Evolutionary game dynamics with non-uniform interaction rates

The classical setting of evolutionary game theory, the replicator equation, assumes uniform interaction rates. The rate at which individuals meet and interact is independent of their strategies. Here we extend this framework by allowing the interaction rates to depend on the strategies. This extension leads to non-linear fitness functions. We show that a strict Nash equilibrium remains uninvadable for non-uniform interaction rates, but the conditions for evolutionary stability need to be modified. We analyze all games between two strategies. If the two strategies coexist or exclude each other, then the evolutionary dynamics do not change qualitatively, only the location of the equilibrium point changes. If, however, one strategy dominates the other in the classical setting, then the introduction of non-uniform interaction rates can lead to a pair of interior equilibria. For the Prisoner's Dilemma, non-uniform interaction rates allow the coexistence between cooperators and defectors. For the snowdrift game, non-uniform interaction rates change the equilibrium frequency of cooperators.     

Evolutionary dynamics in structured populations

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called ‘spatial selection’: cooperators prevail against defectors by clustering in physical or other spaces.     

Evolutionary games and population dynamics: maintenance of cooperation in public goods games

The emergence and abundance of cooperation in nature poses a tenacious and challenging puzzle to evolutionary biology. Cooperative behaviour seems to contradict Darwinian evolution because altruistic individuals increase the fitness of other members of the population at a cost to themselves. Thus, in the absence of supporting mechanisms, cooperation should decrease and vanish, as predicted by classical models for cooperation in evolutionary game theory, such as the Prisoner's Dilemma and public goods games. Traditional approaches to studying the problem of cooperation assume constant population sizes and thus neglect the ecology of the interacting individuals. Here, we incorporate ecological dynamics into evolutionary games and reveal a new mechanism for maintaining cooperation. In public goods games, cooperation can gain a foothold if the population density depends on the average population payoff. Decreasing population densities, due to defection leading to small payoffs, results in smaller interaction group sizes in which cooperation can be favoured. This feedback between ecological dynamics and game dynamics can generate stable coexistence of cooperators and defectors in public goods games. However, this mechanism fails for pairwise Prisoner's Dilemma interactions and the population is driven to extinction. Our model represents natural extension of replicator dynamics to populations of varying densities.     

Interaction characteristics as evolutionary features for the spatial Prisoner’s Dilemma in a population modeled by continuous probabilistic cellular automata and evolutionary algorithm

Evolutionary games usually take into consideration individuals’ strategies as the transformative characteristic which leads to the evolution of the population. Here, besides the strategies, interaction aspects are also considered as evolutionary attributes which can change over time as the replacement dynamic renovates the population choosing locally better individuals to reproduce. The population is modeled by cellular automata, interactions by the Prisoner’s Dilemma game and the replacement process is ruled by two versions of death-birth dynamic. Although the average payoff per game is considered as the fitness for choosing better individuals, the number of games per time step and a maximum radius of interaction with neighbours are also present in the individual’s chromosome which is passed to the next generation. Numerical simulations show that individual interaction properties and cooperation level are linked to the version of death-birth dynamic used and the game payoff. For instance, when the fitness bias is on the death event, individuals have more interactions in a larger radius, and the cooperation level is usually lower than the case where the fitness bias is on the birth event. Also, the individuals’ interaction profiles are heterogeneous, and cooperative individuals form clusters in the lattice to protect themselves.     

Ecological public goods games: cooperation and bifurcation

The Public Goods Game is one of the most popular models for studying the origin and maintenance of cooperation. In its simplest form, this evolutionary game has two regimes: defection goes to fixation if the multiplication factor r is smaller than the interaction group size N, whereas cooperation goes to fixation if the multiplication factor r is larger than the interaction group size N. Hauert et al. [Hauert, C., Holmes, M., Doebeli, M., 2006a. Evolutionary games and population dynamics: Maintenance of cooperation in public goods games. Proc. R. Soc. Lond. B 273, 2565-2570] have introduced the Ecological Public Goods Game by viewing the payoffs from the evolutionary game as birth rates in a population dynamic model. This results in a feedback between ecological and evolutionary dynamics: if defectors are prevalent, birth rates are low and population densities decline, which leads to smaller interaction groups for the Public Goods game, and hence to dominance of cooperators, with a concomitant increase in birth rates and population densities. This feedback can lead to stable co-existence between cooperators and defectors. Here we provide a detailed analysis of the dynamics of the Ecological Public Goods Game, showing that the model exhibits various types of bifurcations, including supercritical Hopf bifurcations, which result in stable limit cycles, and hence in oscillatory co-existence of cooperators and defectors. These results show that including population dynamics in evolutionary games can have important consequences for the evolutionary dynamics of cooperation.     

Evolutionary dynamics of continuous strategy games on graphs and social networks under weak selection

Understanding the emergence of cooperation among selfish individuals has been a long-standing puzzle, which has been studied by a variety of game models. Most previous studies presumed that interactions between individuals are discrete, but it seems unrealistic in real systems. Recently, there are increasing interests in studying game models with a continuous strategy space. Existing research work on continuous strategy games mainly focuses on well-mixed populations. Especially, little theoretical work has been conducted on their evolutionary dynamics in a structured population. In the previous work (Zhong et al., BioSystems, 2012), we showed that under strong selection, continuous and discrete strategies have significantly different equilibrium and game dynamics in spatially structured populations. In this paper, we further study evolutionary dynamics of continuous strategy games under weak selection in structured populations. By using the fixation probability based stochastic dynamics, we derive exact conditions of natural selection favoring cooperation for the death–birth updating scheme. We also present a network gain decomposition of the game equilibrium, which might provide a new view of the network reciprocity in a quantitative way. Finally, we make a detailed comparison between games using discrete and continuous strategies. As compared to the former, we find that for the latter (i) the same selection conditions are derived for the general 2 × 2 game; especially, the rule b/c > k in a simplified Prisoner's Dilemma is valid as well; however, (ii) for a coordination game, interestingly, the risk-dominant strategy is disfavored. Numerical simulations have also been conducted to validate our results.     

Active linking in evolutionary games

In the traditional approach to evolutionary game theory, the individuals of a population meet each other at random, and they have no control over the frequency or duration of interactions. Here we remove these simplifying assumptions. We introduce a new model, where individuals differ in the rate at which they seek new interactions. Once a link between two individuals has formed, the productivity of this link is evaluated. Links can be broken off at different rates. In a limiting case, the linking dynamics introduces a simple transformation of the payoff matrix. We outline conditions for evolutionary stability. As a specific example, we study the interaction between cooperators and defectors. We find a simple relationship that characterizes those linking dynamics which allow natural selection to favour cooperation over defection.     

Oscillatory dynamics in evolutionary games are suppressed by heterogeneous adaptation rates of players

Game dynamics in which three or more strategies are cyclically competitive, as represented by the rock-scissors-paper game, have attracted practical and theoretical interests. In evolutionary dynamics, cyclic competition results in oscillatory dynamics of densities of individual strategists. In finite-size populations, it is known that oscillations blow up until all but one strategies are eradicated if without mutation. In the present paper, we formalize replicator dynamics with players who have different adaptation rates. We show analytically and numerically that the heterogeneous adaptation rate suppresses the oscillation amplitude. In social dilemma games with cyclically competing strategies and homogeneous adaptation rates, altruistic strategies are often relatively weak and cannot survive in finite-size populations. In such situations, heterogeneous adaptation rates save coexistence of different strategies and hence promote altruism. When one strategy dominates the others without cyclic competition, fast adaptors earn more than slow adaptors. When not, mixture of fast and slow adaptors stabilizes population dynamics, and slow adaptation does not imply inefficiency for a player.     

Selection for intermediate mortality and reproduction rates in a spatially structured population

How local interactions influence both population and evolutionary dynamics is currently a key topic in theoretical ecology. We use a 'well-mixed' analytical model and spatially explicit individual-based models to investigate a system where a population is subject to rare disturbance events. The disturbance can only propagate through regions of the population where the density of individuals is sufficiently high and individuals affected by the disturbance die shortly after. We find that populations where individuals are sessile often exhibit very different dynamic behaviour when compared to populations where individuals are mobile and spatially well mixed. When mutations are allowed which affect either offspring birth rates or mortality rates, the well-mixed populations always evolve to a state where a single disturbance event leads to extinction. Populations often persist substantially longer if individuals are sessile and they disperse their offspring locally. We also find that for sessile populations selection may favour short-lived individuals with limited offspring production. Population dynamics are found to be strongly influenced by the host characters that are evolving and the rate at which host variation is introduced into the system.     

Ecological theatre and the evolutionary game: how environmental and demographic factors determine payoffs in evolutionary games

In the standard approach to evolutionary games and replicator dynamics, differences in fitness can be interpreted as an excess from the mean Malthusian growth rate in the population. In the underlying reasoning, related to an analysis of “costs” and “benefits”, there is a silent assumption that fitness can be described in some type of units. However, in most cases these units of measure are not explicitly specified. Then the question arises: are these theories testable? How can we measure “benefit” or “cost”? A natural language, useful for describing and justifying comparisons of strategic “cost” versus “benefits”, is the terminology of demography, because the basic events that shape the outcome of natural selection are births and deaths. In this paper, we present the consequences of an explicit analysis of births and deaths in an evolutionary game theoretic framework. We will investigate different types of mortality pressures, their combinations and the possibility of trade-offs between mortality and fertility. We will show that within this new approach it is possible to model how strictly ecological factors such as density dependence and additive background fitness, which seem neutral in classical theory, can affect the outcomes of the game. We consider the example of the Hawk–Dove game, and show that when reformulated in terms of our new approach new details and new biological predictions are produced.     

Unpredictability induced by unfocused games in evolutionary game dynamics

Evolutionary game theory is a basis of replicator systems and has applications ranging from animal behavior and human language to ecosystems and other hierarchical network systems. Most studies in evolutionary game dynamics have focused on a single game, but, in many situations, we see that many games are played simultaneously. We construct a replicator equation with plural games by assuming that a reward of a player is a simple summation of the reward of each game. Even if the numbers of the strategies of the games are different, its dynamics can be described in one replicator equation. We here show that when players play several games at the same time, the fate of a single game cannot be determined without knowing the structures of the whole other games. The most absorbing fact is that even if a single game has a ESS (evolutionary stable strategy), the relative frequencies of strategies in the game does not always converge to the ESS point when other games are played simultaneously.     

Stochastic sampling of interaction partners versus deterministic payoff assignment

Evolutionary game dynamics describes how successful strategies spread in a population. In well-mixed populations, the usual assumption, e.g. underlying the replicator dynamics, is that individuals obtain a payoff from interactions with a representative sample of the population. This determines their fitness. Here, we analyze a situation in which payoffs are obtained through a single interaction, so that individuals of the same type can have different payoffs. We show analytically that for weak selection, this scenario is identical to the usual approach in which an individual interacts with the whole population. For strong selection, however, differences arise that are reflected in the fixation probabilities and lead to deviating evolutionary dynamics.     

Repeated games and direct reciprocity under active linking

Direct reciprocity relies on repeated encounters between the same two individuals. Here we examine the evolution of cooperation under direct reciprocity in dynamically structured populations. Individuals occupy the vertices of a graph, undergoing repeated interactions with their partners via the edges of the graph. Unlike the traditional approach to evolutionary game theory, where individuals meet at random and have no control over the frequency or duration of interactions, we consider a model in which individuals differ in the rate at which they seek new interactions. Moreover, once a link between two individuals has formed, the productivity of this link is evaluated. Links can be broken off at different rates. Whenever the active dynamics of links is sufficiently fast, population structure leads to a simple transformation of the payoff matrix, effectively changing the game under consideration, and hence paving the way for reciprocators to dominate defectors. We derive analytical conditions for evolutionary stability.     

Consumer-resource dynamics: quantity, quality, and allocation

Background

The dominant paradigm for modeling the complexities of interacting populations and food webs is a system of coupled ordinary differential equations in which the state of each species, population, or functional trophic group is represented by an aggregated numbers-density or biomass-density variable. Here, using the metaphysiological approach to model consumer-resource interactions, we formulate a two-state paradigm that represents each population or group in a food web in terms of both its quantity and quality.

Methodology and Principal Findings

The formulation includes an allocation function controlling the relative proportion of extracted resources to increasing quantity versus elevating quality. Since lower quality individuals senesce more rapidly than higher quality individuals, an optimal allocation proportion exists and we derive an expression for how this proportion depends on population parameters that determine the senescence rate, the per-capita mortality rate, and the effects of these rates on the dynamics of the quality variable. We demonstrate that oscillations do not arise in our model from quantity-quality interactions alone, but require consumer-resource interactions across trophic levels that can be stabilized through judicious resource allocation strategies. Analysis and simulations provide compelling arguments for the necessity of populations to evolve quality-related dynamics in the form of maternal effects, storage or other appropriate structures. They also indicate that resource allocation switching between investments in abundance versus quality provide a powerful mechanism for promoting the stability of consumer-resource interactions in seasonally forcing environments.

Conclusions/Significance

Our simulations show that physiological inefficiencies associated with this switching can be favored by selection due to the diminished exposure of inefficient consumers to strong oscillations associated with the well-known paradox of enrichment. Also our results demonstrate how allocation switching can explain observed growth patterns in experimental microbial cultures and discuss how our formulation can address questions that cannot be answered using the quantity-only paradigms that currently predominate.     

Neighbourhood size,dispersal distance and the complex dynamics of the spatial Ricker model

Amongst the most frequently made assumptions in simple population models are that individuals interact equally with every other individual and that dispersal occurs with equal likelihood to any location. This is especially true for models of a single population (as opposed to a patchy population or metapopulation). For many species of animals and probably for all plant species these assumptions are unlikely to hold true. Here one much-studied population model—the Ricker model—is reformulated such that interactions occur only between individuals located within a certain distance of each other and dispersal distance is finite. Two alternative reformulations are presented. Results demonstrate that both limiting the interaction neighbourhood and reducing dispersal distance tend to stabilise the global population dynamics, although the extent to which this occurs depends upon the reformulation used. Spatial pattern formation is a feature of the simulated population. At lower intrinsic rates of growth (r) these patterns tend to be static, while for higher r, they are dynamic. Both the stabilisation of global dynamics and spatial pattern formation are well-described features of metapopulation models. Here, similar effects are shown to occur on a single contiguous patch of habitat.     

Ideal free distributions, evolutionary games, and population dynamics in multiple-species environments

In this article, we develop population game theory, a theory that combines the dynamics of animal behavior with population dynamics. In particular, we study interaction and distribution of two species in a two-patch environment assuming that individuals behave adaptively (i.e., they maximize Darwinian fitness). Either the two species are competing for resources or they are in a predator-prey relationship. Using some recent advances in evolutionary game theory, we extend the classical ideal free distribution (IFD) concept for single species to two interacting species. We study population dynamical consequences of two-species IFD by comparing two systems: one where individuals cannot migrate between habitats and one where migration is possible. For single species, predator-prey interactions, and competing species, we show that these two types of behavior lead to the same population equilibria and corresponding species spatial distributions, provided interspecific competition is patch independent. However, if differences between patches are such that competition is patch dependent, then our predictions strongly depend on whether animals can migrate or not. In particular, we show that when species are settled at their equilibrium population densities in both habitats in the environment where migration between habitats is blocked, then the corresponding species spatial distribution need not be an IFD. Thus, when species are given the opportunity to migrate, they will redistribute to reach an IFD (e.g., under which the two species can completely segregate), and this redistribution will also influence species population equilibrial densities. Alternatively, we also show that when two species are distributed according to the IFD, the corresponding population equilibrium can be unstable.     

Bipartite graphs as models of population structures in evolutionary multiplayer games

By combining evolutionary game theory and graph theory, "games on graphs" study the evolutionary dynamics of frequency-dependent selection in population structures modeled as geographical or social networks. Networks are usually represented by means of unipartite graphs, and social interactions by two-person games such as the famous prisoner's dilemma. Unipartite graphs have also been used for modeling interactions going beyond pairwise interactions. In this paper, we argue that bipartite graphs are a better alternative to unipartite graphs for describing population structures in evolutionary multiplayer games. To illustrate this point, we make use of bipartite graphs to investigate, by means of computer simulations, the evolution of cooperation under the conventional and the distributed N-person prisoner's dilemma. We show that several implicit assumptions arising from the standard approach based on unipartite graphs (such as the definition of replacement neighborhoods, the intertwining of individual and group diversity, and the large overlap of interaction neighborhoods) can have a large impact on the resulting evolutionary dynamics. Our work provides a clear example of the importance of construction procedures in games on graphs, of the suitability of bigraphs and hypergraphs for computational modeling, and of the importance of concepts from social network analysis such as centrality, centralization and bipartite clustering for the understanding of dynamical processes occurring on networked population structures.     

THE SHORT-TERM BEHAVIORAL REACTIONS OF BOTTLENOSE DOLPHINS TO INTERACTIONS WITH BOATS IN DOUBTFUL SOUND, NEW ZEALAND

Doubtful Sound is home to one of the southernmost resident populations of bottlenose dolphins ( Tursiops sp.). This population regularly interacts with scenic cruises. During these interactions, dolphins tend to horizontally and vertically avoid vessels, especially when the behavior of these vessels is intrusive. This study aimed at understanding the behavioral reactions of individuals to these interactions that lead to the disruption of the school's behavioral state. Observing the behavioral events performed by individuals during an interaction can help define the short-term reactions elicited by the boat presence. I recorded the behavioral events performed by all individuals of focal schools. The frequency of occurrence of all events was compared depending on the presence of vessels, their behavior, and the behavioral state of the focal school. Dolphins tended to perform more side flops while interacting with powerboats, a behavior which may be used as a non-vocal communication tool. Moreover, the movement of dolphins became more erratic during interactions with all types of vessels. These effects increased when the boats were more intrusive while interacting. This study shows that the impact of interaction with boats can be minimized if the vessels respect the guidelines in place.     

Stochastic and spatial dynamics of nematode parasites in farmed ruminants

Host-parasite systems provide powerful opportunities for the study of spatial and stochastic effects in ecology; this has been particularly so for directly transmitted microparasites. Here, we construct a fully stochastic model of the population dynamics of a macroparasite system: trichostrongylid gastrointestinal nematode parasites of farmed ruminants. The model subsumes two implicit spatial effects: the host population size (the spatial extent of the interaction between hosts) and spatial heterogeneity ('clumping') in the infection process. This enables us to investigate the roles of several different processes in generating aggregated parasite distributions. The necessity for female worms to find a mate in order to reproduce leads to an Allee effect, which interacts nonlinearly with the stochastic population dynamics and leads to the counter-intuitive result that, when rare, epidemics can be more likely and more severe in small host populations. Clumping in the infection process reduces the strength of this Allee effect, but can hamper the spread of an epidemic by making infection events too rare. Heterogeneity in the hosts' response to infection has to be included in the model to generate aggregation at the level observed empirically.