Game-Theoretical Models in Biology

By Mark Broom, Jan Rychtar

© 2013 – Chapman and Hall/CRC

520 pages | 86 B/W Illus.

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Hardback: 9781439853214
pub: 2013-03-26
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About the Book

Covering the major topics of evolutionary game theory, Game-Theoretical Models in Biology presents both abstract and practical mathematical models of real biological situations. It discusses the static aspects of game theory in a mathematically rigorous way that is appealing to mathematicians. In addition, the authors explore many applications of game theory to biology, making the text useful to biologists as well.

The book describes a wide range of topics in evolutionary games, including matrix games, replicator dynamics, the hawk-dove game, and the prisoner’s dilemma. It covers the evolutionarily stable strategy, a key concept in biological games, and offers in-depth details of the mathematical models. Most chapters illustrate how to use MATLAB® to solve various games.

Important biological phenomena, such as the sex ratio of so many species being close to a half, the evolution of cooperative behavior, and the existence of adornments (for example, the peacock’s tail), have been explained using ideas underpinned by game theoretical modeling. Suitable for readers studying and working at the interface of mathematics and the life sciences, this book shows how evolutionary game theory is used in the modeling of these diverse biological phenomena.


"… a comprehensive, up-to-date introduction that uniquely blends mathematical clarity and biological intuition.

The first half explores the many ways that the concepts ‘game’ and ‘evolution’ can be realized. … [The authors] discuss not only evolution in terms of competition among existing strategies but also long-term evolution through successive trait substitutions. The authors also examine the effects of population size and structure, providing an entry point into one of the most active and exciting new directions in the field.

The second half brings to life the mathematical frameworks introduced earlier … the authors illuminate not only the behaviors that emerge in these particular contexts but also the process of matching mathematical framework to biological reality, a critical skill for aspiring evolutionary theorists. …

Broom and Rychtár offer a refreshingly clear exposition of kin selection, using a simple model with direct biological interpretation. They present this alongside other, distinct mechanisms for cooperation, such as direct and indirect reciprocity, avoiding the confusion that muddles much of the literature on this topic. …

Students of evolutionary game theory … would do well to read Game-Theoretical Models in Biology all the way to the finish line. This engaging primer demonstrates that there is no tension between mathematical elegance and biological fidelity: both are needed to further our understanding of evolution."

—Benjamin Allen, Emmanuel College, and Martin A. Nowak, Harvard University, Science, Vol. 341, August 2013

"In the biological sciences, there is probably no better time to become a game theorist, and anyone who so aspires will value this text as a guide. Assuming only a modicum of mathematics, Broom and Rychtár lead their readers all the way from the rudiments of evolutionary game theory to the research frontier. They are appealingly candid on how their scope reflects their taste. Nonetheless, their coverage is remarkably wide-ranging, from old standards like the Hawk-Dove game to newer applications such as epidemiology. The authors strike an excellent compromise between breadth and depth by limiting the generality of some theoretical treatments, choosing good examples, and using up-to-date references to round out their coverage. By focusing on evolutionary stable strategy as a "static concept," Broom and Rychtár convincingly demonstrate the power of game-theoretic models to describe real animal behaviour, in ways that mathematicians who specialize on evolutionary dynamics are apt to overlook. Thus the book brings a timely sense of balance to the range of texts now available. At the research frontier, the trail has many forks; but whichever fork readers decide to explore, this book will leave them admirably well prepared for the way ahead."

—Mike Mesterton-Gibbons, Florida State University

"This book should prove to be a valuable addition to the bookshelf of anyone working in, or wishing to work in, the area of application of game theoretical approaches to modelling biological phenomena. The last 39 years, since the seminal paper of Maynard Smith and Price, have seen an explosion of research in this area. The early work concentrated on fairly simple models such as Hawk-Dove and the War of Attrition, but this rapidly expanded to encompass matrix games and multi-player games. Recently, additional complexities through population structure and adaptive dynamics have greatly enhanced the applicability of these models. Here, the authors, who have themselves made several contributions to these more complex models, in particular with respect to kleptoparisitism, have set out to provide both a solid underpinning of the mathematical aspects of the subject, but also to anchor it solidly in its biological context. They have succeeded admirably in both aims, and this book should become the standard reference for this area. I shall certainly find it an invaluable resource, and recommend it highly."

—Chris Cannings, Professor Emeritus, University of Sheffield

"Many books on evolutionary game dynamics are on my shelf — why would I put another one there? The first thing that caught my attention in the new book of Mark Broom and Jan Rychtár are the exercises. Reading a book on mathematics is a fruitless exercise for me without a pencil and paper and maybe a computer. The wonderful exercises in this book are sufficiently deep to provoke some serious thinking, but not so difficult that the reader turns them down in despair. Broom and Rychtár cover a multitude of subjects, from the very basics of game theory to concrete biological applications. Some of them have not found their place into the minds of many young scientists, such as food competition and the ideal free distribution, but it pays to learn about these classical applications. The book focuses on static aspects. However, with complex dynamical considerations, theoreticians are often going too far and it is absolutely fascinating to see how far you can get from such a focus on static properties of games. I highly recommend this book—not for your shelf, but for your desk!"

—Arne Traulsen, Max Planck Gesellschaft

"Evolutionary game theory originally flourished some thirty years ago as a means to predict individual behavior in biological systems through its static game-theoretic solution concept of an evolutionarily stable strategy (ESS). Since then, the advancement of the theory has increasingly occurred in other disciplines. In their timely book, Broom and Rychtár have returned the theory to its biological roots and illustrated the relevance of these recent developments to biology. Their careful mathematical analysis of diverse biological models based on such current topics as nonlinear and multi-player games, structured population effects and the stochastic effects of finite populations provides a comprehensive set of applications. The book will serve both as an important resource for researchers in the field and as a valuable text for students at a graduate or senior undergraduate level. Students will especially appreciate the extensive set of exercises for each chapter and that the book is exceptionally well-written and self-contained."

—Ross Cressman, Wilfrid Laurier University

Table of Contents


The history of evolutionary games

The key mathematical developments

The range of applications

Reading this book

What Is a Game?

Key game elements

Games in biological settings

Two Approaches to Game Analysis

The dynamical approach

The static approach—evolutionarily stable strategies (ESSs)

Dynamics versus statics

Some Classical Games

The hawk-dove game

The prisoner’s dilemma

The war of attrition

The sex ratio game

The Underlying Biology

Darwin and natural selection


Games involving genetics

Fitness, strategies and players

Selfish genes: how can non-beneficial genes propagate?

The role of simple mathematical models

Matrix Games

Properties of ESSs

ESSs in a 2 × 2 matrix game

Haigh’s procedure to locate all ESSs

ESSs in a 3 × 3 matrix game

Patterns of ESSs

Extensions to the hawk-dove game

Nonlinear Games

Overview and general theory

Linearity in the focal player strategy and playing the field

Nonlinearity due to non-constant interaction rates

Nonlinearity in the strategy of the focal player

Some differences between linear and nonlinear theory

Asymmetric Games

Selten’s theorem for games with two roles

Bimatrix games

Uncorrelated asymmetry—the owner-intruder game

Correlated asymmetry

Multi-Player Games

Multi-player matrix games

The multi-player war of attrition

Structures of dependent pairwise games

Extensive Form Games and Other Concepts in Game Theory

Games in extensive form

Perfect, imperfect and incomplete information

Repeated games

State-Based Games

State-based games

A question of size

Life history theory

Games in Finite and Structured Populations

Finite populations and stochastic games

Evolution on graphs

Spatial games and cellular automata

Adaptive Dynamics

Introduction and philosophy

Fitness functions and the fitness landscape

Pairwise invasibility and evolutionarily singular strategies

Adaptive dynamics with multiple traits

The assumptions of adaptive dynamics

The Evolution of Cooperation

Kin selection and inclusive fitness

Greenbeard genes

Direct reciprocity: developments of the prisoner’s dilemma


Indirect reciprocity and reputation dynamics

The evolution of cooperation on graphs

Multi-level selection

Group Living

The costs and benefits of group living

Dominance hierarchies: formation and maintenance

The enemy without: responses to predators

The enemy within: infanticide and other anti-social behavior

Mating Games

Introduction and overview

Direct conflict

Indirect conflict and sperm competition

The battle of the sexes

Selecting mates: signaling and the handicap principle

Other signaling scenarios

Food Competition


Ideal free distribution for a single species

Ideal free distribution for multiple species

Distributions at and deviations from the ideal free distribution

Compartmental models of kleptoparasitism

Compartmental models of interference

Producer-scrounger models

Predator-Prey and Host-Parasite Interactions

Game-theoretical predator-prey models

The evolution of defense and signaling

Brood parasitism

Parasitic wasps and the asymmetric war of attrition

Complex parasite lifecycles

Epidemic Models

SIS and SIR models

The evolution of virulence

Viruses and the prisoner's dilemma


Types of evolutionary games used in biology

What makes a good mathematical model?

Future developments

Appendix: Intro to MATLAB



MATLAB, Further Reading, and Exercises appear at the end of each chapter.

About the Authors

Mark Broom is a professor of mathematics at City University London. For 20 years, he has carried out mathematical research in game theory applied to biology. His major research themes include multi-player games, patterns of evolutionarily stable strategies, models of parasitic behavior (especially kleptoparasitism), the evolution of defense and signaling, and evolutionary processes on graphs. He received his PhD in mathematics from the University of Sheffield.

Jan Rychtár is an associate professor of mathematics at the University of North Carolina at Greensboro, where he helped start a math-biology research group involving faculty and students from the biology and mathematics departments. He works on game theoretical models and mathematical models of kleptoparasitism. His recent research themes focus on models of brood parasitism in dung beetles, models of mate selection in honey bees, and evolutionary processes on graphs. He received his PhD in mathematics from the University of Alberta.

About the Series

Chapman & Hall/CRC Mathematical and Computational Biology

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Combinatorics
SCIENCE / Life Sciences / Evolution