The Ideal Free Distribution with travel costs.

This article studies the effect of travel costs on population distribution in a patchy environment. The Ideal Free Distribution with travel costs is defined in the article as the distribution under which it is not profitable for individuals to move, i.e., the movement between patches ceases. It is shown that depending on the travel costs between patches, the Ideal Free Distribution may be unique, there may be infinitely many possible IFDs, or no Ideal Free Distribution exists. In the latter case, animal distribution can converge to an equilibrium of distributional dynamics at which individuals do disperse, but the net movement between patches ceases. Such distributional equilibrium corresponds to balanced dispersal.

The asymmetric Hawk-Dove game with costs measured as time lost.

The classic Hawk-Dove game is a symmetric game in that it does not distinguish between the winners and losers of Hawk-Hawk or Dove-Dove contests. Either of the two interacting Hawks or the two interacting Doves have the same probability to win/lose the contest. In addition, all pairwise interactions take the same time and after disbanding, the individuals pair instantaneously again. This article develops an asymmetric version of the Hawk-Dove model where all costs are measured by the time lost. These times are strategy dependent and measure the length of the conflict and, when a fight occurs between two interacting Hawks, the time an individual needs to recover and pair again. These recovery times depend on whether the Hawk won or lost the contest so that we consider an asymmetric Hawk-Dove game where we distinguish between winners and losers. However, the payoff matrix for this game does not correspond to the standard bimatrix game, because some entries are undefined. To calculate strategy payoffs we consider not only costs and benefits obtained from pairwise contests but also costs when individuals are disbanded. Depending on the interacting and recovery times, the evolutionary outcomes are: Hawk only, both Hawk and Dove, and a mixed strategy. This shows that measuring the cost in time lost leads to a new prediction since, in the classic (symmetric) Hawk-Dove model that does assume positive cost (C>0), both Hawk and Dove strategy is never an evolutionary outcome.

Reducing courtship time promotes marital bliss: The Battle of the Sexes game revisited with costs measured as time lost.

Classic bimatrix games, that are based on pair-wise interactions between two opponents belonging to different populations, do not consider the cost of time. In this article, we build on an old idea that lost opportunity costs affect individual fitness. We calculate fitnesses of each strategy for a two-strategy bimatrix game at the equilibrium distribution of the pair formation process that includes activity times. This general approach is then applied to the Battle of the Sexes game where we analyze the evolutionary outcome by finding the Nash equilibria (NE) of this time-constrained game when courtship and child rearing costs are measured by time lost. While the classic Battle of the Sexes game has either a unique strict NE (specifically, all males exhibit Philanderer behavior and either all females are Coy or all are Fast depending on model parameters), or a unique interior NE where both sexes exhibit mixed behavior, including time costs for courtship and child rearing changes this prediction. First, (Philanderer, Coy) is never a NE. Second, if the benefit of having offspring is independent of parental strategies, (Philanderer, Fast) is the unique strict NE but a second stable interior NE emerges when courtship time is sufficiently short. In fact, as courtship time becomes shorter, this mixed NE (where most males are Faithful and the Coy female population is increasing) attracts almost all initial population configurations. Third, this latter promotion of marital bliss also occurs when parents who share in child rearing receive a higher benefit from their offspring than those that don't. Finally, for courtship time of moderate duration, the same phenomenon occurs when the population size increases.

When optimal foragers meet in a game theoretical conflict: A model of kleptoparasitism.

Kleptoparasitism can be considered as a game theoretical problem and a foraging tactic at the same time, so the aim of this paper is to combine the basic ideas of two research lines: evolutionary game theory and optimal foraging theory. To unify these theories, firstly, we take into account the fact that kleptoparasitism between foragers has two consequences: the interaction takes time and affects the net energy intake of both contestants. This phenomenon is modeled by a matrix game under time constraints. Secondly, we also give freedom to each forager to avoid interactions, since in optimal foraging theory foragers can ignore each food type (we have two prey types: either a prey item in possession of another predator or a free prey individual is discovered). The main question of the present paper is whether the zero-one rule of optimal foraging theory (always or never select a prey type) is valid or not, in the case where foragers interact with each other? In our foraging game we consider predators who engage in contests (contestants) and those who never do (avoiders), and in general those who play a mixture of the two strategies. Here the classical zero-one rule does not hold. Firstly, the pure avoider phenotype is never an ESS. Secondly, the pure contestant can be a strict ESS, but we show this is not necessarily so. Thirdly, we give an example when there is mixed ESS.

Emergence of communities and diversity in social networks.

Communities are common in complex networks and play a significant role in the functioning of social, biological, economic, and technological systems. Despite widespread interest in detecting community structures in complex networks and exploring the effect of communities on collective dynamics, a deep understanding of the emergence and prevalence of communities in social networks is still lacking. Addressing this fundamental problem is of paramount importance in understanding, predicting, and controlling a variety of collective behaviors in society. An elusive question is how communities with common internal properties arise in social networks with great individual diversity. Here, we answer this question using the ultimatum game, which has been a paradigm for characterizing altruism and fairness. We experimentally show that stable local communities with different internal agreements emerge spontaneously and induce social diversity into networks, which is in sharp contrast to populations with random interactions. Diverse communities and social norms come from the interaction between responders with inherent heterogeneous demands and rational proposers via local connections, where the former eventually become the community leaders. This result indicates that networks are significant in the emergence and stabilization of communities and social diversity. Our experimental results also provide valuable information about strategies for developing network models and theories of evolutionary games and social dynamics.

Bimatrix games that include interaction times alter the evolutionary outcome: The Owner-Intruder game.

Classic bimatrix games, that are based on pair-wise interactions between two opponents in two different roles, do not consider the effect that interaction duration has on payoffs. However, interactions between different strategies often take different amounts of time. In this article, we further develop a new approach to an old idea that opportunity costs lost while engaged in an interaction affect individual fitness. We consider two scenarios: (i) individuals pair instantaneously so that there are no searchers, and (ii) searching for a partner takes positive time and populations consist of a mixture of singles and pairs. We describe pair dynamics and calculate fitnesses of each strategy for a two-strategy bimatrix game that includes interaction times. Assuming that distribution of pairs (and singles) evolves on a faster time scale than evolutionary dynamics described by the replicator equation, we analyze the Nash equilibria (NE) of the time-constrained game. This general approach is then applied to the Owner-Intruder bimatrix game where the two strategies are Hawk and Dove in both roles. While the classic Owner-Intruder game has at most one interior NE and it is unstable with respect to replicator dynamics, differences in pair duration change this prediction in that up to four interior NE may exist with their stability depending on whether pairing is instantaneous or not. The classic game has either one (all Hawk) or two ((Hawk,Dove) and (Dove,Hawk)) stable boundary NE. When interaction times are included, other combinations of stable boundary NE are possible. For example, (Dove,Dove), (Dove,Hawk), or (Hawk,Dove) can be the unique (stable) NE if interaction time between two Doves is short compared to some other interactions involving Doves.

Revisiting the "fallacy of averages" in ecology: Expected gain per unit time equals expected gain divided by expected time.

Fitness is often defined as the average payoff an animal obtains when it is engaged in several activities, each taking some time. We point out that the average can be calculated with respect to either the time distribution, or to the event distribution of these activities. We show that these two averages lead to the same fitness function. We illustrate this result through two examples from foraging theory, Holling II functional response and the diet choice model, and one game-theoretic example of Hamilton's rule applied to the time-constrained Prisoner's dilemma (PD). In particular, we show that in these models, fitness defined as expected gain per unit time equals fitness defined as expected gain divided by expected time. We also show how these fitnesses predict the optimal outcome for diet choice and the prevalence of cooperation in the repeated PD game.

Beyond replicator dynamics: From frequency to density dependent models of evolutionary games.

Game theoretic models of evolution such as the Hawk-Dove game assume that individuals gain fitness (which is a proxy of the per capita population growth rate) in pair-wise contests only. These models assume that the equilibrium distribution of phenotypes involved (e.g., Hawks and Doves) in the population is given by the Hardy-Weinberg law, which is based on instantaneous, random pair formation. On the other hand, models of population dynamics do not consider pairs, newborns are produced by singles, and interactions between phenotypes or species are described by the mass action principle. This article links game theoretic and population approaches. It shows that combining distribution dynamics with population dynamics can lead to stable coexistence of Hawk and Dove population numbers in models that do not assume a priori that fitness is negative density dependent. Our analysis shows clearly that the interior Nash equilibrium of the Hawk and Dove model depends both on population size and on interaction times between different phenotypes in the population. This raises the question of the applicability of classic evolutionary game theory that requires all interactions take the same amount of time and that all single individuals have the same payoff per unit of time, to real populations. Furthermore, by separating individual fitness into birth and death effects on singles and pairs, it is shown that stable coexistence in these models depends on the time-scale of the distribution dynamics relative to the population dynamics. When explicit density-dependent fitness is included through competition over a limited resource, the combined dynamics of the Hawk-Dove model often lead to Dove extinction no matter how costly fighting is for Hawk pairs.

The ESS and replicator equation in matrix games under time constraints.

Recently, we introduced the class of matrix games under time constraints and characterized the concept of (monomorphic) evolutionarily stable strategy (ESS) in them. We are now interested in how the ESS is related to the existence and stability of equilibria for polymorphic populations. We point out that, although the ESS may no longer be a polymorphic equilibrium, there is a connection between them. Specifically, the polymorphic state at which the average strategy of the active individuals in the population is equal to the ESS is an equilibrium of the polymorphic model. Moreover, in the case when there are only two pure strategies, a polymorphic equilibrium is locally asymptotically stable under the replicator equation for the pure-strategy polymorphic model if and only if it corresponds to an ESS. Finally, we prove that a strict Nash equilibrium is a pure-strategy ESS that is a locally asymptotically stable equilibrium of the replicator equation in n-strategy time-constrained matrix games.

Unlimited niche packing in a Lotka-Volterra competition game.

A central question in the study of ecology and evolution is: "Why are there so many species?" It has been shown that certain forms of the Lotka-Volterra (L-V) competition equations lead to an unlimited number of species. Furthermore, these authors note how any change in the nature of competition (the competition kernel) leads to a finite or small number of coexisting species. Here we build upon these works by further investigating the L-V model of unlimited niche packing as a reference model and evolutionary game for understanding the environmental factors restricting biodiversity. We also examine the combined eco-evolutionary dynamics leading up to the species diversity and traits of the ESS community in both unlimited and finite niche-packing versions of the model. As an L-V game with symmetric competition, we let the strategies of individuals determine the strength of the competitive interaction (like competes most with like) and also the carrying capacity of the population. We use a mixture of analytic proofs (for one and two species systems) and numerical simulations. For the model of unlimited niche packing, we show that a finite number of species will evolve to specific convergent stable minima of the adaptive landscape (also known as species archetypes). Starting with a single species, faunal buildup can proceed either through species doubling as each diversity-specific set of minima are reached, or through the addition of species one-by-one by randomly assigning a speciation event to one of the species. Either way it is possible for an unlimited number or species to evolve and coexist. We examine two simple and biologically likely ways for breaking the unlimited niche-packing: (1) some minimum level of competition among species, and (2) constrain the fundamental niche of the trait space to a finite interval. When examined under both ecological and evolutionary dynamics, both modifications result in convergent stable ESSs with a finite number of species. When the number of species is held below the number of species in an ESS coalition, we see a diverse array of convergent stable niche archetypes that consist of some species at maxima and some at minima of the adaptive landscape. Our results support those of others and suggest that instead of focusing on why there are so many species we might just as usefully ask, why are there so few species?

Interaction times change evolutionary outcomes: Two-player matrix games.

Two most influential models of evolutionary game theory are the Hawk-Dove and Prisoner's dilemma models. The Hawk-Dove model explains evolution of aggressiveness, predicting individuals should be aggressive when the cost of fighting is lower than its benefit. As the cost of aggressiveness increases and outweighs benefits, aggressiveness in the population should decrease. Similarly, the Prisoner's dilemma models evolution of cooperation. It predicts that individuals should never cooperate despite cooperation leading to a higher collective fitness than defection. The question is then what are the conditions under which cooperation evolves? These classic matrix games, which are based on pair-wise interactions between two opponents with player payoffs given in matrix form, do not consider the effect that conflict duration has on payoffs. However, interactions between different strategies often take different amounts of time. In this article, we develop a new approach to an old idea that opportunity costs lost while engaged in an interaction affect individual fitness. When applied to the Hawk-Dove and Prisoner's dilemma, our theory that incorporates general interaction times leads to qualitatively different predictions. In particular, not all individuals will behave as Hawks when fighting cost is lower than benefit, and cooperation will evolve in the Prisoner's dilemma.

Homophilic replicator equations.

Tags are conspicuous attributes of organisms that affect the behaviour of other organisms toward the holder, and have previously been used to explore group formation and altruism. Homophilic imitation, a form of tag-based selection, occurs when organisms imitate those with similar tags. Here we further explore the use of tag-based selection by developing homophilic replicator equations to model homophilic imitation dynamics. We assume that replicators have both tags (sometimes called traits) and strategies. Fitnesses are determined by the strategy profile of the population, and imitation is based upon the strategy profile, fitness differences, and similarity in tag space. We show the characteristics of resulting fixed manifolds and conditions for stability. We discuss the phenomenon of coat-tailing (where tags associated with successful strategies increase in abundance, even though the tags are not inherently beneficial) and its implications for population diversity. We extend our model to incorporate recurrent mutations and invasions to explore their implications upon tag and strategy diversity. We find that homophilic imitation based upon tags significantly affects the diversity of the population, although not the ESS. We classify two different types of invasion scenarios by the strategy and tag compositions of the invaders and invaded. In one scenario, we find that novel tags introduced by invaders become more readily established with homophilic imitation than without it. In the other, diversity decreases. Lastly, we find a negative correlation between homophily and the rate of convergence.

The replicator equation and other game dynamics.

The replicator equation is the first and most important game dynamics studied in connection with evolutionary game theory. It was originally developed for symmetric games with finitely many strategies. Properties of these dynamics are briefly summarized for this case, including the convergence to and stability of the Nash equilibria and evolutionarily stable strategies. The theory is then extended to other game dynamics for symmetric games (e.g., the best response dynamics and adaptive dynamics) and illustrated by examples taken from the literature. It is also extended to multiplayer, population, and asymmetric games.

Optimal forager against ideal free distributed prey.

The introduced dispersal-foraging game is a combination of prey habitat selection between two patch types and optimal-foraging approaches. Prey's patch preference and forager behavior determine the prey's survival rate. The forager's energy gain depends on local prey density in both types of exhaustible patches and on leaving time. We introduce two game-solution concepts. The static solution combines the ideal free distribution of the prey with optimal-foraging theory. The dynamical solution is given by a game dynamics describing the behavioral changes of prey and forager. We show (1) that each stable equilibrium dynamical solution is always a static solution, but not conversely; (2) that at an equilibrium dynamical solution, the forager can stabilize prey mixed patch use strategy in cases where ideal free distribution theory predicts that prey will use only one patch type; and (3) that when the equilibrium dynamical solution is unstable at fixed prey density, stable behavior cycles occur where neither forager nor prey keep a fixed behavior.

Two-patch population models with adaptive dispersal: the effects of varying dispersal speeds.

The population-dispersal dynamics for predator-prey interactions and two competing species in a two patch environment are studied. It is assumed that both species (i.e., either predators and their prey, or the two competing species) are mobile and their dispersal between patches is directed to the higher fitness patch. It is proved that such dispersal, irrespectively of its speed, cannot destabilize a locally stable predator-prey population equilibrium that corresponds to no movement at all. In the case of two competing species, dispersal can destabilize population equilibrium. Conditions are given when this cannot happen, including the case of identical patches.

Cooperation and evolutionary dynamics in the public goods game with institutional incentives.

The one-shot public goods game is extended to include institutional incentives (i.e. reward and/or punishment) that are meant to promote cooperation. It is shown that the Nash equilibrium (NE) outcomes predict either partial or fully cooperative behavior in these extended multi-player games with a continuous strategy space. Furthermore, for some incentive schemes, multiple NE outcomes are shown to emerge. Stability of all these equilibria under standard evolutionary dynamics (i.e. the replicator equation and the canonical equation of adaptive dynamics) is characterized.

Costly punishment does not always increase cooperation.

In a pairwise interaction, an individual who uses costly punishment must pay a cost in order that the opponent incurs a cost. It has been argued that individuals will behave more cooperatively if they know that their opponent has the option of using costly punishment. We examined this hypothesis by conducting two repeated two-player Prisoner's Dilemma experiments, that differed in their payoffs associated to cooperation, with university students from Beijing as participants. In these experiments, the level of cooperation either stayed the same or actually decreased when compared with the control experiments in which costly punishment was not an option. We argue that this result is likely due to differences in cultural attitudes to cooperation and punishment based on similar experiments with university students from Boston that found cooperation did increase with costly punishment.

The effects of opportunistic and intentional predators on the herding behavior of prey.

In this article, we study how predator behavior influences the aggregation of prey into herds. Game-theoretic models of herd formation are developed based on different survival probabilities of solitary prey and prey that join the herd and on the predator's preference of what type of prey to search for. For an intentional predator that will only pursue its preferred type of prey, a single herd with no solitaries cannot emerge unless the herd acts as a prey refuge. If neither prey choice provides a refuge, it is shown that an equilibrium always exists where there are both types of prey and the predator does not always search for the same type of prey (i.e., a mixed equilibrium exists). On the other hand, if the predator is opportunistic in that it sometimes shifts to pursue the type of prey that is observed first, there may be a single herd equilibrium that does not act as a prey refuge when there is a high level of opportunistic behavior. For low opportunistic levels, a mixed equilibrium is again the only outcome. The evolutionary stability of each equilibrium is tested to see if it predicts the eventual herding behavior of prey in its corresponding model. Our analysis confirms that both predator and prey preferences (for herd or solitary) have strong effects on why prey aggregate. In particular, in our models, only the opportunistic predator can maintain all prey in a single herd that is under predation risk.

Multilayer network structure enhances the coexistence of competitive species.

The concept of a multiplex network can be used to characterize the dispersal paths and states of different species in a patch habitat system. The multiplex network is one of three types of multilayer networks. In this study, the effect of a multiplex network on the long-term stable coexistence of species is investigated using the concept of metapopulation. Based on the mean field approximation, the stability analysis of a two-species system shows that, compared to the single layer network, the multiplex network is more conducive to the stable coexistence of species when one species has a stronger colonization ability. That is, in such a patch habitat system, if the dispersal paths of the stronger species are different than those of the weaker species, then the larger the heterogeneity of the dispersal network of the stronger species is, the more likely the long-term stable coexistence of species. This result provides a different perspective for understanding the biodiversity in heterogeneous habitats.

CSS, NIS and dynamic stability for two-species behavioral models with continuous trait spaces.

Static continuously stable strategy (CSS) and neighborhood invader strategy (NIS) conditions are developed for two-species models of frequency-dependent behavioral evolution when individuals have traits in continuous strategy spaces. These are intuitive stability conditions that predict the eventual outcome of evolution from a dynamic perspective. It is shown how the CSS is related to convergence stability for the canonical equation of adaptive dynamics and the NIS to convergence to a monomorphism for the replicator equation of evolutionary game theory. The CSS and NIS are also shown to be special cases of neighborhood p(*)- superiority for p(*) equal to one half and zero, respectively. The theory is illustrated when each species has a one-dimensional trait space.