## 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

Robert Austin
Physics, Princeton University
David Basanta
Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center
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
Ruchira Datta
Mathematical Biosciences Institute, The Ohio State University
Rick Durrett
Department of Mathematics, Duke University
Ted Galanthay
Mathematics, Ithaca College
Jeff Gore
Physics, MIT
Christopher Griffin
Applied Research Laboratory, The Pennsylvania State University
Christoph Hauert
Department of Mathematics, University of British Columbia
Burt Kotler
Mitrani Department of Desert Ecology, Ben-Gurion University
Yuan Lou
Department of Mathematics, The Ohio State University
Johan Metz
Plant Ecology and Phytochemistry, Analysis and Dynamical Systems, Institute of Biology, Mathematical Institute
Jacek Miekisz
Institute of Applied Mathematics, University of Warsaw
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
Department of Ecology and Evolutionary Biology, University of Tennessee
Sebastian Schreiber
Department of Evolution and Ecology, University of California, Davis
Barry Sinervo
Ecology and Evolutionary Biology, UC Santa Cruz
Kateřina Staňková
Department of Knowledge Engineering, Maastricht University
Jan van Gils
Marine Ecology, NIOZ Royal Netherlands Institute for Sea Research
Monday, April 27, 2015
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:15 AM

Greetings and info from MBI - Marty Golubitsky

09:15 AM
09:30 AM

Welcome and Overview - The Organizers

09:30 AM
10:15 AM
Susan Riechert - 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.

10:15 AM
10:45 AM

Break

10:45 AM
11:30 AM
Joel Brown - Plants play games too: How the tragedy of the commons explains much about the vegetation we see

Plant communities offer conspicuous displays of woody stems, masses of leaves, and often several layers of such vegetation. Plants in their quest to compete and reproduce seem to produce a lot of biomass.Plant€™ play games for nutrients (belowground) and light (aboveground). The solutions to these games result from three sources of a tragedy of the commons. First, the plants over-produce roots to pre-empt each others access to water and nitrogen. Second, the plants do the same with their leaves to pre-empt access to light.And third, the plants may invest heavily in stems because the lion€™s share of light goes to the tallest plant. We begin with a simple game of belowground root production, we can then examine how asymmetric competition for light amplifies the tragedy of the commons, and finally using a Cobb-Douglas production function we can integrate roots, leaves and stem into a single model of resource allocation in response to competition. Such models can be placed within the context of population dynamics, plant number, total plant biomass and ultimately new avenues for species coexistence. Not only does evolutionary game theory assist in understanding plants, arguable a game theoretic approach may be the only way to understand some of the most important features of plants and their communities.

11:30 AM
12:15 PM
Burt Kotler - Foraging games between gerbils and their predators

Sand dune dwelling gerbils interact with foxes, owls, and horned vipers in an environment in which resource patches renew and deplete daily. There, gerbils face tradeoffs of food and safety and must use the tools of time allocation and vigilance to manage risk. Predators must contend with gerbil behavior and manage fear using the tools of time allocation and daring. For gerbils, this means optimal patch use and optimal vigilance levels in a depleting environment over the course of the night, i.e, their harvest rates in resource patches must balance energetic, predation, and missed opportunity costs throughout the night, and their vigilance levels must balance predator encounter rate, predator lethality, and the effectiveness of vigilance and decline throughout the night as resources deplete. For predator, this means that they must choose their activity to equalize opportunity throughout the night. The consequences of these are that gerbil activity declines throughout the night in lock-step with predator activity and the apprehensiveness of the gerbils. Furthermore, a complete theory the predator-prey foraging game in gerbils needs to account for the following. 1. Foraging decisions of gerbils are responsive to their own state and that of their predators; owls are responsive only to their own state. 2. The state of a gerbil affects it foraging decisions, and it foraging decisions affect its state. This feedback is necessary to understand risk management by gerbils over a lunar cycle. 3. Gerbils enjoy safety in numbers, and gerbils show density-dependent patch use and habitat selection. This creates a 'risk pump' across habitats as gerbils carry safety with them as they alter habitat use. 4. Sight lines affect the quality of vigilance and risk management in response to different predators.

Mechanism of species coexistence with GP???

• Empirical field behavior from Kotler et al 2002

• Numbered List of experimental results a complete theory must include

• Feedback of state and behavior

• Full state. Gerbils respond to own state and that of the owls; owls respond only to own

• Temporal month, night

• Spatial including risk pump

• Sight lines

• Owls and activity

12:15 PM
02:00 PM

Lunch Break

02:00 PM
02:45 PM
Amos Bouskila - 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.

02:45 PM
03:30 PM
Bill Mitchell - Game theory of interactions among predators and groups of prey

Abstract not submitted.

03:30 PM
03:55 PM

Break

03:55 PM
04:40 PM
Jeff Gore - 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.

04:40 PM
05:15 PM

General Discussion

05:15 PM
06:45 PM

Reception and Poster Session

06:45 PM

Shuttle pick-up from MBI

Tuesday, April 28, 2015
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:45 AM
Douglas Morris - Testing games of habitat selection

All organisms use habitat so it is reasonable to assume that most, if not all, species engage in evolutionary games of habitat selection. A large variety of taxa occupy habitat in ways consistent with theory and habitat selection thus appears universal, at least for motile organisms with sensory capabilities. All organisms also consume resources, so it should be possible to test theories of habitat selection with foraging behavior. Coarse-grained field experiments confirm that invasion landscapes based on foraging behavior predict the relative abundance of meadow voles in replicated habitats. But foraging behavior in fine-grained experiments that manipulated the risks and rewards of foraging patches was not uniquely predicted by the activity-density of the voles. Harvest rates in safe versus risky patches within foraging sites mirrored visitation rates to those patches, but not at the intermediate scale where a habitat’s quality was determined by more than one site. Variation in density thus appears to dictate foraging behavior at coarse-grained scales where habitat selection is resolved through dispersal from one habitat to another. At fine-grained scales, however, variation in risk and reward appear to dictate local patch use.

09:45 AM
10:30 AM
Ted Galanthay - 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.

10:30 AM
11:00 AM

Break

11:00 AM
11:45 AM
Barry Sinervo - The rock-paper-scissors game is everywhere in nature

The rock-paper-scissors game in its pure form has each strategy losing to one strategy, while beating another. Here I show that this one population game is often modified by another population that can leverage fitness by varying present-generation choices, given the highly predictable nature of RPS cycles at future time points. For example, females should be selected to prefer rare sires to produce rare sons, given their higher fitness than common sons in the next generation of RPS competition. The action of the second population's choices can often convert the RPS game in the RPS population to an apostatic RPS game in which rare strategies are favored, but the three strategies still exhibit RPS intransitivity. I develop analytical tools for analyzing such two population interactions, and review the literature for other 3 population games. Several fish species, two birds, insects, isopods and literally hundreds of lizard species all play variations of the RPS. I also show how other two population games generate conditions whereby one population can enforce an RPS in the other population. An example of predators feeding on cryptic types, aposematic model (defend warning coloration) and aposematic mimic (undefended cheater) is shown to perhaps reflect an RPS game when viewed from the predators perspective.

11:45 AM
12:30 PM
Andrew Belmonte - Spatial patterns and interactions in public goods games

Abstract not submitted.

12:30 PM
02:00 PM

Lunch Break

02:00 PM
02:45 PM
Rick Durrett - Spatial Evolutionary Games

Evolutionary games first arose in the work of Maynard Smith and Price in the 70s, who introduced the concept into ecology in order to explain why conflicts over territory between male animals of the same species are usually of the “limited war” type and do not cause serious damage. A second important application, which involves the famousPrisoner's dilemma game, is to understand the persistence of altruistic behavior. There are many other applications, including recent work seeking to understand the competition (and cooperation) of different types of cells in cancer.

Most of the analyses of evolutionary game dynamics assume a homogeneously mixing population. However twenty years ago, Nowak and May, and Durrett and Levin showed that space could drastically change the outcome of evolutionary games, for instance allowing cooperators to persist in Prisoner's dilemma. There is now an extensive literature on spatial games, but much of it is based on heuristic principles or approximate analyses. In this talk we will explain how recent work of Cox, Durrett, and Perkins for voter model perturbations can be applied to study spatial evolutionary games in which all relative fitness are close to 1, a situation which covers many applications to cancer.

The main result is that the effect of space is equivalent to (i) changing the entries of the game matrix and (ii) replacing the replicator ODE by a related PDE. The first idea is due to Ohtsuki and Nowak (for the pair approximation) while the second is well known in the theory of stochastic spatial processes. A remarkable aspect of our result is that the limiting PDE depends on the kernel which dictates the interaction between players only through the values of two simple probabilities associated with it (an idea initially proposed by Corina Tarnita et al. Due to results of Aronson and Weinberger, and Fife and McLeod, we can analyze any 2x2 game. However, when there are three strategies the limiting object is a system of reaction diffusion equations. Many results can be derived using techniques from my AMS Memoir “Mutual Invadability implies Coexistence” but it is important open problem to understand what happens in the spatial game when the replicator dynamics show bistability.

02:45 PM
03:30 PM
Yuan Lou - ESS for dispersal in heteregeneous environments

From habitat degradation and climate change to spatial spread of invasive species, dispersal plays a central role in determining how organisms cope with a changing environment. How should organisms disperse "optimally" in heterogeneous environments? I will discuss some recent development on the evolution of dispersal, focusing on finding evolutionarily stable strategies (ESS) for dispersal. This talk is based on joint works with Steve Cantrell, Chris Cosner and Adrian Lam.

03:30 PM
03:45 PM

Break

03:45 PM
04:30 PM
Kateřina Staňková - Understanding the occurrence of cry-wolf plants in a tri-trophic system

This talk will focus on modeling a tri-trophic system consisting of plants, herbivores and predators, in which plants release herbivory-induced chemical signals betraying herbivores to their predators. In this system, so-called “cry wolf” plants occur, which produce signals even when they harbor no or only few herbivores. Initially, these cheating plants enjoy being protected even when they are not attacked, but the predators gradually learn to avoid these plants because they produce a different signal from the other, “honest” plants. This can then be followed by a change of signal of the “cry-wolf” plants, so that they are again visited by the predators.

We propose to model the system in two ways: i. a differential game of a Lotka-Volterra type with time-varying decisions of plants having a time-varying probability of being cheaters or honest, and decisions of the predators having a time-varying probability of visiting cheating plants; ii. A local dynamic game played on a finite lattice, described by the same set of equations as model i., where each cell can be inhabited by plants, herbivores, carnivores or empty spaces with certain probability.

We analyze both models and compare their predictions with field and laboratory data. We then hypothesize which elements are important for coexistence of cheating and honest plants in the system and discuss (among others) whether spatial models are necessary to explain this coexistence.

04:30 PM
05:00 PM

General Discussion

05:00 PM

Shuttle pick-up from MBI

Wednesday, April 29, 2015
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:45 AM
Johan Metz - 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.

09:45 AM
10:30 AM
Christoph Hauert - Origin and Structure of Social Networks Based on Cooperative Actions

Abstract not submitted.

10:30 AM
11:00 AM

Break

11:00 AM
11:45 AM
Mark Broom - 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.

11:45 AM
12:30 PM
Christopher Griffin - The Emergence of Stable Non-Selfish Behaviors in Evolutionary Games because of External Influences

Strict dominance and Nash equilibria of non-cooperative behaviors in classical games (e.g., Prisoner's Dilemma, Hawk-Dove, public goods etc.) challenges our observations of cooperation in the natural world. In this talk, we examine the role external influences can play on a species when its intra-species strategy choice is affected by another species also playing an intra-species game. Under simple conditions, we show that intra-species cooperation becomes evolutionarily stable. Our model is inherently simple: two species (predator and prey) each play a prisoner’s dilemma game within species. They interact with each other through a zero-sum game whose outcome affects intra-species strategic choice and generalizes the notion of a Bayesian game. We extend this study to settings with dynamic population size and more complex strategy spaces and illustrate the resulting strategy evolution. Examples outside of biology are also considered, including network bandwidth sharing.

12:30 PM
02:00 PM

Lunch Break

02:00 PM
02:45 PM
Sebastian Schreiber - Evolution and co-evolution of habitat choice in stochastic environments

Habitat selection by individuals can profoundly influence population persistence in heterogenous landscapes, stability of predator-prey interactions, and geographical shifts in species distributions in response to climate change. While there are significant and extensive advances in the evolutionary theory of habitat selection for populations living in spatially heterogeneous environments, the combined effects of temporal and spatial variation on the evolution of habitat selection is less well understood. Given the ubiquity of temporal variation and its notable impacts on demography, I will introduce a multispecies framework for studying evolutionarily stable strategies (ESSs) for habitat selection using systems of stochastic differential equations (SDEs). I will illustrate how spatial-temporal variation can select for sink populations, opposing habitat preferences for predators and their prey, and niche overlap for competing species. Part of this work is in collaboration with Steve Evans (UC Berkeley) and Alex Hening (Oxford).

02:45 PM
03:30 PM
Zhijun Wu - Evolution of Social Cliques

Species make social contacts and form social networks. The latter may have great impacts on the evolution of a population, such as preserving certain genetic features, sharing knowledge and information, preventing invasions, etc. In this talk, we show that the evolution of a population over a social network can be modeled as a symmetric evolutionary game. Its equilibrium states can therefore be obtained and analyzed by solving an optimization problem called the generalized knapsack problem. We show that an equilibrium state often corresponds to a social clique, when the population is distributed evenly on the clique. However, an equilibrium state may or may not be evolutionarily stable, whether it is on a clique or not. Only those stable ones may be observable or sustainable in nature. We analyze several different types of equilibrium states and prove a set of conditions for their stabilities. We show in particular that the equilibrium states on cliques are evolutionarily stable except for special circumstances, while non-clique equilibrium states are unstable in general. Therefore, the optimal clique strategies should have an evolutionary advantage over the non-clique ones.

03:30 PM
03:45 PM

Break

03:45 PM
04:30 PM
Kalle Parvinen - Unifying adaptive dynamics and inclusive fitness: Evolution of dispersal and cooperation in an extended Hamilton-May model

In the model of Hamilton and May (1977) there are infinitely many habitats, which can support one adult individual. In the beginning of the season, each adult gets a large number of offspring and dies. Some proportion of the offspring stay in the natal habitat, whereas others disperse randomly to other patches, unless they die during dispersal, which happens with probability p. After immigration, the individual to become adult is randomly chosen among the juveniles in the patch. The classical result of Hamilton and May (1977) is that the evolutionarily stable dispersal strategy is 1/(2 âˆ’ p).

We consider an extended version of the Hamilton-May model, so that habitats can support n individuals, and the relative fecundity of individuals depends on their behavioral strategy. Recently, Wakano and Lehmann (2014) investigated the evolution of cooperation in such a model, and used inclusive fitness arguments to give a condition when evolutionary branching can occur in the model. On the other hand, Metz and Gyllenberg (2001) have presented the concept of metapopulation reproduction ratio, which can be used to study adaptive dynamics in metapopulations. Here we will explain how this concept can be applied to the extended Hamilton-May model. Furthermore, we present the fitness gradient, and the second derivative of the metapopulation reproduction ratio with respect to a general strategy in an explicit form, so that detailed analytic investigations are possible. Earlier, the metapopulation reproduction ratio approach has mainly been applied to such complex metapopulation models, that only numerical analyses have been possible. Finally, we apply our result to the model of evolution of cooperation investigated by Wakano and Lehmann (2014), and find perfect agreement of the branching condition. We also apply our result to the evolution of dispersal, and obtain the evolutionarily stable dispersal strategy in an explicit form, which for n = 1 agrees with the classical result of Hamilton and May (1977).

Our result thus provides an unifying investigation of the inclusive fitness approach and the adaptive dynamics & metapopulation reproduction ratio approach.

04:30 PM
05:00 PM

General Discussion

05:00 PM

Shuttle pick-up from MBI

Thursday, April 30, 2015
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:45 AM
Robert Austin - Cancer and Evolutionary Game Theory

The progression of cancer in vivo comes not from mutant cancer cells growing alone in well-mixed cultures but rather from very complex and ill-understood interactions between a heterogenous cancer metapopulation, the surrounding and often inter-dispersed stromal cells, and a vasculature supplying nutrients, immune system cells and possibly exchange of cancer cells with the eternal environment. It is a staggering task to understand at a functional level these interacting populations. One of the most robust characteristics of cancer is the phenomena of dormancy: after initial surgery and/or chemotherapy, the cancer apparently ceases to grow and is said to be in remission. Unfortunately often the cancer returns after this dormant period and resumes growth, often now resistant to the initial chemotherapy that was used. We provide here using a combination of experiments in complex environments with cancer and stromal cells, and a non-linear model of interacting cells, a generalized picture of how dormancy develops and suggestions as to how dormancy could be extended.

09:45 AM
10:30 AM
David Basanta - Using game theory as a first order approximation to cancer evolution

Abstract not submitted.

10:30 AM
11:00 AM

Break

11:00 AM
11:45 AM
John Nagy - Coevolution of Cancer Hallmarks

Angiogenesis and dysregulated tissue homeostasis (proliferation) are classical, and related, characteristics of cancer--angiogenesis, the growth of new blood vessels, favors tumor expansion via cell proliferation. Both traits are thought to arise by clonal selection--essentially natural selection acting on tumor cell populations. However, angiogenesis evolution is not well understood. Here we show that angiogenic capacity in tumors is likely to evolve by selective sweeps acting on other traits, rather than direct clonal selection. We study an established mathematical model of tumor growth with angiogenesis. In the model, competing clones vary in how they partition ATP for proliferation, angiogenic signaling, and cell maintenance. Adaptive dynamics analysis predicts that selection invariably ends up favoring less angiogenic clones, which essentially act as free-riders on angiogenic clones. The result is necrosis by vascular hypoplasia, a sort of "Tumor-on-a-tumor" predicted in previous work. In contrast, simulation results from an analogous stochastic model show that these deterministic predictions are unlikely. Selection acts more strongly on proliferation than on angiogenesis, so by comparison angiogenic capability is nearly selectively neutral. Therefore, whatever angiogenic phenotype the best-fit proliferative clone expresses comes to dominate the tumor. In principle, this dominance is only temporary, since given enough time a clone with a highly adapted combination of proliferation and angiogenesis phenotypes will arise. However, in most simulations, tumors reaching lethal mass expressed poorly-adapted angiogenic phenotypes because the probability of an "optimal" combination arising prior to death is low. This result highlights the need to construct a theory of coevolution of cancer hallmarks.

11:45 AM
12:30 PM
Jan van Gils - Playing games in the mud: interactions between shorebirds and their benthic prey

Abstract not submitted.

12:30 PM
02:00 PM

Lunch Break

02:00 PM
02:45 PM
Tim Reluga - Challenges in the unification of life history theory and evolutionary game theory

In my talk, I'll review the approach to population games my collaborators and I have used to study epidemic games over the last 10 years. I'll review the basic derivations, highlight the strengths and weaknesses of the approach, and discuss some of the current mathematical challenges we face in generalizing the results obtained so far.

02:45 PM
03:30 PM
Rosalyn Rael - Multiscale Evolutionary Game Theory Modeling of Food Webs

Food webs, or ecological networks representing consumer-resource relationships, are widely used tools for studying relationships between ecosystem stability, resilience, and diversity. Combined with dynamic models, food webs have been used to understand how many types of ecological interactions play a role in shaping ecosystem structure. However, these relationships can change over longer time scales as ecosystems themselves evolve and undergo natural changes in composition as species going extinct and new ones emerge. In this talk I will present a model that combines deterministic ecological dynamics with a stochastic speciation process and discuss patterns in abundance, extinction, trait distribution, and structural changes that emerge as food webs evolve. Many of the 500 simulated food webs in this study persist over hundreds of speciation events, exhibiting varying degrees of diversity and connectance over time. During the food web evolution process, some food webs steadily increase in richness and exhibit infrequent extinction events. Others exhibit cascading extinctions over a short period, and in many cases the food webs crash entirely. One main goal of this work in progress is to characterize properties of species and their food web networks that correlate with robustness of a food web to extinction and introduction of new species. I will discuss how evolutionary game theory will be used to evaluate the evolutionary stability of species coexisting at the same trophic level in the context of a whole ecosystem, and quantify the sensitivity of ecological and evolutionary equilibria to changes in food web structure. This work will build on previous studies using evolutionary game theory models of competition and predation in small-scale food web subsets or modules consisting of only a few species. Working from the whole-ecosystem perspective gives an idea of what scenarios can emerge in the process of assembly, and allows for the investigation of how rules and evolutionary strategies that emerge in small-scale games are influenced by the ecosystem as a whole and feed back into it.

03:30 PM
05:00 PM

General Discussion / Working in groups

05:00 PM

Shuttle pick-up from MBI

06:00 PM
06:30 PM

Cash Bar at Crowne Plaza

06:30 PM
08:00 PM

Banquet at Crowne Plaza

Friday, May 1, 2015
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:45 AM
Jacek Miekisz - Strategy dependent time delays in replicator dynamics

It is usually assumed that interactions between individuals immediately affect the state of population. In reality, in biological models, results of interactions may appear in the future, and in social models, individuals or players may act, that is choose appropriate strategies, on the basis of the information concerning events in the past.

It is well known that time delays may cause oscillations in dynamical systems. We will show that the presence of oscillations in such systems depends on particular causes of time delays. In particular, we will discuss two evolutionary game models with the same payoff matrix and with a stable and unstable interior stationary point.

We modify above models to allow time delays to be strategy-dependent. They exhibit a novel behavior: after transient oscillations, the population settles at an equilibrium which depends on time delays.

We will discuss stability of stationary states in stochastic models of finite populations with time delays.

09:45 AM
10:30 AM
Vlastimil Krivan - The habitat selection game

In my talk I will discuss the Habitat selection game, a gametheoretical concept aiming to describe animal distributions in space. This concept generalizes the Ideal Free Distribution of Fretwell andLucas in several directions. For a single population, it providescharacterization under which the IFD is evolutionarily stable. I willbriefly discuss examples with the Allee type population growth, costof dispersal and some applications for optimal harvesting. Extensionsfor two species (either competing or in predator-prey relation) willbe discussed too.

10:30 AM
10:45 AM

Break

10:45 AM
11:30 AM
Sasha Dall - Variation, social information use and personality

No abstract has been provided.

11:30 AM
12:15 PM
Ruchira Datta - Emergent Node Tree Structures for Multi-scale Modeling of the Dynamics of Interacting Agents

We describe ENTs: emergent node tree structures to decompose the dynamics of systems of interacting agents. These decompose the systems into groups; however this notion of grouping is independent of cooperation. Indeed members of a group can be entirely in conflict; more generally, members of a group can have a combination of shared and competing interests. Rather, roughly speaking, players are grouped if the effect of their actions outside the group is of lower dimension than the effects of their actions on one another. This condition is less stringent than subgraph decomposition in graphical games; all players can interact with all other players, including those outside of their group. The reduction in dimensionality has the potential to support simplified solution concepts and methods, allowing us to handle the dynamics of large-scale systems. This research is in progress.

12:15 PM
12:20 PM

Closing

12:20 PM

Shuttle pick-up from MBI (One to airport and one back to hotel)

Name Email Affiliation
Austin, Robert austin@princeton.edu Physics, Princeton University
Basanta, David david@cancerevo.org Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center
Belmonte, Andrew andrew.belmonte@gmail.com Department of Mathematics, Pennsylvania State University
Bichara, Derdei derdei.bichara@asu.edu Simon A. Levin Mathematical, Computational & Modeling Sciences Center, Arizona 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
Chowell, Diego dchowel1@asu.edu Applied Mathematics, Arizona State University
Dall, Sasha S.R.X.Dall@exeter.ac.uk Centre for Ecology & Conservation, University of Exeter
Datta, Ruchira datta.53@mbi.osu.edu Mathematical Biosciences Institute, The Ohio State University
deForest, Russ russell.f.deforest@gmail.com Mathematics, Pennsylvania State University
Durrett, Rick rtd@math.duke.edu Department of Mathematics, Duke University
Galanthay, Ted tgalanthay@ithaca.edu Mathematics, Ithaca College
Ghim, Cheol-Min cmghim@unist.ac.kr School of Life Sciences, Ulsan National Institute of Science and Technology
Glover, David glover.61@osu.edu EEOB, The Ohio State University
Gore, Jeff gore@mit.edu Physics, MIT
Griffin, Christopher griffinch@ieee.org Applied Research Laboratory, The Pennsylvania State University
Halloway, Abdel abdel.halloway@gmail.com Biological sciences, University of Illinois at Chicago
Hamilton, Ian hamilton.598@osu.edu EEOB/Mathematics, The Ohio State University
Hauert, Christoph christoph.hauert@math.ubc.ca Department of Mathematics, University of British Columbia
Hellmann, Jennifer hellmann.13@osu.edu Evolution, Ecology, and Organismal Biology, The Ohio State University
Hilbe, Christian hilbe@fas.harvard.edu Program for Evolutionary Dynamics, Harvard University
Ibrahim, Mohammed moibraheem@yahoo.com Mathematics, University of Ilorin
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
Li, Wuchen lwc2017@gmail.com School of Math, Georgia tech
Li, Aming amingli2011@gmail.com Center for Complex Network Reserch, Northeastern University
Lou, Yuan lou@math.ohio-state.edu Department of Mathematics, The Ohio State University
Malicki, Maciej mamalicki@gmail.com Department of Mathematics, California Institute of Technology
Marschall, Elizabeth marschall.2@osu.edu Evolution, Ecology, and Organismal Biology, The Ohio State University
McNickle, Gordon gmcnickle@wlu.ca Biology, Wilfrid Laurier University
Metz, Johan 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
Munther, Daniel danielsmunther@gmail.com Mathematics, Cleveland State University
Nagy, John john.nagy@scottsdalecc.edu Life Sciences, Scottsdale Community College
Nova, Nicole nicole.nova@duke.edu Biology, Duke University
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 Department of Ecology and Evolutionary Biology, University of Tennessee
Robertson, Suzanne srobertson7@vcu.edu Department of Mathematics and Applied Mathematics, Virginia Commonwealth University
Schmidt, Kenneth kenneth.schmidt@ttu.edu Biological Sciences, Texas Tech University
Schreiber, Sebastian sschreiber@ucdavis.edu Department of Evolution and Ecology, University of California, Davis
Sinervo, Barry lizardrps@gmail.com Ecology and Evolutionary Biology, UC Santa Cruz
Stankova, Katerina k.stankova@maastrichtuniversity.nl Department of Knowledge Engineering, Maastricht University
Udiani, Oyita oudiani@asu.edu Simon A. Levin Mathematical Computational & Modeling Sciences Center, Arizona State University
van Gils, Jan Jan.van.Gils@nioz.nl Marine Ecology, NIOZ Royal Netherlands Institute for Sea Research
Ventura, Rafael rhv3@duke.edu Philosophy Department, Duke University
Vital, Dieff dieff.vital001@mymdc.net Engineering, Miami Dade College
Wang, Min mwang@iastate.edu Mathematics & Statistics, Iowa State University
Wang, Chi-Jen cjwang.math@gmail.com Mathematics, Georgia Institute of Technology
Wu, Zhijun zhijun@iastate.edu Math, Bioinformatics, & Computational Biology, Iowa State University
Zimmerman, Mark zimmermanmp@vcu.edu Mathematics and Applied Mathematics, Virginia Commonwealth University
Cancer and Evolutionary Game Theory

The progression of cancer in vivo comes not from mutant cancer cells growing alone in well-mixed cultures but rather from very complex and ill-understood interactions between a heterogenous cancer metapopulation, the surrounding and often inter-dispersed stromal cells, and a vasculature supplying nutrients, immune system cells and possibly exchange of cancer cells with the eternal environment. It is a staggering task to understand at a functional level these interacting populations. One of the most robust characteristics of cancer is the phenomena of dormancy: after initial surgery and/or chemotherapy, the cancer apparently ceases to grow and is said to be in remission. Unfortunately often the cancer returns after this dormant period and resumes growth, often now resistant to the initial chemotherapy that was used. We provide here using a combination of experiments in complex environments with cancer and stromal cells, and a non-linear model of interacting cells, a generalized picture of how dormancy develops and suggestions as to how dormancy could be extended.

Using game theory as a first order approximation to cancer evolution

Abstract not submitted.

Spatial patterns and interactions in public goods games

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.

Plants play games too: How the tragedy of the commons explains much about the vegetation we see

Plant communities offer conspicuous displays of woody stems, masses of leaves, and often several layers of such vegetation. Plants in their quest to compete and reproduce seem to produce a lot of biomass.Plantâ€™ play games for nutrients (belowground) and light (aboveground). The solutions to these games result from three sources of a tragedy of the commons. First, the plants over-produce roots to pre-empt each others access to water and nitrogen. Second, the plants do the same with their leaves to pre-empt access to light.And third, the plants may invest heavily in stems because the lionâ€™s share of light goes to the tallest plant. We begin with a simple game of belowground root production, we can then examine how asymmetric competition for light amplifies the tragedy of the commons, and finally using a Cobb-Douglas production function we can integrate roots, leaves and stem into a single model of resource allocation in response to competition. Such models can be placed within the context of population dynamics, plant number, total plant biomass and ultimately new avenues for species coexistence. Not only does evolutionary game theory assist in understanding plants, arguable a game theoretic approach may be the only way to understand some of the most important features of plants and their communities.

Variation, social information use and personality

No abstract has been provided.

Emergent Node Tree Structures for Multi-scale Modeling of the Dynamics of Interacting Agents

We describe ENTs: emergent node tree structures to decompose the dynamics of systems of interacting agents. These decompose the systems into groups; however this notion of grouping is independent of cooperation. Indeed members of a group can be entirely in conflict; more generally, members of a group can have a combination of shared and competing interests. Rather, roughly speaking, players are grouped if the effect of their actions outside the group is of lower dimension than the effects of their actions on one another. This condition is less stringent than subgraph decomposition in graphical games; all players can interact with all other players, including those outside of their group. The reduction in dimensionality has the potential to support simplified solution concepts and methods, allowing us to handle the dynamics of large-scale systems. This research is in progress.

Spatial Evolutionary Games

Evolutionary games first arose in the work of Maynard Smith and Price in the 70s, who introduced the concept into ecology in order to explain why conflicts over territory between male animals of the same species are usually of the “limited war” type and do not cause serious damage. A second important application, which involves the famousPrisoner's dilemma game, is to understand the persistence of altruistic behavior. There are many other applications, including recent work seeking to understand the competition (and cooperation) of different types of cells in cancer.

Most of the analyses of evolutionary game dynamics assume a homogeneously mixing population. However twenty years ago, Nowak and May, and Durrett and Levin showed that space could drastically change the outcome of evolutionary games, for instance allowing cooperators to persist in Prisoner's dilemma. There is now an extensive literature on spatial games, but much of it is based on heuristic principles or approximate analyses. In this talk we will explain how recent work of Cox, Durrett, and Perkins for voter model perturbations can be applied to study spatial evolutionary games in which all relative fitness are close to 1, a situation which covers many applications to cancer.

The main result is that the effect of space is equivalent to (i) changing the entries of the game matrix and (ii) replacing the replicator ODE by a related PDE. The first idea is due to Ohtsuki and Nowak (for the pair approximation) while the second is well known in the theory of stochastic spatial processes. A remarkable aspect of our result is that the limiting PDE depends on the kernel which dictates the interaction between players only through the values of two simple probabilities associated with it (an idea initially proposed by Corina Tarnita et al. Due to results of Aronson and Weinberger, and Fife and McLeod, we can analyze any 2x2 game. However, when there are three strategies the limiting object is a system of reaction diffusion equations. Many results can be derived using techniques from my AMS Memoir “Mutual Invadability implies Coexistence” but it is important open problem to understand what happens in the spatial game when the replicator dynamics show bistability.

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.

The Emergence of Stable Non-Selfish Behaviors in Evolutionary Games because of External Influences

Strict dominance and Nash equilibria of non-cooperative behaviors in classical games (e.g., Prisoner's Dilemma, Hawk-Dove, public goods etc.) challenges our observations of cooperation in the natural world. In this talk, we examine the role external influences can play on a species when its intra-species strategy choice is affected by another species also playing an intra-species game. Under simple conditions, we show that intra-species cooperation becomes evolutionarily stable. Our model is inherently simple: two species (predator and prey) each play a prisoner’s dilemma game within species. They interact with each other through a zero-sum game whose outcome affects intra-species strategic choice and generalizes the notion of a Bayesian game. We extend this study to settings with dynamic population size and more complex strategy spaces and illustrate the resulting strategy evolution. Examples outside of biology are also considered, including network bandwidth sharing.

Origin and Structure of Social Networks Based on Cooperative Actions

Abstract not submitted.

Foraging games between gerbils and their predators

Sand dune dwelling gerbils interact with foxes, owls, and horned vipers in an environment in which resource patches renew and deplete daily. There, gerbils face tradeoffs of food and safety and must use the tools of time allocation and vigilance to manage risk. Predators must contend with gerbil behavior and manage fear using the tools of time allocation and daring. For gerbils, this means optimal patch use and optimal vigilance levels in a depleting environment over the course of the night, i.e, their harvest rates in resource patches must balance energetic, predation, and missed opportunity costs throughout the night, and their vigilance levels must balance predator encounter rate, predator lethality, and the effectiveness of vigilance and decline throughout the night as resources deplete. For predator, this means that they must choose their activity to equalize opportunity throughout the night. The consequences of these are that gerbil activity declines throughout the night in lock-step with predator activity and the apprehensiveness of the gerbils. Furthermore, a complete theory the predator-prey foraging game in gerbils needs to account for the following. 1. Foraging decisions of gerbils are responsive to their own state and that of their predators; owls are responsive only to their own state. 2. The state of a gerbil affects it foraging decisions, and it foraging decisions affect its state. This feedback is necessary to understand risk management by gerbils over a lunar cycle. 3. Gerbils enjoy safety in numbers, and gerbils show density-dependent patch use and habitat selection. This creates a 'risk pump' across habitats as gerbils carry safety with them as they alter habitat use. 4. Sight lines affect the quality of vigilance and risk management in response to different predators.

Mechanism of species coexistence with GP???

• Empirical field behavior from Kotler et al 2002

• Numbered List of experimental results a complete theory must include

• Feedback of state and behavior

• Full state. Gerbils respond to own state and that of the owls; owls respond only to own

• Temporal month, night

• Spatial including risk pump

• Sight lines

• Owls and activity

The habitat selection game

In my talk I will discuss the Habitat selection game, a gametheoretical concept aiming to describe animal distributions in space. This concept generalizes the Ideal Free Distribution of Fretwell andLucas in several directions. For a single population, it providescharacterization under which the IFD is evolutionarily stable. I willbriefly discuss examples with the Allee type population growth, costof dispersal and some applications for optimal harvesting. Extensionsfor two species (either competing or in predator-prey relation) willbe discussed too.

ESS for dispersal in heteregeneous environments

From habitat degradation and climate change to spatial spread of invasive species, dispersal plays a central role in determining how organisms cope with a changing environment. How should organisms disperse "optimally" in heterogeneous environments? I will discuss some recent development on the evolution of dispersal, focusing on finding evolutionarily stable strategies (ESS) for dispersal. This talk is based on joint works with Steve Cantrell, Chris Cosner and Adrian Lam.

A foraging game predicts net primary productivity and the global distribution of biomes.

Plants remove CO2 from the atmosphere by photosynthesis, and as a result are a major component of the global climate system. It is therefore critical to understand controls on plant productivity, and how to model the most important processes that limit plant productivity. Most previous models focus almost entirely on physiological processes and environmental correlations, but here we argue that since plant-plant competition can reduce plant production by several orders of magnitude, ecological processes might actually be more important than physiological processes. We explore this by analyzing a simple model plant growth as an evolutionary game where plant success is a function of competition for nitrogen and access to carbon via root, stem and leaf production simultaneously. With as few as 5 parameters, accurately predicts global patterns in net primary productivity (total or root/stem/leaf individually). Furthermore, the inclusion of competitive processes allows the model can predict the distribution of the major biome types from first principals. The ability to predict the type of vegetation (e.g. tundra vs tropical forest) from first principals is one of the most powerful aspects of the model because most previous models require the vegetation type to be known and specified a priori. The model presented here is simultaneously the simplest and most accurate model of plant production to date, and we believe that the use of such a model will dramatically enhance our ability to predict and forecast future climate change.

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.

Strategy dependent time delays in replicator dynamics

It is usually assumed that interactions between individuals immediately affect the state of population. In reality, in biological models, results of interactions may appear in the future, and in social models, individuals or players may act, that is choose appropriate strategies, on the basis of the information concerning events in the past.

It is well known that time delays may cause oscillations in dynamical systems. We will show that the presence of oscillations in such systems depends on particular causes of time delays. In particular, we will discuss two evolutionary game models with the same payoff matrix and with a stable and unstable interior stationary point.

We modify above models to allow time delays to be strategy-dependent. They exhibit a novel behavior: after transient oscillations, the population settles at an equilibrium which depends on time delays.

We will discuss stability of stationary states in stochastic models of finite populations with time delays.

Game theory of interactions among predators and groups of prey

Abstract not submitted.

Testing games of habitat selection

All organisms use habitat so it is reasonable to assume that most, if not all, species engage in evolutionary games of habitat selection. A large variety of taxa occupy habitat in ways consistent with theory and habitat selection thus appears universal, at least for motile organisms with sensory capabilities. All organisms also consume resources, so it should be possible to test theories of habitat selection with foraging behavior. Coarse-grained field experiments confirm that invasion landscapes based on foraging behavior predict the relative abundance of meadow voles in replicated habitats. But foraging behavior in fine-grained experiments that manipulated the risks and rewards of foraging patches was not uniquely predicted by the activity-density of the voles. Harvest rates in safe versus risky patches within foraging sites mirrored visitation rates to those patches, but not at the intermediate scale where a habitat’s quality was determined by more than one site. Variation in density thus appears to dictate foraging behavior at coarse-grained scales where habitat selection is resolved through dispersal from one habitat to another. At fine-grained scales, however, variation in risk and reward appear to dictate local patch use.

Coevolution of Cancer Hallmarks

Angiogenesis and dysregulated tissue homeostasis (proliferation) are classical, and related, characteristics of cancer--angiogenesis, the growth of new blood vessels, favors tumor expansion via cell proliferation. Both traits are thought to arise by clonal selection--essentially natural selection acting on tumor cell populations. However, angiogenesis evolution is not well understood. Here we show that angiogenic capacity in tumors is likely to evolve by selective sweeps acting on other traits, rather than direct clonal selection. We study an established mathematical model of tumor growth with angiogenesis. In the model, competing clones vary in how they partition ATP for proliferation, angiogenic signaling, and cell maintenance. Adaptive dynamics analysis predicts that selection invariably ends up favoring less angiogenic clones, which essentially act as free-riders on angiogenic clones. The result is necrosis by vascular hypoplasia, a sort of "Tumor-on-a-tumor" predicted in previous work. In contrast, simulation results from an analogous stochastic model show that these deterministic predictions are unlikely. Selection acts more strongly on proliferation than on angiogenesis, so by comparison angiogenic capability is nearly selectively neutral. Therefore, whatever angiogenic phenotype the best-fit proliferative clone expresses comes to dominate the tumor. In principle, this dominance is only temporary, since given enough time a clone with a highly adapted combination of proliferation and angiogenesis phenotypes will arise. However, in most simulations, tumors reaching lethal mass expressed poorly-adapted angiogenic phenotypes because the probability of an "optimal" combination arising prior to death is low. This result highlights the need to construct a theory of coevolution of cancer hallmarks.

Unifying adaptive dynamics and inclusive fitness: Evolution of dispersal and cooperation in an extended Hamilton-May model

In the model of Hamilton and May (1977) there are infinitely many habitats, which can support one adult individual. In the beginning of the season, each adult gets a large number of offspring and dies. Some proportion of the offspring stay in the natal habitat, whereas others disperse randomly to other patches, unless they die during dispersal, which happens with probability p. After immigration, the individual to become adult is randomly chosen among the juveniles in the patch. The classical result of Hamilton and May (1977) is that the evolutionarily stable dispersal strategy is 1/(2 âˆ’ p).

We consider an extended version of the Hamilton-May model, so that habitats can support n individuals, and the relative fecundity of individuals depends on their behavioral strategy. Recently, Wakano and Lehmann (2014) investigated the evolution of cooperation in such a model, and used inclusive fitness arguments to give a condition when evolutionary branching can occur in the model. On the other hand, Metz and Gyllenberg (2001) have presented the concept of metapopulation reproduction ratio, which can be used to study adaptive dynamics in metapopulations. Here we will explain how this concept can be applied to the extended Hamilton-May model. Furthermore, we present the fitness gradient, and the second derivative of the metapopulation reproduction ratio with respect to a general strategy in an explicit form, so that detailed analytic investigations are possible. Earlier, the metapopulation reproduction ratio approach has mainly been applied to such complex metapopulation models, that only numerical analyses have been possible. Finally, we apply our result to the model of evolution of cooperation investigated by Wakano and Lehmann (2014), and find perfect agreement of the branching condition. We also apply our result to the evolution of dispersal, and obtain the evolutionarily stable dispersal strategy in an explicit form, which for n = 1 agrees with the classical result of Hamilton and May (1977).

Our result thus provides an unifying investigation of the inclusive fitness approach and the adaptive dynamics & metapopulation reproduction ratio approach.

Multiscale Evolutionary Game Theory Modeling of Food Webs

Food webs, or ecological networks representing consumer-resource relationships, are widely used tools for studying relationships between ecosystem stability, resilience, and diversity. Combined with dynamic models, food webs have been used to understand how many types of ecological interactions play a role in shaping ecosystem structure. However, these relationships can change over longer time scales as ecosystems themselves evolve and undergo natural changes in composition as species going extinct and new ones emerge. In this talk I will present a model that combines deterministic ecological dynamics with a stochastic speciation process and discuss patterns in abundance, extinction, trait distribution, and structural changes that emerge as food webs evolve. Many of the 500 simulated food webs in this study persist over hundreds of speciation events, exhibiting varying degrees of diversity and connectance over time. During the food web evolution process, some food webs steadily increase in richness and exhibit infrequent extinction events. Others exhibit cascading extinctions over a short period, and in many cases the food webs crash entirely. One main goal of this work in progress is to characterize properties of species and their food web networks that correlate with robustness of a food web to extinction and introduction of new species. I will discuss how evolutionary game theory will be used to evaluate the evolutionary stability of species coexisting at the same trophic level in the context of a whole ecosystem, and quantify the sensitivity of ecological and evolutionary equilibria to changes in food web structure. This work will build on previous studies using evolutionary game theory models of competition and predation in small-scale food web subsets or modules consisting of only a few species. Working from the whole-ecosystem perspective gives an idea of what scenarios can emerge in the process of assembly, and allows for the investigation of how rules and evolutionary strategies that emerge in small-scale games are influenced by the ecosystem as a whole and feed back into it.

Challenges in the unification of life history theory and evolutionary game theory

In my talk, I'll review the approach to population games my collaborators and I have used to study epidemic games over the last 10 years. I'll review the basic derivations, highlight the strengths and weaknesses of the approach, and discuss some of the current mathematical challenges we face in generalizing the results obtained so far.

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.

Evolution and co-evolution of habitat choice in stochastic environments

Habitat selection by individuals can profoundly influence population persistence in heterogenous landscapes, stability of predator-prey interactions, and geographical shifts in species distributions in response to climate change. While there are significant and extensive advances in the evolutionary theory of habitat selection for populations living in spatially heterogeneous environments, the combined effects of temporal and spatial variation on the evolution of habitat selection is less well understood. Given the ubiquity of temporal variation and its notable impacts on demography, I will introduce a multispecies framework for studying evolutionarily stable strategies (ESSs) for habitat selection using systems of stochastic differential equations (SDEs). I will illustrate how spatial-temporal variation can select for sink populations, opposing habitat preferences for predators and their prey, and niche overlap for competing species. Part of this work is in collaboration with Steve Evans (UC Berkeley) and Alex Hening (Oxford).

The rock-paper-scissors game is everywhere in nature

The rock-paper-scissors game in its pure form has each strategy losing to one strategy, while beating another. Here I show that this one population game is often modified by another population that can leverage fitness by varying present-generation choices, given the highly predictable nature of RPS cycles at future time points. For example, females should be selected to prefer rare sires to produce rare sons, given their higher fitness than common sons in the next generation of RPS competition. The action of the second population's choices can often convert the RPS game in the RPS population to an apostatic RPS game in which rare strategies are favored, but the three strategies still exhibit RPS intransitivity. I develop analytical tools for analyzing such two population interactions, and review the literature for other 3 population games. Several fish species, two birds, insects, isopods and literally hundreds of lizard species all play variations of the RPS. I also show how other two population games generate conditions whereby one population can enforce an RPS in the other population. An example of predators feeding on cryptic types, aposematic model (defend warning coloration) and aposematic mimic (undefended cheater) is shown to perhaps reflect an RPS game when viewed from the predators perspective.

Understanding the occurrence of cry-wolf plants in a tri-trophic system

This talk will focus on modeling a tri-trophic system consisting of plants, herbivores and predators, in which plants release herbivory-induced chemical signals betraying herbivores to their predators. In this system, so-called “cry wolf” plants occur, which produce signals even when they harbor no or only few herbivores. Initially, these cheating plants enjoy being protected even when they are not attacked, but the predators gradually learn to avoid these plants because they produce a different signal from the other, “honest” plants. This can then be followed by a change of signal of the “cry-wolf” plants, so that they are again visited by the predators.

We propose to model the system in two ways: i. a differential game of a Lotka-Volterra type with time-varying decisions of plants having a time-varying probability of being cheaters or honest, and decisions of the predators having a time-varying probability of visiting cheating plants; ii. A local dynamic game played on a finite lattice, described by the same set of equations as model i., where each cell can be inhabited by plants, herbivores, carnivores or empty spaces with certain probability.

We analyze both models and compare their predictions with field and laboratory data. We then hypothesize which elements are important for coexistence of cheating and honest plants in the system and discuss (among others) whether spatial models are necessary to explain this coexistence.

Playing games in the mud: interactions between shorebirds and their benthic prey

Abstract not submitted.

Evolution of Social Cliques

Species make social contacts and form social networks. The latter may have great impacts on the evolution of a population, such as preserving certain genetic features, sharing knowledge and information, preventing invasions, etc. In this talk, we show that the evolution of a population over a social network can be modeled as a symmetric evolutionary game. Its equilibrium states can therefore be obtained and analyzed by solving an optimization problem called the generalized knapsack problem. We show that an equilibrium state often corresponds to a social clique, when the population is distributed evenly on the clique. However, an equilibrium state may or may not be evolutionarily stable, whether it is on a clique or not. Only those stable ones may be observable or sustainable in nature. We analyze several different types of equilibrium states and prove a set of conditions for their stabilities. We show in particular that the equilibrium states on cliques are evolutionarily stable except for special circumstances, while non-clique equilibrium states are unstable in general. Therefore, the optimal clique strategies should have an evolutionary advantage over the non-clique ones.

Abstract not submitted.

It is usually assumed that interactions between individuals immediately affect the state of population. In reality, in biological models, results of interactions may appear in the future, and in social models, individuals or players may

No abstract has been provided.

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

Abstract not submitted.

Plant communities offer conspicuous displays of woody stems, masses of leaves, and often several layers of such vegetation. Plants in their quest to compete and reproduce seem to produce a lot of biomass.Plant€™ play games

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