## CTW: Evolutionary Game Theory

### Organizers

Andrew Belmonte
Department of Mathematics, Pennsylvania State University
Vlastimil Krivan
Mathematics and Biomathematics, Faculty of Science, University of South Bohemia
John Nagy
Life Sciences, Scottsdale Community College
Zhijun Wu
Math, Bioinformatics, & Computational Biology, Iowa State University

Evolutionary game theory, along with replicator equations, has been applied successfully to modeling evolution of various biological or social systems, ranging from virus infection to bacteria development, from plant succession to animal breeding, and from trace of evolutionary history to study of biodiversity and ecology. Applications in areas such as population genetics, animal behaviors, and evolution of social cooperation have especially seen great developments and impacts. In evolutionary game theory, species are considered as if they are players in a game, competing for resources, for survival, and for reproduction. A mathematical (game) model can then be established for study of any given population of competing species, and for analysis of population changes and prediction of equilibrium states and their stabilities. The theory involves such mathematical branches as game theory, optimization theory, and ordinary differential equations, and further extends to graph theory, stochastic processes, and partial differential equations as appropriate. Although emerged as a powerful mathematical tool for evolutionary and ecological modeling, the evolutionary game theory is still in the stage of early development. Theoretical issues remain to be addressed and computational methods need to be developed, for equilibrium computation, dynamic simulation, and stability analysis. Application problems are arising and yet to be investigated in many critical fields of biology such as development of energy-efficient or nutrition-rich plants and animals, analysis of human microbiome genomic data, control of infectious diseases, modeling immune-defense systems of biological species, etc. This workshop is to bring an interdisciplinary group of experts as well as biologists and mathematicians who are interested in evolutionary game modeling, to have an extensive discussion on current and future development of evolutionary game theory and applications. Topics include reviews or reports on recent theoretical or computational developments, or critical applications. The goal of the workshop is to increase communications among researchers and especially between biologists and mathematicians, in order to have a better understanding of the theory, to identify challenges and applications of the field, to promote interdisciplinary collaborations, and to accelerate future developments of the field.

### Accepted Speakers

Athena Aktipis
Psychology, Arizona State University
Priyanga Amarasekare
Ecology and Evolutionary Biology, University of California Los Angeles
Robert Austin
Physics, Princeton University
Amos Bouskila
Life Sciences, Ben-Gurion Univ. of the Negev
Mark Broom
Department of Mathematics, City University London
Joel Brown
Biological Sciences, University of Illinois at Chicago
Chris Cannings
school of maths and stats, university
Ross Cressman
Jim Cushing
Mathematics, University of Arizona
Ted Galanthay
Mathematics, Ithaca College
Jeff Gore
Physics, MIT
Christoph Hauert
Burt Kotler
Mitrani Department of Desert Ecology, Ben-Gurion University
Yuan Lou
Department of Mathematics, The Ohio State University
Johan (=Hans) Metz
Plant Ecology and Phytochemistry, Analysis and Dynamical Systems, Institute of Biology, Mathematical Institute
Bill Mitchell
Biology, Indiana State University
Douglas Morris
Kalle Parvinen
Department of Mathematics and Statistics, University of Turku
Rosalyn Rael
Center for Bioenvironmental Research, Tulane University
Tim Reluga
Department of Mathematics, Pennsylvania State University
Susan Riechert
Barry Sinervo
Ecology and Evolutionary Biology, UC Santa Cruz
Kateřina Staňková
Department of Knowledge Engineering, Maastricht University
Monday, April 27, 2015
Time Session
Tuesday, April 28, 2015
Time Session
Wednesday, April 29, 2015
Time Session
Thursday, April 30, 2015
Time Session
Friday, May 1, 2015
Time Session
Name Email Affiliation
Aktipis, Athena aktipis@asu.edu Psychology, Arizona State University
Alonzo, Suzanne shalonzo@ucsc.edu
Amarasekare, Priyanga amarasek@ucla.edu Ecology and Evolutionary Biology, University of California Los Angeles
Austin, Robert austin@princeton.edu Physics, Princeton University
Belmonte, Andrew andrew.belmonte@gmail.com Department of Mathematics, Pennsylvania State University
Bouskila, Amos bouskila@bgu.ac.il Life Sciences, Ben-Gurion Univ. of the Negev
Broom, Mark mark.broom.1@city.ac.uk Department of Mathematics, City University London
Brown, Joel squirrel@uic.edu Biological Sciences, University of Illinois at Chicago
Cannings, Chris C.Cannings@shef.ac.uk school of maths and stats, university
Cressman, Ross rcressman@wlu.ca
Cushing, Jim cushing@math.arizona.edu Mathematics, University of Arizona
Dall, Sasha S.R.X.Dall@exeter.ac.uk Centre for Ecology & Conservation, University of Exeter
Durrett, Rick rtd@math.duke.edu Department of Mathematics, Duke University
Galanthay, Ted tgalanthay@ithaca.edu Mathematics, Ithaca College
Gore, Jeff gore@mit.edu Physics, MIT
Griffin, Christopher griffinch@ieee.org Applied Research Laboratory, The Pennsylvania State University
Hauert, Christoph christoph.hauert@math.ubc.ca
Kang, Yun yun.kang@asu.edu Applied Sciences and Mathematics, Arizona State University
Kotler, Burt kotler@bgu.ac.il Mitrani Department of Desert Ecology, Ben-Gurion University
Krivan, Vlastimil vlastimil.krivan@gmail.com Mathematics and Biomathematics, Faculty of Science, University of South Bohemia
Lou, Yuan lou@math.ohio-state.edu Department of Mathematics, The Ohio State University
Metz, Johan (=Hans) j.a.j.metz@biology.leidenuniv.nl Plant Ecology and Phytochemistry, Analysis and Dynamical Systems, Institute of Biology, Mathematical Institute
Miekisz, Jacek miekisz@mimuw.edu.pl Institute of Applied Mathematics, University of Warsaw
Mitchell, Bill William.Mitchell@indstate.edu Biology, Indiana State University
Nagy, John john.nagy@scottsdalecc.edu Life Sciences, Scottsdale Community College
Parvinen, Kalle kalle.parvinen@utu.fi Department of Mathematics and Statistics, University of Turku
Rael, Rosalyn Rosalyn.rael@gmail.com Center for Bioenvironmental Research, Tulane University
Reluga, Tim treluga@math.psu.edu Department of Mathematics, Pennsylvania State University
Riechert, Susan sriecher@utk.edu
Sinervo, Barry lizardrps@gmail.com Ecology and Evolutionary Biology, UC Santa Cruz
Stankova, Kateřina k.stankova@maastrichtuniversity.nl Department of Knowledge Engineering, Maastricht University
van Gils, Jan Jan.van.Gils@nioz.nl Marine Ecology, NIOZ Royal Netherlands Institute for Sea Research
Wu, Zhijun zhijun@iastate.edu Math, Bioinformatics, & Computational Biology, Iowa State University
Zhou, Wen (Rick) rickzhouwen@gmail.com Statistics, Colorado State University
Need-based transfers and cooperation in uncertain environments

Abstract not submitted.

Different approaches to modeling foraging and predator-prey games among animals

Understanding principles and processes in ecology and evolution is not easy. Generating hypotheses and predictions in these disciplines is often not intuitive due, in part, to the many factors that may affect the outcomes of processes. Moreover, some of the situations involve games among various organisms that may lead to unintuitive results. Theoretical models may not provide proofs that we reached full understanding of the system, but they can generate testable hypotheses and predictions and can assist in the understanding of experimental results. Here I describe different modeling approaches we have used to investigate animal decisions in regard to foraging under the risk of predation in two systems. In the first, we interpret the escape strategy of a lizard from an avian predator with a simple decision tree model. The second system describes games among rodents and between rodents and their predators. This system begs for a game theoretic model, and two approaches will be exemplified. A static game has the advantage of simplicity. It can often be solved analytically and its results are relatively easy to interpret. Nevertheless, the simplicity has its costs in terms of realism. Some simplifications embedded in the static approach can be relaxed in a dynamic state-variable game model. These models provide refined insights and more specific predictions, taking into consideration variation in the state of the animals and its temporal dynamics.

Modelling evolution in structured populations involving multi-player interactions

Within the last ten years, models of evolution have begun to incorporate structured populations, including spatial structure, through the modelling of evolutionary processes on graphs (evolutionary graph theory). One limitation of this otherwise quite general framework is that interactions are restricted to pairwise ones, through the edges connecting pairs of individuals. Yet many animal interactions can involve many individuals, and theoretical models also describe such multi-player interactions. We shall discuss a more general modelling framework of interactions of structured populations, including the example of competition between territorial animals. Depending upon the behaviour concerned, we can embed the results of different evolutionary games within our structure, as occurs for pairwise games such as the Prisoner's Dilemma or the Hawk-Dove game on graphs. For a population to evolve we also need an evolutionary dynamics, and we demonstrate a birth-death dynamics for our framework. Finally we discuss some examples together with some important differences between this approach and evolutionary graph theory.

Evolution of Animal Networks

Joint work with Prof Mark Broom, City University, London.

Consider a populations of individuals with pairwise relationships between those individuals. We represent this population by a simple graph G = (V, E), where V is the set of vertices representing the individuals and E V V representing the existing relationships. The degree of a vertex x, d(x), is the number of vertices to which x is linked. We suppose that, labeling the vertices as {x1,x2,...,xn}, vertex xi has a target ti, which represents the degree which individual xi would ideally possess.A sequence {t1,t2,...,tn} is said to be graphic if there exists a simple graph which has precisely those degrees. In that case each individual could satisfy its target. On the other hand in general there will be some deviation from the set of targets. We consider the evolution of the network with the the following Markov Chain; at time t pick a vertex i with probability 1/n (independently of previous picks), if the vertex has degree equal to its target (a Neutral) then nothing happens, if the vertex has degree less than its target (a Joiner) then an edge (i,j) is added picking a possible j with equal probabilities over those available, while if the vertex has degree greater than its target (a Breaker) an edge (i,j) is removed picking a j with equal probability from those available. A graph G has deviation d(G) = Σi {1, 2, . . . , n}|d(xi) − t(xi)|. It is proved that the set of graphs with minimal deviation is connected under the above Markov Chain, and that the chain converges to that set. The Markov Chain over that minimal set is reversible, so satisfies the complete balance condition. Some examples are given. In a further model we allow the individuals to select the change they induce when selected so that they improve their total payoff, (d(x) − t(x)), over the resulting stationary distribution compared with the current one. Variants of this model will be considered and some examples given.

Some bifurcation theorems for evolutionary game theoretic versions of matrix models for structured population dynamics

I’ll describe some theorems concerning the fundamental bifurcations that can occur as r (or R0) increases through 1 for evolutionary versions of matrix models for structured population dynamics. I’ll illustrate the theorems with applications motivated by recent observations from field studies of nesting colonies of gulls on Protection Island National Wildlife Refuge in the Strait Juan de Fuca (between Vancouver Island and the state of Washington). These observations concern changes in life history strategies (such as a rise in egg cannibalism and changes in the timing of egg laying) which have been correlated with an increase in mean sea temperature in the Strait.

Optimal information use in habitat selection

How might organisms constrained by perceptual limitations or imperfect information use available information optimally in habitat selection? To begin to answer this question, we study a general ordinary differential equation model of a single species in a two-patch heterogeneous environment in which organisms have access to resource information. There exists a global evolutionarily stable strategy, which depends on the magnitude of the constraints and the heterogeneity of the resources, which leads to the ideal free distribution (IFD). When organisms pay a cost to travel between patches, this strategy is no longer evolutionarily stable, but a strategy that incorporates these costs and does not lead to the IFD is convergent stable.

Cooperation, cheating, and collapse in biological populations

Natural populations can suffer catastrophic collapse in response to small changes in environmental conditions, and recovery can be difficult even after the environment is restored to its original condition. We have used laboratory microbial ecosystems to directly measure theoretically proposed early warning signals of impending population collapse based on critical slowing down. Our experimental yeast populations cooperatively break down sugar the sugar sucrose, meaning that below a critical size the population cannot sustain itself. The cooperative nature of yeast growth on sucrose makes the population susceptible to "cheater" cells, which do not contribute to the public good and reduce the resilience of the population.

Origin and Structure of Social Networks Based on Cooperative Actions

Abstract not submitted.

Foraging games between gerbils and their predators

Abstract not submitted.

Evolutionary branching in the multivariate case

Over the last two decades evolutionary branching has emerged as a possible mathematical paradigm for explaining the origination of phenotypic diversity. Although branching is well understood for one-dimensional trait spaces, a similarly detailed understanding for higher dimensional trait spaces was still lacking. However, we recently arrived at some, surprising, first insights. In particular, we have shown that, as long as the evolutionary trajectory stays within the reign of the local quadratic approximation of the fitness function, any initial small scale polymorphism around an attracting invadable evolutionarily singular strategy (ess) will evolve towards a dimorphism. That is, if the trajectory does not pass the boundary of the domain of dimorphic coexistence and falls back to monomorphism (after which it moves again towards the singular strategy and from there on to a small scale polymorphism, etc.). To reach these results we analyzed in some detail the behaviour of the solutions of the coupled Lande-equations purportedly satisfied by the phenotypic clusters of a quasi-$n$-morphism, and give a precise characterisation of the local geometry of the set $mathcal D$ in traitspace squared harbouring protected dimorphisms. Another matter is that in higher dimensional trait spaces an attracting invadable ess needs not connect to $mathcal D$ at all. However, for the practically important subset of strongly attracting ess-es (i.e., ess-es that robustly locally attract the (quasi-)monomorphic evolutionary dynamics for all possible non-degenerate mutational (or genetic) covariance matrices) invadability implies that the ess connects to $mathcal D$, however without the guarantee that the polymorphic evolutionary trajectory will not revert to monomorphism still within the reign of the local quadratic approximation for the invasion fitnesses.

Evolutionary suicide and the adaptive dynamics of cooperation and dispersal

Abstract not submitted.

Maynard Smith & Parker's (1976) Rule Book for Animal Contests, Mostly

Maynard Smith & Parker' 1976 paper on asymmetric games offered animal behaviorists and behavioral ecologists a theoretical framework/guide to understanding animal behavior in competitive contexts. In this essay I trace the influence of this 'contest rule book' from the factors that led the two researchers to develop a treatise on the logic of the asymmetric game to empirical tests of the contest rules and theoretical additions made to the basic model and its underlying assumptions. Over a thousand studies cite this paper directly and thousands more cite work spurred by the original paper. The vast majority of these studies confirm the evolutionarily stable strategy (ESS) predictions made by Maynard Smith & Parker. Theoretical and empirical deviations from EES can largely be explained by the need for further structuring of the analyses into sub games and investigation of less obvious asymmetries than apparent size and resource value. To date, much progress has been made in three areas of interest to behaviorists: (1) understanding of the strategic nature of contests between conspecifics over limited resources; (2) modelling developments that deal with how information about potential asymmetries is gained; and (3) evaluation of the question of honest signaling with specific reference to threat displays. I propose suggestions for future work, much of which will either require collaboration with mathematicians, or require that students interested in animal behavior obtain a strong foundation in biomathematics. My preference is for the latter strategy.

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