to generate a random variable from a Bernoulli distribution. This distribution is a discrete distribution that takes the value 1 with probability p, and the value of 0 with probability q = 1 − p. This is frequently used in Monte Carlo simulation models when dealing with binary variables, for example to randomly draw the sex of an individual (male or female), whether a host is found or not by a parasitoid female, whether a host can escape or not from a parasitoid attack, etc. The cumulative distribution function in this case is similar to the one shown in the left panel of Fig. 3.5, but with only two possible outcomes. Therefore, the procedure to draw a random number having a probability p to appear (e.g. to decide if a host is attacked on not) is simply the following: first generate a random value U uniformly distributed in the [0, 1] interval. Then, if U ≤ p return 1, otherwise return 0.

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Fig. 3.5 Two examples of the inverse-transform method, with a semi-quantitative trait (left panel) or a continuously distributed trait (right panel), that can be used to draw a random number from any kind of statistical distribution. In both cases, the cumulative distribution function F(x) is presented. A value U is drawn uniformly from the [0, 1] interval, and the delivered value X is given by the inverse function F−1 (U). In the left panel, the pi values are the frequencies in which each value xi is or can be observed. Please note that the left panel could also represent the distribution of a purely qualitative trait. In this case, the method will work a similar way

The inverse-transform method can be applied for any kind of statistical distribution. For some of them, which are theoretically well known, more advanced (i.e. more efficient) methods are sometimes used instead. Boxes 3.1, 3.2, and 3.3 give the usual algorithmic procedures used to draw random numbers from an Exponential, a Normal, or a Poisson distribution, respectively.

Box 3.1 Drawing random numbers from an Exponential distribution. Exponential distributions describe the distribution of times between events. In Monte Carlo simulations, such a distribution can be used, for example, to randomly draw the time elapsed between two host attacks by a parasitoid female. Exponential distributions are defined by only one parameter, the so-called rate parameter λ, and the expected value (i.e. the mean of the distribution) is known to be λ−1 . To draw a random number in an Exponential distribution having a rate parameter λ, the following algorithmic procedure can be used: • Choose a parameter λ (λ > 0); • Randomly draw a value U from a Uniform distribution between 0 and 1; (continued)

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Box 3.1 (continued) • Compute X = − λ−1 ln (U); • Return X. If we want to draw a random number in an Exponential distribution having a mean μ instead, we just have to replace λ−1 by μ in the procedure above.

Box 3.2 Drawing random numbers from a Normal distribution. Normal distributions are very common continuous distributions that can be used to describe many quantitative traits. In Monte Carlo simulations, they can be used, for example, to randomly draw traits such as length, size, weight, or distance. There are several methods that can be used to generate normally distributed random values. The most well known one is the so-called Box-Muller method that allows one to simultaneously draw two random, non-correlated values from a Normal distribution having a mean of 0 and a variance of 1. This works according to the following algorithmic procedure: • Randomly draw two independent values U1 and U2 from a Uniform distribution between 0 and 1; • Compute and return two random numbers X1 and X2 using the following equations:  .

 X1 = cos (2π U2 )  −2 log (U1 ) . X2 = sin (2π U2 ) −2 log (U1 )

If we want to draw numbers from a Normal distribution having a mean μ and a variance σ 2 instead, the obtained values must be additionally transformed as μ + X1 σ and μ + X2 σ , respectively.

Box 3.3 Drawing random numbers from a Poisson distribution. Poisson distributions are used to describe the discrete distribution of the number of events appearing during a given time interval. In Monte Carlo simulations, they can be used, for example, to randomly draw the number of eggs laid by a parasitoid female, the number of females a male can mate, etc. Poisson distributions are based on a single parameter λ > 0 that corresponds (continued)

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Box 3.3 (continued) to the mean of the distribution. The following algorithmic procedure can be used to draw a random number from a Poisson distribution with mean λ: • Choose λ (λ > 0); • Let n = 0 and a = 1; • While a ≥ e−λ do:   Draw a value U from a Uniform distribution between 0 and 1  .  a = aU  n = n + 1 • Compute X = n − 1; • Return X.

Other algorithmic procedures are available to draw random numbers from any kind of distribution. There are even procedures that are available to draw random vectors of values from multidimensional distributions when several (e.g. correlated) values are needed in a Monte Carlo simulation. The interested readers should consult available textbooks, for example, Rubinstein and Kroese (2017).

3.5 Analysing the Obtained Results Once the results of all simulations are collected, there is the need to analyse them both to check whether the computations produced suspicious outputs (pointing to some mistakes in the computer programming code) and to understand the messages delivered. For this, the results of each simulation run (or only averages and variances) are first saved. Then standard statistical descriptive tools can be used, such as summary statistics (means, ranges, variances, correlations, etc.). In this respect, graphic outputs are always needed, such as histograms, or simple plots showing the effects of different values of the forcing parameters on the mean and variance of the state parameters. Figure 3.3 gives an example of such a graph. Another related descriptive tool that can be used is termed ‘sensitivity analysis’. The goal is to quantify the sensitivity of the model to changes in specific parameters. For this, the model is re-run fixing all parameters but the one we are interested in. For that one, different values are used sequentially, and changes in the corresponding average outputs are analysed to quantify the importance of this parameter on the model’s outputs. Repeating such a procedure for all parameters of the model can

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lead to a better understanding of the scientific question for which the Monte Carlo model has been designed. Once all results are collected and described, it is tempting to perform statistical analyses to test the effect of, for example, a variation in one or several forcing parameters on the values of the state parameters of the model. Several authors take this approach, which is also sometimes requested by referees or editors of international journals. Although this is a continuously debated question, it is not valid to perform statistical comparisons on simulated data sets. Statistical tests are designed to be used when we do not know the real values of the parameters we are estimating, and parameter estimation should be done by sampling the populations. In this case, we need a statistical procedure (usually based on the parameters’ distributions) to compare them. In a simulation work, however, parameters such as means are explicitly known. We only have to run a sufficient amount of simulations to know them exactly. There is thus no distribution (since parameters are not estimated from some sampling procedures), and thus statistical comparison procedures are not needed. In other words, by running simulations several times, we always end up with a statistically significant test, since the standard errors will mechanically tend to zero. The only thing that can be done is to present effects, with the descriptive statistical tools mentioned above, discussing trends, etc. (see, e.g. White et al., 2014).

3.6 Looking for Optimised Values For some scientific questions, we are willing to address using Monte Carlo simulations, and once the simulations have been run and the results obtained and analysed, we can go one step forward. We sometimes need to know what values of the state parameters maximise some pre-defined objective criterion. In behavioural ecology, for example, we would often like to identify the set of state parameters that define the behaviour of the simulated animals maximising the overall number of progeny produced. In defining an efficient biological control programme against a crop pest, we might alternatively be interested in identifying the value of the parameters that maximise the number of hosts killed per time unit or the crop production, etc. There are different numerical ways to achieve this goal. One popular method is the use of genetic algorithms. This approach is efficient, simple, and relatively easy to implement. The method was invented in the early 1970s to mimic the process of natural evolution (Holland, 1975; Goldberg, 1989) and has been used since then to solve several problems in the field of behavioural ecology and evolutionary biology (see, e.g. Sumida et al., 1990; Mitchell, 1998; Hoffmeister & Wajnberg, 2008; Ruxton & Beauchamp, 2008; Wajnberg et al., 2012; Hamblin, 2013; Wajnberg et al., 2013; Plouvier & Wajnberg, 2018). The main processes involved in a genetic algorithm are shown in Fig. 3.6. At the beginning, a set (or ‘population’) of possible solutions are drawn randomly,

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Fig. 3.6 Flowchart showing the essential components of a genetic algorithm (see text for explanations)

using the procedures described in the sections above. The word ‘solution’ actually means a list of values for all state parameters (that are called here ‘genes’) of the Monte Carlo simulation problem. These are arranged sequentially along a so-called chromosome. Running the simulation model on each of the initial chromosomes leads to an estimate of their ‘fitness’ through the pre-defined objective function of the model. Then, pairs of ‘parents’ for the next generation are selected randomly in the initial population of chromosomes using a probability proportional to their fitness. Hence, chromosomes with higher fitness have a higher chance to contribute to the next generation. There are several methods to implement this. The interested reader can consult Hoffmeister and Wajnberg (2008) or Hamblin (2013) for additional explanations. Once the pairs of parents are identified, they may undergo crossing over events, which means that recombination is performed, like on real chromosomes in biology (Hoffmeister & Wajnberg, 2008; Hamblin, 2013). Finally, each parameter (gene) of the new children obtained can pass through a mutation process that usually implies adding to their value an adjustment drawn from a Normal distribution having a mean of 0.0. Then, the new chromosomes (children) are used to build a new generation and the process is repeated until a stopping criterion is reached. Here again, several stopping criteria can be used. For example, the looping process can be stopped when a fixed number of generations have been completed (e.g. Barta et al., 1997; Wajnberg et al., 2013) or after a fixed computation time. Usually, however, the performance of the genetic algorithm in producing an optimal solution is followed over the course of multiple generations, and the process is then stopped if a fixed number of generations has passed without a substantial improvement in the fitness of the best chromosome.

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Fig. 3.7 Optimised (with the use of a genetic algorithm) mean linear walking speed (expressed in cell unit; left panel) and SD of the angular speed (expressed in radian; right panel) maximising the fitness (i.e. number of hosts discovered and attacked per time unit) of simulated parasitoid females foraging in a 2D space for hosts present in different density, and having different spatially aggregate levels. The host aggregation level ranges from 0.0, indicating that hosts are randomly distributed, to 1.0, in which all hosts are clumped into a single patch. Increasing values of this parameter correspond to a decreasing number of host patches in the environment

Such an optimisation procedure has been used for the simple didactic example presented in the first section of this chapter (simulating the walking behaviour of a single parasitoid female foraging for hosts having an aggregate distribution in a 2D space), and the optimal solution found is shown in Fig. 3.3. The same Monte Carlo simulation model was then run with different host densities, and with different host aggregation patterns in the environment, and a genetic algorithm was used in each situation to find the optimised linear walking speed and SD of the angular speed, which describes the tendency of the animal to walk on a straight line or not. The results obtained are presented in Fig. 3.7. They indicate that optimised parasitoid females foraging for hosts in a 2D space should walk slower and less on a straight line when there are most hosts available in the environment. When hosts become more aggregated, optimised females should also walk slower and less on a straight line.

3.7 A More Advanced Example Plouvier and Wajnberg (2018) developed a Monte Carlo simulation coupled with the use of a genetic algorithm to find what biological and ecological features of natural enemies should optimise their pest control efficacy on a field crop. The economic income collected by the farmer when releasing biological control agents was also considered. For this, the model was designed to optimise a criterion corresponding

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to the total crop yield (expressed as plant biomass), taking into account damage done by the pest, minus the overall cost of producing and releasing natural enemies. The simulations were done using a spatially explicit framework simulating a field represented by a grid in which crop plants are grown in rows, one every second row. The state parameters optimised represented eight life-history and behavioural traits of the released natural enemies: (1) longevity (expressed as time steps before dying), (2) fecundity (expressed as the total number of eggs corresponding to the total number of hosts killed if the natural enemy is a solitary parasitoid), (3) pest attack rate, per time unit, (4) level of local intra-specific competition (i.e. interference) leading to a reduction in the pest attack rate, (5) pest handling time (expressed in time steps), (6) probability to move/disperse, and (7) the mean and (8) SD of the (Normal) distribution of the distance moved (both expressed in grid cell unit). Five forcing variables were analysed: (1) the number of sites in the field in which the natural enemies are released (1 vs. 2), (2) the number of individuals released (10 vs. 20), (3) the timing of natural enemies release (6 vs. 10 time steps after the crop has been sown), (4) the cost of producing and releasing the natural enemy (200 vs. 300 arbitrary units), and (5) the growth rate of the plant (r = 0.05 vs. r = 0.06). Finally, several other features were added to the model. For example, a negative trade-off between longevity and fecundity of the natural enemies was considered, as this is observed in real situations (see, e.g. Miyatake, 1997). The plant development has been modelled with an exponential growth function with a decreasing rate. Pests arrive on the crop at time step 5 (after the crop has been sown) and are distributed randomly over the entire crop. Natural enemies are attacking pests following a type II functional response (Holling, 1959), and they can disperse at each time step with a probability that increases with the local density of competitors, but decreases with the local density of pests. Briefly, the results obtained demonstrated that ideal natural enemies, that is, ones that optimise the income collected by the farmer using a biological control approach, should have a shorter longevity, and hence a higher fecundity. They should also have a lower pest handling time leading to increase the overall number of pests attacked on each plant. Finally, they should have a higher tendency to disperse when the local pest density is low. On the other hand, pest attack rate and the level of competition (i.e. interference) between natural enemies surprisingly appear to be less important. Results obtained also indicated that the more important ecological and behavioural features defining efficient natural enemies are correlated, demonstrating a kind of an optimised ‘behavioural syndrome’ (Sih et al., 2004a, b). Also, most of the optimised parameters actually had a bimodal distribution, indicating that there were actually two optimised pest control strategies. The first one, called ‘incremental strategy’, corresponds to low pest handling time, associated to a low tendency to disperse. In this case, the natural enemies are ‘cleaning’ the plants from pests before going to another plant. The second one, called ‘decremental strategy’,

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on the contrary, is associated to a high pest handling time with a high dispersion tendency, corresponding to a natural enemy that attacks fewer hosts in a local area before leaving but covers a larger crop surface area. The ability of each of these two pest control strategies to perform well depends on the number of natural enemies released and on the number of release points (Plouvier & Wajnberg, 2018).

3.8 Conclusion Monte Carlo simulations have been demonstrated to be a very efficient modelling tool that can be used when the problems to be addressed are too complicated (e.g. based on too many parameters) to be handled with standard, analytical modelling approaches. In this respect, this chapter gives a general overview on how such a modelling framework can be implemented, for example using a programming language like C or C++, with minimum skill. There are currently some other computer languages that might sometimes be more convenient to use for developing such simulations with minimal programming effort. For example, in the R statistical and programming environment (R Core Team, 2020), there are built-in functions enabling users to draw random numbers from all basic statistical distributions, hence without having to code this explicitly. However, using such computer languages usually produce simulation models that need more computer time to run, and that are not easily distributed on several computers simultaneously. In this respect, another point that must be mentioned here is that computers are still regularly becoming more and more powerful and able to provide faster processing times. Moreover, nowadays computers are interconnected through networks in local or even worldwide computer grids offering powerful computing resources providing a way to distribute simultaneously long-lasting simulations leading to delivery of results in reasonable times. Such distributed computer resources enable researchers to address highly complex scientific problems that are especially, but not only, developed to understand the ecology of systems involving agricultural pests and their natural enemies. Acknowledgements All computations done to prepare this chapter were achieved using the biomed virtual organisation of the EGI infrastructure, with the dedicated support of resource centres BEINJING-LCG2, IN2P3-IRES, OBSPM, INFN-FERRARA, GRIF, INFN-CATANIA, INFN-ROMA3, INFN-BARI, CREATIS-INSA-LYON, NCG-INGRID-PT, INFN-PISA, CESNET-MCC, and CLOUFIN, resource centres in UK hosted by GridPP collaboration, and the additional support of the resource centres listed here: http://operations-portal.egi.eu/vapor/ resources/GL2Browser?VOfilter=biomed. I also thank Andrei Tsaregorodtsev for his help in using this computer grid, and George Heimpel for his comments on a previous version of the manuscript. Finally, this work was supported by the Israel Institute for Advanced Studies, in Jerusalem, in which a major part of this chapter was written.

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References Augusta de Moraes, R. C., & Selvatti, A. P. (2018). Bootstrap and rogue identification tests for phylogenetic analyses. Molecular Biology and Evolution, 35, 2327–2333. Barta, Z., Flynn, R., & Giraldeau, L. A. (1997). Geometry for a selfish foraging group: A genetic algorithm approach. Proceedings of the Royal Society of London Series B Biological Science, 264, 1233–1238. Bruins, E. B. A. W., Wajnberg, E., & Pak, G. A. (1994). Genetic variability in the reactive distance in Trichogramma brassicae after automatic tracking of the walking path. Entomologia Experimentalis et Applicata, 72, 297–303. Clark, C. W., & Mangel, M. (2000). Dynamic state variable models in ecology – Methods and applications. Oxford University Press. Giró, A., Padró, J. A., Valls, J., & Wagensberg, J. (1985). Monte Carlo simulation of an ecosystem: A matching between two levels of observation. Bulletin of Mathematical Biology, 47, 111–122. Giró, A., Valls, J., Padró, J. A., & Wagensberg, J. (1986). Monte Carlo simulation program for ecosystems. Computer Applications in the Biosciences (Cabios), 2, 291–296. Godfray, H. C. J. (1994). Parasitoids. Behavioral and evolutionary ecology. Princeton University Press. Goldberg, D. E. (1989). Genetic algorithms in search, optimization, and machine learning. Addison-Wesley, Longman Publishing, Inc. Hamblin, S. (2013). On the practical usage of genetic algorithms in ecology and evolution. Methods in Ecology and Evolution, 4, 184–194. Hoffmeister, T. S., & Wajnberg, E. (2008). Finding optimal behaviors with genetic algorithms. In E. Wajnberg, C. Bernstein, & J. van Alphen (Eds.), Behavioral ecology of insect parasitoids – From theoretical approaches to field applications (pp. 384–401). Blackwell Publishing. Holland, J. H. (1975). Adaptation in natural and artificial systems. The University of Michigan Press. Holling, C. S. (1959). Some characteristics of simple types of predation and parasitism. The Canadian Entomologist, 91, 385–398. Houston, A. I., & McNamara, J. M. (1999). Models of adaptive behaviour – An approach based on state. Cambridge University Press. L’Ecuyer, P. (1999). Good parameters and implementations for combined multiple recursive random number generators. Operations Research, 47, 159–164. L’Ecuyer, P., & Simard, R. (2007). TestU01: A C library for empirical testing of random number generators. ACM Transactions on Mathematical Software, 33, 22. Lehmer, D. H. (1951). Mathematical methods in large-scale computing units. Annals of the Computation Laboratory of Harvard University, 26, 141–146. Lewis, P. A., Goodman, A. S., & Miller, J. M. (1969). A pseudo-random number generator for the system/360. IBM Systems Journal, 8, 136–146. Maigne, L., Hill, D., Calvat, P., Breton, V., Reuillon, R., Lazaro, D., Legre, Y., & Donnarieix, D. (2004). Parallelization of Monte Carlo simulations and submission to a grid environment. Parallel Processing Letters, 14, 177–196. Maynard-Smith, J. (1982). Evolution and the theory of games. Cambridge University Press. Metropolis, N. (1987). The beginning of the Monte Carlo method. Los Alamos Science. Special Issue, 1987, 125–130. Mitchell, M. (1998). An introduction to genetic algorithms. MIT Press. Miyatake, T. (1997). Genetic trade-off between early fecundity and longevity in Bactrocera cucurbitae (Diptera: Tephritidae). Heredity, 78, 93–100. Plouvier, N. W., & Wajnberg, E. (2018). Improving the efficiency of augmentative biological control with arthropod natural enemies: A modeling approach. Biological Control, 125, 121– 130. R Core Team. (2020). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://www.r-project.org/

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Roitberg, B. D. (1985). Search dynamics in fruit-parasitic insects. Journal of Insect Physiology, 31, 865–872. Rubinstein, R. Y., & Kroese, D. P. (2017). Simulation and the Monte Carlo method (3rd ed.). Wiley. Ruxton, G. D., & Beauchamp, G. (2008). The application of genetic algorithms in behavioural ecology, illustrated with a model of anti-predator vigilance. Journal of Theoretical Biology, 250, 435–448. Sih, A., Bell, A., & Johnson, J. C. (2004a). Behavioral syndromes: An ecological and evolutionary overview. Trends in Ecology & Evolution, 19, 372–378. Sih, A., Bell, A. M., Johnson, J. C., & Ziemba, R. E. (2004b). Behavioral syndromes: An integrative overview. The Quarterly Review of Biology, 79, 241–277. Sumida, B. H., Houston, A. I., McNamara, J. M., & Hamilton, W. D. (1990). Genetic algorithms and evolution. Journal of Theoretical Biology, 147, 59–84. Wajnberg, E., Bernstein, C., & van Alphen, J. (2008). Behavioral ecology of insect parasitoids – From theoretical approaches to field applications. Blackwell Publishing. Wajnberg, E., Coquillard, P., Vet, L. E. M., & Hoffmeister, T. (2012). Optimal resource allocation to survival and reproduction in parasitic wasps foraging in fragmented habitats. PLoS ONE, 7(6), e38227. Wajnberg, E., Hoffmeister, T. S., & Coquillard, P. (2013). Optimal within-patch movement strategies for optimising patch residence time: An agent-based modelling approach. Behavioral Ecology and Sociobiology, 67, 2053–2063. White, J. W., Rassweiler, A., Samhouri, J. F., Stier, A. C., & White, C. (2014). Ecologists should not use statistical significance tests to interpret simulation model results. Oikos, 123, 385–388.

Chapter 4

Representing Insect Movement in Agricultural Areas using Spatially Explicit Models Adriano Gomes Garcia, Igor Daniel Weber, and Maysa Pereira Tomé

Abstract Understanding the movement of organisms is fundamental for comprehending how life thrives and the changes in population dynamics of a species. It is also crucial for decision-making in insect pest management, as the organism’s interaction with its environment plays a key role in the efficacy of a particular management strategy. In augmentative biological control, for instance, a parasitoidhost relationship may be intensively studied, and a high parasitism rate may be found under laboratory conditions. However, what if the natural agent is unable to locate the host in the field? Another example: refuge areas are established to promote the survival of susceptible insects when GM crops are adopted, thereby delaying the evolution of resistance in target pests. Nonetheless, if the pest’s movement is disregarded, the strategy may lead to complete failure. In this chapter, we examine different cases of insect movement in agricultural areas with significant implications for pest management. We demonstrate how spatially explicit models can contribute to our understanding of this process and, consequently, become powerful tools for decision-makers. Keywords Biological control · Bt crops · Landscape ecology · Movement modelling · R software

4.1 Introduction Since the early days of ecology, we have tried to understand the spatial distribution of organisms. Why is life so scarce toward the poles? And why is biodiversity so high in tropical regions? Whereas the only option for some forms of life is to perish under detrimental conditions, some animals, plants, and even microorganisms are able to leave undesirable conditions, actively or passively, to explore better places,

A. G. Garcia () · I. D. Weber · M. P. Tomé Department of Entomology and Acarology, Luiz de Queiroz College of Agriculture (ESALQ), University of São Paulo, Piracicaba, SP, Brazil © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. A. Moral, W. A. C. Godoy (eds.), Modelling Insect Populations in Agricultural Landscapes, Entomology in Focus 8, https://doi.org/10.1007/978-3-031-43098-5_4

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whose mild climate conditions allow the establishment and growth of perennial populations (Martin et al., 2007; Robinson et al., 2009; Runge et al., 2014). Not only that, but in the case of animals, they explore the environment to breed, escape predation and competition, habitat fragmentation, and the arrival of invasive species (Villa-Martín et al., 2015; Doherty & Driscoll, 2018). The movement ecology field rose from the need to develop a general theory of organismal movements unifying conceptual, methodological, and empirical frameworks (Nathan et al., 2008; Shimatani et al., 2012). In this subdiscipline of ecology, the movement of organisms is studied. However, organisms move in a complex, heterogeneous, and structured space; in other words, a landscape. In this sense, landscape ecology covers the study of the composition, functions, and structure of landscapes, as well as the interaction between the landscape elements, to identify spatial patterns and ecological processes (Turner et al., 2001). Understanding the movement of organisms in a complex mosaic is fundamental to comprehend how life thrives, modifying the organization of populations, communities, and ecosystems (Nathan et al., 2008). Agricultural environments are one of the main systems investigated by integrating movement and landscape ecology studies since these approaches allow a holistic management approach of crop systems (Deffontaines et al., 1995; Steingröver et al., 2010). In this context, insect pest management is highlighted. The growing concern about the impact of intensive use of chemicals on the environment and human health, and the increase of insect resistance to pesticides have motivated the search for alternative control methods to reduce insect pest populations, using an Integrated Pest Management (IPM) approach (Garcia et al., 2014). Landscape ecology is present in different IPM systems, such as those involving intercropping, refuge configurations when using GMOs (genetically modified organisms), fragmented landscapes, and metapopulations, among others (Veith et al., 1996; Garcia & Godoy, 2017; Garcia et al. 2020). Given the complexity of the landscape, it is a challenging task to isolate all the components to study a particular feature. An alternative consists in the use of computational models to reduce time and resources. Recently, modelling has grown in importance worldwide, given the COVID-19 pandemic that has impacted the world, but it also provides immense support in agriculture, especially in prediction models used to describe the dynamics of insect pests (Garcia et al., 2022). Spatial models are frequently used to investigate insect dynamics in agricultural fields from a perspective of movement. These models can be structured to incorporate movement and population growth, by using parameters such as the net reproductive rate or degree-day approaches (Turchin, 1998). The evolution of insect resistance to GMOs is an illustrative example. Refuge areas are one of the strategies adopted to delay the evolution of insect resistance to Bt (Bacillus thuringiensis) toxins. Refuge areas consist of planting non-transgenic crops to promote the survival of susceptible insects that will dilute the resistance allele, preventing an increase in the proportion of the resistant phenotype when

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resistance is a recessive trait (Crowder & Carrière, 2009). However, insect movement can delay or hasten the rise of a resistant population. The relationship between larval movement and evolution of resistance to transgenic plants has been intensively studied by ecological modelers (Carroll et al., 2012, 2013). Under a high intensity of larval movement, some refuge configurations, such as random refuge, can benefit the evolution of resistance instead of the opposite (Carroll et al., 2012; Garcia et al., 2016). Another illustrative example is intercropping systems. The ability of an insect exploring different food resources can give them an advantage when some crop is absent, since they utilize different habitat patches in response to changes in crop availability, changing its spatial pattern (Kennedy & Storer, 2000; Garcia & Godoy, 2017). Fully understanding different stage preferences can help farmers to design the landscape and habitat management strategies. The configuration of the landscape can influence even the motherhood behavior of some insect pests. Diabrotica speciosa (Germar, 1824) (Coleoptera: Chrysomelidae) is a polyphagous insect pest occurring in South America that feeds on maize and soybean crops, among others (Garcia et al., 2014). However, different insect stages maximize their survival in different hosts. This flexibility to explore different habitats and difference among the biological parameters according to the life stage and host creates a parent-offspring conflict: Should parents look for a host to maximize their survival and fertility or should they look for a better host for their offspring? Such issue was investigated by using a computational model, comparing the effects of different strategies, “mother knows best” and “optimal bad motherhood” (Garcia et al., 2020). The landscape can even play an important role in plant diseases transmitted by insects. In citrus crops, for instance, HLB (huanglongbing or greening) is a bacterial disease transmitted mainly by the Asian citrus psyllid Diaphorina citri Kuwayama, 1908 (Hemiptera: Psyllidae) (Cocuzza et al., 2017). It causes chlorosis of the leaves, production of yellow shoots, and death of a citrus tree within 2– 3 years after infection (Lee et al., 2015). A key factor that determines the intensity of an infection is the arrangement of non-commercial patches with hosts around the commercial area (Diniz et al., 2020). It has been identified that the inoculum sources of HLB are located outside the farms, in orchards, abandoned groves, and ranches (Bergamin-Filho et al., 2016). Therefore, there is a need for management strategies focused in external areas avoiding insects to carry the bacterium to citrus farms to control HLB (Boina & Bloomquist, 2015). Since the external area can be arranged in different ways, the study on insect movement combined to computational modelling can support the investigation on the impact of different management strategies that compose an IPM program (Garcia et al., 2022). In this chapter, we discuss the ecological theory behind key events that can drive the movement of insects, using real case studies and bringing the perspective of agricultural entomology on movement research. Here, our framework adopts the

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concept of movement ecology for individuals, assuming that organismal movements provide “mobile links” between habitats or ecosystems, connecting resources, genes, and processes among separate locations, reflecting more realistically what happens to the populations (Jeltsch et al., 2013). This chapter is divided as follows. In Sect. 4.2, we show the process to model insect movement step by step using computational modelling in the R environment (R Core Team, 2022), illustrating the process with schematization and snippets of R code. Our goal here is to provide a guide for readers that are willing to create their first computational code involving insect movement, giving support to future studies in this area. In Sect. 4.3, we introduce two important case studies that were recently published and illustrate the importance of merging movement ecology, landscape ecology, and computational modelling. Given the growth of augmentative biological control in Brazil, our first case study (Weber et al., 2022) uses computational modelling to mimic the behavior and movement of a host and its natural enemy in a crop area, providing details on how the computational code should be structured. In this chapter specifically, we explain in detail the dispersal functions used to simulate the movement of stinkbugs and the foraging behavior of their parasitoid, Telenomus podisi Ashmead 1893 (Hymenoptera: Scelionidae), represented via computational algorithms to reproduce the movement dynamics of the system under investigation. The second case study focuses on the management of Spodoptera frugiperda (J.E. Smith, 1797) (Lepidoptera: Noctuidae) (FAW). It is known that S. frugiperda is one of the main targets of Bt technology in maize farms. However, resistant populations have brought concern to producers, and pest management measures need to be associated with resistance management as well. Tomé et al. (2022) combined knowledge of ecological traits, insect movement, and biological data and wrote a computational model to investigate whether a neighborhood with grasslands and off-season periods could influence the frequency of the resistance allele in the population of FAW in heterogeneous fields.

4.2 Modelling Insect Movement Step by Step In this section, we show in detail the basic processes involved in the elaboration of a computational code programmed using R (R Core Team, 2022) to describe the movement of insects in a particular area. We have chosen to use an explicit spatial model, whose area can be represented by a matrix composed by a grid of cells (Garcia et al., 2014, 2020). Based on this matrix, the simulated space can become increasingly complex, enabling a creative design process using just a few lines of code. Considering an area composed by two different patches, 1 and 2, for instance, we can structure the area in different ways, for example, blocks, strips, and random. See below for snippets of R code producing different areas with two patches.

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require(plot.matrix) landscape 0, and nonlinear dispersal was given by a function .di : Rn+ → (0, 1), for .i, j ∈ {1, 2}, i /= j . The authors found that even though one patch may go extinct in isolation, if the two patches are connected by heterogeneous dispersal, one can prevent the declining patch from going extinct and also ensure that both patches are persistent. For further details on this model, see (de Godoy et al., 2023). For more on nonlinear dispersal models see (Vortkamp et al., 2020) and on stochastic dispersal models see (Kirkland et al., 2006; Li & Schreiber, 2006; Tejo et al., 2021).

8.6.3 Application: Invasion Dynamics As an application of the above results, we will investigate how matrix stability relates to the population dynamics of H. armigera between two agricultural patches.

8.6.3.1

Background

H. armigera, a member of the subfamily Heliothinae, is known to be a pervasive moth species around the world (Anderson et al., 2016). This pest is polyphagous (consumes more than one resource), is highly mobile, and has high fecundity. For these reasons studying the population dynamics or potential for invasive spread is important for such a species. Using a (st)age-structured two-patch model, we will investigate such an insects’ dynamics in a spatial context. The dataset used in this case study corresponds to count observations of H. armigera moths on a farm in Palmeira, Bahia, Brazil recorded from 2015 to 2019. The farm consists of two main areas where cotton crops are grown. Our aim was to see how the matrix stability types discussed above may be applied to gain some initial qualitative insights into this system.

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Dispersaldriven growth

1

1

>1

12 1

21 2

0, .

m1 − m2 + m12 ≤ m1 and m12 − m21 ≥ 0.

Dennis et al. (2015) highlight that when one of the inequalities above is violated, in most of the cases the estimate for .λ is very large, whereas the detection probability, p, is very small. In their simulations, the greater the detection probability and the number of periods of observations, the less the inequalities are violated. However, there are cases where the inequalities are satisfied, but the estimate for .λ is still

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large. In contrast, situations where .λ is finite and the inequalities are violated are also found. Nonetheless, they recommend using the following covariance diagnostic to determine whether unrealistic estimates may arise. For more than two sampling occasions (.T > 2), cov(y1 , ..., yT ) =

.

2(y1 y2 , . . . , yT −1 yT )  y1 + . . . + yT 2 , − T T (T − 1)

where .yt = (yit , . . . , yRT ) and .y1¯y2 denotes the mean of the product .y1 y2 over R sites. A negative value for this covariance diagnostic suggests that the issue of infinite estimates of .λ may arise. An implementation of this covariance diagnostic in R is available in Sect. 10.6.3.

10.3 Model Extensions Following the introduction of the N-mixture model by Royle (2004), many extensions have been proposed to estimate abundances using a broad range of data. This includes, but is not limited to, data in which species abundances are permitted to vary with time, data containing abundances for multiple species, data pertaining to species that are observed very rarely and data with a large proportion of zerocounts. In this section, we will examine the methodologies proposed for some of these extensions.

10.3.1 N-Mixture Models for Multiple Species The N-mixture model introduced by Royle (2004) provides estimates of abundance for single species. There have since been several extensions made to this model that allow us to examine abundances of multiple species simultaneously. Golding et al. (2017) proposed a multi-species extension to the N-mixture where the imperfect detection is addressed in the form of false-positive errors, by combining the N-mixture model with a dependent double-observer data collection framework. This framework involves a primary observer recording the number of individuals that they observe and a secondary observer verifying the observations of the first observer. This observation method has three possible outcomes, with three associated probabilities: 1. The primary observer observes an individual with probability .p1 . 2. The secondary observer observes an individual that the primary observer missed with probability .(1 − p1 )p2 . 3. Both observers miss an individual with probability .(1 − p1 )(1 − p2 ).

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Because this process has multiple possible outcomes, the observation process in this case cannot be modelled using a binomial distribution as in the Royle (2004) paper but is instead a multinomial process. The abundance of species s at site i and sampling occasion t is then modelled as Yits ∼ Multinomial(Ni, pits ),

.

   where . i t s pits = 1 and .pits represents the survey outcomes. This model may prove useful in scenarios where the possibility of misidentification or doublecounting of individuals is a concern, though it requires the presence of a second observer, which may not be practical for every data collection programme. Gomez et al. (2018) proposed a multi-species extension to the N-mixture model which involves the use of a beta distribution for the detection probability of lowabundance species. The abundance of species s at site i and sampling occasion t is in this modelling framework represented by Yits ∼ Binomial(Nis , ps ),

.

ps ∼ Beta(τ p, ¯ τ (1 − p)), ¯ where .p¯ is the mean detection probability among species, and .τ is a precision parameter that measures the similarity in detection probabilities. The use of the beta distribution to model the detection probability allows for sharing strength between species, using information for abundant species to inform detection probabilities for less abundant species. In turn, this can allow the estimation of abundance for species whose rarity may otherwise preclude them from examination. Moral et al. (2018) developed a method that allows for the estimation of abundances of two species, as well as a measurement of the relationship between them. This is achieved through the inclusion of a parameter in the abundance of one species which links it to the other species. At site i and sampling occasion t, the abundance of one species is allowed to depend on the other species as follows: Yit1 ∼ Binomial(Ni1 ),

.

Ni1 ∼ Poisson(λi1 ), Yit2 ∼ Binomial(Ni2 ), Ni2 ∼ Poisson(ψi + λi2 Ni1 ), where .λi2 > 0 allows for a positive impact of the abundance of one species on the other, while .λi2 = 0 suggests that one species does not impact the other. In this case, the parameter .ψi allows the species to be independently modelled. This modelling framework provides the opportunity to make inferences as to the relationship between species with the inclusion of the .λi2 parameter, though it focuses on pairs of species, rather than the community as a whole.

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Mimnagh et al. (2022) proposed a multi-species N-mixture (MNM) model. This is a methodology that allows for the measurement of abundances of multiple species and the relationships between them through the estimation of inter-species correlations, which are introduced using a multivariate normal random effect in the abundance. For species s at site i and sampling occasion t, this model may be summarised as Yits ∼ Binomial(Nis , pits ),

.

Nis ∼ Poisson(λis ), log(λis ) = ais + x⏉ i βs , ai ∼ MVN(μa , Σ a ), where .MVN(·, ·) denotes a multivariate normal distribution and .xi represents a vector of covariates at the site level that can be used to better predict the latent abundance .Nis . The unstructured covariance matrix .Σ a allows for the estimation of positive and negative inter-species correlations in abundance and allows the user to make inferences as to the relationships that these species have with one another. A disadvantage common to all of the models described in this section is an inability to deal with species present in the study area which are not observed during the survey. This is an area of interest that has been examined in terms of occupancy modelling, but a solution remains to be found for abundance modelling.

10.3.2 N-Mixture Models for an Open Population The N-mixture model introduced by Royle (2004) assumes that the population at each site is closed since it considers that the latent population, .Ni , does not change over time. However, this assumption can be violated in studies where animals are observed during many years or even decades. For these cases, Kéry et al. (2009) and Dail et al. (2011) proposed extensions to deal with open populations. In this section, we focus on the model proposed by Dail et al. (2011), as the model proposed by Kéry et al. (2009) can be viewed as a specific case. The generalised N-mixture for open populations considers that the site abundance varies throughout the periods of observation. The counts .Yit ∼ binomial(Nit , p), where .Nit denotes the population size at site i and time t. Hence, the integrated likelihood function is given in the form of L(p, θ |{Yit }) =

R 

.

i=1

⎡ ⎣

∞ Ni1 =Yi1

T   ∞  Nit Yit p (1 − p)Ni −Yit × ··· Yit Nit =Yit

t=1

 π(Ni1 , . . . , Nit , θ )

.

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The estimation of the parameters is carried out assuming that the abundance at each site and time has a first-order Markovian structure, i.e., that .Nit depends only on .Ni(t−1) . Thus, the distribution for the abundance can be written as  .π(Ni1 , . . . , NiT , θ ) = π(Ni1 , θ ) Tt=2 π(Nit , Ni(t−1) , θ ), where .π(Ni1 , θ ) is the distribution of the initial abundance at site i and time 1. In addition, .π(Nit , Ni(t−1) , θ ) is modelled through migration decomposition (Nichols et al., 2000) as a sum of two independent random variables, .Sit and .Git , where .Sit represents the animals at site i and time t who survived from .t − 1 and .Git denotes gains (new animals, e.g., due to births and/or immigration) at site i since time .t − 1. In probabilistic terms, these variables are represented as Git |Ni(t−1) ∼ Poisson(γ (Ni(t−1 ))),

.

Sit |Ni(t−1) , ω ∼ Binomial(Ni(t−1) , ω), where .γ (Ni(t−1) ) is the rate of the new arrivals at site i, which can be a function of the site abundance in the previous time, and .ω represents the survival probability. In this case, the discrete convolution, .Pa,b , that is used to represent the prior distribution from state .Ni(t−1) = a to .Nit = b, for .t > 1, is given by Pa,b =

min{a,b}

.

Binomial(c; a, ω) × Poisson(b − c; γ (a))

c=0

=

min{a,b} 



a c γ (a)b−c e−γ (a) ω (1 − ω)b−a × . c (b − c)!

c=0

Assuming that .π(Ni1 ; θ ) can be any discrete positive distribution, e.g., a Poisson.(λ), the integrated likelihood in (10.4) can be rewritten as L(p, λ, ω, α|{Yit }) =

R 

.

i=1

⎡ ⎣

∞ Ni1 =Yi1

  ∞   T  Nit Yit Ni −Yit p (1 − p) × ··· Yit Nit =Yit

t=1

 T λNi1 e−λ  PNi(t−1) ,Nit . Ni1 ! t=2

(10.4) Although the sum over .Nit is infinite, in practice it is necessary to set an upper bound, L, large enough that the remainder sum does not impact significantly the parameter estimates. For the simulations and examples, Dail et al. (2011) set .L = 200. However, they mention that the ideal choice may depend on the observed counts, and its choice varies according to the problem at hand. As in the N-mixture model for closed population, to estimate the parameters via classical inference,

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numerical optimisation methods are required, as it is not possible to find a closedform expression for the integrated likelihood. Additionally, Dail et al. (2011) proposed a closure test to verify whether the population under analysis is from a closed population or not. As the model proposed by Royle (2004) is a particular case when .ω = 1 and .γ = 0, it is possible to utilise, for T sufficiently large, the asymptotic test introduced by Self et al. (1987) to test .{H0 : γ = 0 and ω = 1} versus .{H1 : γ /= 0 and 0 ≤ ω < 1}. As the asymptotic distribution of the test is based on mixtures of chi-squared distributions and depends on the Fisher information matrix, they recommend the use of the observed information matrix, as the expected one cannot be obtained analytically. However, under the Bayesian perspective, the results of the asymptotic test are based only on the posterior distributions of .ω and .γ , and it is not necessary to obtain an information matrix. Due to the Markovian structure in the estimation of the parameters, the abundance estimate at each time, assuming .π(Ni1 ; θ ) = Poisson(λ), is obtained as ˆ Nˆ .1 = R λ,

.

Nˆ .t = ωˆ Nˆ .t−1 + R γˆ , where R is the number of sites, .Nˆ .1 is the initial abundance at time 1 (regardless ˆ .ωˆ and .γˆ are the estimates of of the site), .Nˆ .t is the abundance at time t and .λ, the parameters .λ, .ω and .γ , respectively. In addition, an estimate of trend in the abundances can be obtained dividing .Nˆ .t−1 by .Nˆ .t . If there is no interest in obtaining ˆ the abundance per period, the total abundance can be computed as .Nˆ = R λ. While Dail et al. (2011) partition the open-population model into survival .Sit and recruitment .Git , Hostetler and Chandler (2015) explain that this partitioning is not always possible if sites are not closed with respect to movement. In this case, it is suggested that we might replace the mechanistic model of Dail et al. (2011) with a classical population growth model. In an unlimited environment, population growth may be simply modelled using an exponential growth model, where r is the maximum per capita rate of increase: Nit ∼ Poisson(er Ni(t−1) ).

.

For scenarios in which a limit on population size exists, density-dependent versions of this model are also possible. If K is the stable equilibrium of a population and r is the population growth rate at low population density, abundance may be given by    Ni(t−1)  r 1− K Nit ∼ Poisson e Ni(t−1) .

.

Further to this, immigration models that allow for population growth following extinction may be implemented through the addition of a term .ν that describes the average number of immigrants per year:

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  Nit ∼ Poisson er Ni(t−1) + ν .

.

The density-dependent models may also incorporate immigration in the same way. The multi-species N-mixture model by Mimnagh et al. (2022) described in Sect. 10.3.1 may also be extended to allow for a relaxation of the closure assumption through the introduction of an autoregressive component on the abundance. Abundance is collected at i sites and t sampling occasions, for s species over k years, and is modelled as Yitks ∼ Binomial(Niks , pitks ),

.

Niks ∼ Poisson(λiks ). If .k = 1, then .λi1s is defined as log(λi1s ) = ais + x⏉ i βs .

.

For .k > 1, .λiks is allowed to depend on the latent abundance at year .k − 1: log(λiks ) = ais + x⏉ i β s + φs log(Ni(k−1)s + 1).

.

This model allows the estimation of inter-species correlations that vary by year, which can allow for inter-species relationships that change in time, though the current specification of .φs is restricted to correlations whose sign does not change from one year to the next.

10.3.3 N-Mixture Models for Zero-Inflated Data The original N-mixture model assumes that the latent abundance can be described using a Poisson distribution. This may not always be justified, particularly when data contain a large number of zero-counts; a common scenario to encounter when working with animal observation data which arises from surveying unoccupied sites. A negative binomial distribution allows for extra-Poisson variation by allowing the mean abundance to vary stochastically, and so a substitution of the Poisson distribution on abundance for a negative binomial distribution may accommodate a limited amount of zero-inflation. However, many datasets contain a larger number of zero-counts than may be modelled using the negative binomial distribution. If the negative binomial distribution proves unsuitable, distributions that accommodate zero-inflation in the data, including the zero-inflated Poisson distribution and zero-altered (hurdle) Poisson distribution, may be used as an alternative. The modelling frameworks described in this section accommodate zero-inflation by first determining whether a site is occupied and subsequently estimating abundance of occupied sites.

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Wenger and Freeman (2008) proposed a method that allows for the use of the N-mixture model to simultaneously model occurrence and abundance by specifying a zero-inflated distribution for the abundance. This is done by specifying a binomial distribution for the occupancy, .Oi , and introducing a variable .Ki , which is the realised abundance at site i, given presence, as Oi ∼ Bernoulli(φi ),

.

Ki ∼ Poisson(λi ), Ni = Oi × Ki . This modelling framework was not intended for use specifically with count data that contain a large number of zero-counts, but rather as an alternative to the original Nmixture model for any count dataset, with the aim of obtaining both occupancy and abundance estimates from a single model. This model retains the assumptions inherent under the original N-mixture model and so may be used to estimate singlespecies abundance for a closed population. Joseph et al. (2009) developed a framework that uses the zero-inflated Poisson and zero-inflated negative binomial models. These are both a combination of a Bernoulli process, which determines occupancy, and a negative binomial or Poisson process, which determines abundance. To implement the zero-inflated Poisson model, the site occupancy is given by Oi ∼ Bernoulli(1 − θ ),

.

where .Oi represents the occupancy at site i, and .θ is the probability of obtaining a zero-count. If the site is deemed to be occupied, the abundance is then estimated using a Poisson distribution as in the original N-mixture model. Hostetler and Chandler (2015) also proposed an extension that uses a zeroinflated Poisson distribution to model excess zero-counts. This is achieved by specifying the true abundance at site i and sampling occasion t as follows:  Nit ∼

Poisson(0)

with probability γ

Poisson(λ)

with probability (1 − γ ),

.

where .γ represents the proportion of excess zero-counts. The MNM model proposed by Mimnagh et al. (2022) may also be used to model excess zeros in the observations through the use of a hurdle-Poisson (or zero-altered) model in the abundance. The observations .Yits are assigned a binomial distribution, as described in Sect. 10.3.1. The hurdle-Poisson distribution then consists of two separate processes. The first is a Bernoulli process, which determines whether a site is occupied (abundance is non-zero) or unoccupied (abundance is zero). If the abundance is non-zero, a second random variable with a zero-truncated Poisson distribution determines the value of the abundance, i.e.,

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Ois ∼ Bernoulli(1 − θ ),

.

Cis ∼ zero-truncated Poisson(λis ), where .Ois represents the occupancy of species s at site i, .θ is the probability of obtaining a zero-count and .Cis represents the abundance of species s at site i. This abundance .Cis is only estimated at sites that are occupied by a particular species (i.e., .Ois = 1). The abundance .Nis is then defined as  Nis =

.

0,

if Ois = 0

Cis , if Ois = 1

.

This may be written as Nis ∼ hurdle-Poisson(λis , θ ).

.

The hurdle-Poisson model described by Mimnagh et al. (2022) differs from the zero-inflated Poisson model described by Joseph et al. (2009) in the assumptions required for abundance estimation. The use of a zero-truncated Poisson distribution for abundance assumes that all zeros arising in the data arise from the occupancy process, while the zero-inflated Poisson distribution allows zeros to arise from both the occupancy and abundance processes. The decision as to which model to use will depend on the goal of the user. If the user is interested in examining true and false zeros (i.e., zeros produced because the site is unoccupied, and zeros produced because, though the site is occupied, no observations were made), then a zeroinflated model is an appropriate choice. If the analysis is concerned with whether a count is zero or non-zero and the user is uninterested in the origin of the zero-counts, then all zeros may be assigned to the occupancy process and the hurdle-Poisson model may be used. We now implement some of the different extensions to N-mixture models on a real dataset in the case study described below.

10.4 Case Study: Bee Abundance Wild bee species play major roles in pollination, increasing the yield of approximately 85% of all cultivated crops (Zattara & Aizen, 2021). Abundance and diversity of bee species are reported to be in decline on a near-global level (TheisenJones & Bienefeld, 2016; Pettis et al., 2010; Leonhardt et al., 2013), with economic and ecological repercussions inherent in this decline. In order to make decisions concerning management of bee populations (i.e., conservation, use and monitoring, according to Caughley (1994)), it is beneficial for us to be able to estimate species abundance.

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In the following section, we examine how bee population sizes may be estimated using the original N-mixture model (Royle, 2004) and the multi-species N-mixture model (Mimnagh et al., 2022). This analysis is motivated by data collected as part of the BeeWalk Survey Scheme (Comont et al., 2021), a programme established in 2008 by the Bumblebee Conservation Trust, which involves transects being surveyed by volunteers across the UK on a monthly basis. By the end of the 2019 data collection period, data had been collected for approximately 70 bee species, at over 1300 sites in the UK. Here we examine bee observation data collected in 2016 and 2019. To ensure that we are comparing data collected from the same seasonal cycles, we examine data collected in June of both years. The models described in this section are implemented using a Bayesian framework. Each of the models was implemented in R (R Core Team, 2022) through the probabilistic programming software JAGS (Plummer, 2003, 2017) using four chains with 20,000 iterations each, of which the first 5000 were discarded as burn-in, with a thinning of five to reduce autocorrelation in the MCMC samples. Parameter convergence was determined using the potential scale reduction factor ˆ a diagnostic criteria proposed by Gelman and Rubin (1992). An .Rˆ value that is (.R), very close to one is an indication that the four chains have mixed well. If .Rˆ value was less than 1.05, the chains were considered to have mixed properly, and the posterior estimates of the parameters were considered reliable. Prior to modelling this data to estimate bee abundance, we must check to confirm that we should not encounter issues regarding infinite estimates of abundance, using the covariance diagnostic proposed by Dennis et al. (2015), and detailed in Sect. 10.2; see the code to run this diagnostic in Sect. 10.6.3. A negative value for this diagnostic would suggest that problems with infinite parameter estimates may occur. Covariance diagnostic values obtained for the data utilised in this section were all positive and so do not suggest that we should expect issues with parameter estimates. The original N-mixture model (Royle, 2004) may be used to estimate abundance for a single species and assumes that populations are closed. This analysis will focus on data collected in June of 2019 at 60 sites for the common carder bumblebee (Bombus pascuorum). Transects examined in this study range in length from 167 to 3670 metres. It was thought that perhaps longer transects may provide more opportunity for bee observations, which may in turn affect abundance estimates. To account for possible effects on abundance due to transect length, the length in metres of each transect was included in the linear predictor for the abundance parameter. Additionally, to account for transect location, the latitude, longitude and their interaction term latitude .× longitude were examined. All covariates were scaled to have zero-mean and unit variance. The estimates for the effect of each of these covariates on carder bumblebee abundance are available in Table 10.1. The effect of longitude and latitude on carder bumblebee estimates may be difficult to discern, due to the use of the interaction term latitude .× longitude, though we can see from Fig. 10.1 that abundance estimates do appear larger in the south of the UK and appear to decrease toward the north. Additionally, transect length appears to have a positive effect on estimates of abundance, with a mean estimate of .0.16

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Table 10.1 Mean parameter estimates and .95% credible intervals for the original N-mixture model applied to Common Carder Bumblebee count data collected in June 2019 Species Carder Bumblebee

Intercept 4.27 (4.12, 4.43)

Latitude .−0.09

(.−0.28, 0.08)

Longitude 0.39 (0.18, 0.61)

Latitude .× Longitude 0.12 (.−0.05, 0.28)

Length 0.16 (0.03, 0.29)

60.0

57.5

Latitude

Abundance 150 100 55.0 50

52.5

50.0 −10.0

−7.5

−5.0

−2.5

0.0

Longitude

Fig. 10.1 Common Carder Bumblebee (Bombus pascuorum) abundance estimates for June 2019 at 60 sites in the UK, obtained using the original N-mixture model

(i.e., abundance estimates associated with longer transects are greater than those associated with shorter transects). The estimates of abundance obtained for the common carder bumblebee using this original N-mixture model are shown in Fig. 10.1. The abundance estimates provided here may be considered as an estimate of the size of the population that is currently foraging at this site. It cannot be viewed as representative of the full carder bee abundance in the area, or the abundance of the local carder bee colony as a whole, as only approximately .30% of a bee colony’s population will forage at a certain time. Additionally, bees may travel several kilometres while foraging (Greenleaf et al., 2007), so it is impossible that the bees observed at each site may not be local to the area and may have travelled a distance from their colony to forage.

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Table 10.2 Mean parameter estimates and .95% credible intervals for the MNM model applied to count data for eight bee species, collected in June 2016 and 2019 Species White-tailed Bumblebee Buff-tailed Bumblebee Garden Bumblebee Red-tailed Bumblebee Tree Bumblebee Early Bumblebee Carder Bumblebee European Honeybee

Intercept 2.72 (2.01, 3.40) 3.50 (3.02, 3.93) 1.72 (1.03, 2.35) 3.67 (3.16, 4.14) 2.92 (2.36, 3.45) 3.15 (2.67, 3.57) 4.03 (3.73, 4.31) 0.98 (.−0.20, 1.99)

Latitude 1.02 (0.41, 1.59) 0.28 (.−0.13, 0.69) 0.04 (.−0.62, 0.74) .−0.58 (.−1.04, .−0.17) 0.04 (.−0.54, 0.67) 0.30 (.−0.12, 0.73) 0.09 (.−0.21, 0.41) .−0.42 (.−1.45, 0.61)

Longitude 0.67 (0.07, 1.25) 0.12 (.−0.05, 0.62) .−0.45 (.−1.19, 0.27) .−0.03 (.−0.49, 0.42) 0.38 (.−0.25, 1.04) 0.24 (.−0.23, 0.76) 0.22 (.−0.11, 0.55) 0.87 (.−0.11, 2.01)

Latitude .× Longitude .−0.03 (.−0.61, 0.51) .−0.13 (.−0.29, 0.31) 0.33 (.−0.33, 1.01) 0.26 (.−0.18, 0.71) 0.83 (0.24, 1.41) 0.74 (0.29, 1.21) 0.13 (.−0.17, 0.42) 0.45 (.−0.44, 1.45)

Length .−0.13

(.−0.27, 0.28) 0.29 (0.16, 0.65) .−0.04 (.−0.56, 0.45) 0.64 (0.29, 1.01) 0.66 (0.26, 1.07) 0.54 (0.20, 0.89) 0.21 (0.01, 0.43) 0.60 (.−0.15, 1.35)

For works on estimates of bee colony abundances, we refer the reader to McGrady et al. (2021), Russo et al. (2015) and Kuhlman et al. (2021). This allows us to examine differences in abundance estimates from June 2016 to June 2019 for the European honeybee (Apis mellifera) and seven species of bumblebee: the white-tailed bumblebee (Bombus lucorum), the buff-tailed bumblebee (Bombus terrestris), the garden bumblebee (Bombus hortorum), the tree bumblebee (Bombus hypnorum), the early bumblebee (Bombus pratorum), the redtailed bumblebee (Bombus lapidarius) and the common carder bumblebee (Bombus pascuorum). The initial examination of this data reveals that .54% of observations (1049 out of a total of 1920 observations) are composed of zero-counts. For this reason, the MNM model with a hurdle component, described in Sect. 10.3.3, is employed. For appropriate comparison with the N-mixture model implementation detailed above, the same sites are examined here using the MNM model. The effect on abundance estimates of the covariates latitude, longitude, latitude .× longitude and transect length is provided in Table 10.2. As with the results in Table 10.1, the presence of the interaction term latitude .× longitude makes it difficult to interpret the effect of latitude and longitude on abundance estimates. For this reason, these results could be displayed in a similar manner to those displayed in Fig. 10.1, with a separate abundance map per species, though we do not present those maps here.

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Table 10.3 The number of sites at which abundance estimates (obtained using the MNM model) for each bee species increased, decreased or remained unchanged between 2016 and 2019 Species White-tailed Bumblebee Buff-tailed Bumblebee Red-tailed Bumblebee Garden Bumblebee Tree Bumblebee Early Bumblebee Common Carder Bumblebee European Honeybee

Increase 32 34 28 25 26 17 18 35

Decrease 22 21 28 32 28 34 29 20

No change 6 5 4 3 6 9 13 5

It appears that the length of the transect has a positive effect on abundance in the case of the buff-tailed bumblebee, the red-tailed bumblebee, the tree bumblebee, the early bumblebee and the common carder bumblebee, as was demonstrated in Table 10.1. It may appear at first glance from the mean estimates that transect length also has a positive effect on abundance estimates for the European honeybee and a negative effect for the white-tailed bumblebee and the garden bumblebee. However, as the .95% credible intervals associated with the effect of transect length for these species contain 0, we cannot say with certainty that these mean estimates are reliable and instead conclude that it seems that transect length does not have an effect on abundance estimates for these species. We can also examine how abundance estimates at each site change between 2016 and 2019. Table 10.3 shows the number of sites (out of the total 60 sites examined) at which abundances increased, decreased or remained unchanged between 2016 and 2019. Species such as the European honeybee and the buff-tailed bumblebee appear to have experienced abundance increases at a large number of sites, while species such as the early bumblebee and common carder bumblebee appear to have experienced a decrease in abundance at a majority of sites. Figure 10.2 displays inter-species abundance correlations, which may allow for inferences to be made as to the relationships that these species have with one another or with their environments. For example, we can see that the early bumblebee has a strong positive correlation with the garden bumblebee, which suggests that the abundances of these species may be increasing or decreasing together, while the early bumblebee has a slightly negative correlation with the whitetailed bumblebee, which suggests that one of these species may be experiencing an increase in abundance while the other decreases. These correlations seem to correspond with the results obtained in Table 10.3, as both the early bumblebee and garden bumblebee experience a decrease at a large number of sites, while the white-tailed bumblebee experiences an increase in abundance at a majority of sites examined.

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0.55

Garden Bumblebee

0.38

0.30

Tree Bumblebee

0.39

0.37

0.43

Early Bumblebee

−0.12 −0.02

0.68

0.43

Common Carder Bumblebee

−0.09 −0.03

0.46

0.23

0.56

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Garden Bumblebee

Tree Bumblebee

Early Bumblebee

Common Carder Bumblebee

Red−tailed Bumblebee

0.05

0.29

0.37

0.24

0.60

0.12

European Honeybee

0.43

0.60

0.49

0.60

0.34

0.23

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−0.8 −0.6 −0.4 −0.2

0

0.2

0.4

0.6

Red−tailed Bumblebee

0.07

0.8

1

Fig. 10.2 Correlation between abundance estimates for seven bee species obtained using the MNM model

10.5 Discussion In this chapter, we have examined several approaches to modelling abundance data, beginning with the original N-mixture model (Royle, 2004) and continuing to explore model extensions, which allow us to estimate animal abundance using data collected in a range of scenarios. The N-mixture family of models are widely used due to their ability to estimate both abundance and detection probability. We have also addressed previous work, which has demonstrated that N-mixture models can sometimes suffer from issues with identifiability (Dennis et al., 2015), which lead to very small estimates for detection probability and very large estimates of abundance. This is an issue that must be kept in mind when using N-mixture models, and the covariance diagnostic provided by Dennis et al. (2015) is a useful tool in assessing whether an N-mixture model is appropriate for use with a certain dataset. We finished by demonstrating how the N-mixture model by Royle (2004) and an N-mixture model for multiple species (Mimnagh et al., 2022) may be implemented to estimate foraging bee abundance using the software R and the probabilistic programming language JAGS. Results of this analysis (Fig. 10.1) suggest that foraging bee populations may be larger in the South of England and decrease as we travel through the North of England and into Scotland. As mentioned previously, due to bee colony dynamics, this abundance estimate does not represent the total bee abundance in the area, but rather the foraging abundance and can be thought of as an index of the local population size.

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10.6 Code Below, we provide R code to implement the N-mixture models discussed in this chapter using JAGS and code to calculate the covariance diagnostic proposed by Dennis et al. (2015).

10.6.1 Bayesian N-Mixture Models for Closed Populations in JAGS In this section, we present an implementation of the Bayesian N-mixture model for closed population using the software R and the probabilistic programming language JAGS. Here, we present a simple Bayesian N-mixture model which assumes that Yi |Ni , p ∼ Bin(Ni , p),

.

Ni ∼ Poisson(λ), λ ∼ Gamma(ζ = 1, η = 0.1), p ∼ Beta(ν = 1, ξ = 1).

(10.5)

The prior on .λ is set up such that it is non-information since a priori .E(λ) = 10 and .Var(λ) = 100. In practice, that means that we believe that the true value of .λ is around 10, but we are not sure about it, which is demonstrated by the large variance of the prior distribution. The prior on the detection probability, p, is also non-informative in the sense that it gives equal prior probability to any possible value in the range .[0, 1]. If we were to make our own implementation of the Bayesian N-mixture model for closed population, we would need to find the full conditional of .λ and p since the joint posterior does have a closed-form distribution. Given the model presented above, the full conditionals of .λ and p are π(λ|y, p) ∝ p(y|N, p) × π(λ),   ∝ Bin(Yi |Ni , p) × Poisson(λ|ζ, η)

.

i

and π(p|y, λ) ∝ p(y|N, p) × π(p),   ∝ Bin(Yi |Ni , p) × Beta(p|ν, ξ ).

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However, with tools like JAGS, we do not need to find the full conditionals associated with the parameters of interest or manually implement any MCMC scheme to sample from the posterior distribution. Instead, our model can be specified as follows: library(rjags) library(R2jags) # Simulate synthetic data ------------------set.seed(1234) T