Evolutionary game theory and population dynamics

In this review, we summarize recent developments in stochastic evolutionary game dynamics of finite populations.

We study effects of strategy-dependent time delays on equilibria of evolving populations. It is well known that time delays may cause oscillations in dynamical systems. Here we report a novel behavior. We show that microscopic models of evolutionary games with strategy-dependent time delays lead to a new type of replicator dynamics. It describes the time evolution of fractions of the population playing given strategies and the size of the population. Unlike in all previous mo...

To our knowledge, the populations are generally assumed to be homogeneous in the traditional approach to evolutionary game dynamics. Here, we focus on the inhomogeneous populations. A simple model which can describe the inhomogeneity of the populations and a microscopic process which is similar to Moran Process are presented. By studying the replicator dynamics, it is shown that this model also keeps the fixed points unchanged and can affect the speed of converging to the equ...

Existing theoretical models of evolution focus on the relative fitness advantages of different mutants in a population while the dynamic behavior of the population size is mostly left unconsidered. We here present a generic stochastic model which combines the growth dynamics of the population and its internal evolution. Our model thereby accounts for the fact that both evolutionary and growth dynamics are based on individual reproduction events and hence are highly coupled an...

Evolutionary game dynamics is one of the most fruitful frameworks for studying evolution in different disciplines, from Biology to Economics. Within this context, the approach of choice for many researchers is the so-called replicator equation, that describes mathematically the idea that those individuals performing better have more offspring and thus their frequency in the population grows. While very many interesting results have been obtained with this equation in the thre...

Evolutionary game dynamics with two 2-strategy games in a finite population has been investigated in this study. Traditionally, frequency-dependent evolutionary dynamics are modeled by deterministic replicator dynamics under the assumption that the population size is infinite. However, in reality, population sizes are finite. Recently, stochastic processes in finite populations have been introduced into evolutionary games in order to study finite size effects in evolutionary ...

A new mathematical model for evolutionary games on graphs is proposed to extend the classical replicator equation to finite populations of players organized on a network with generic topology. Classical results from game theory, evolutionary game theory and graph theory are used. More specifically, each player is placed in a vertex of the graph and he is seen as an infinite population of replicators which replicate within the vertex. At each time instant, a game is played by ...

Evolutionary game theory has become one of the most diverse and far reaching theories in biology. Applications of this theory range from cell dynamics to social evolution. However, many applications make it clear that inherent non-linearities of natural systems need to be taken into account. One way of introducing such non-linearities into evolutionary games is by the inclusion of multiple players. An example is of social dilemmas, where group benefits could e.g.\ increase le...

Imitation dynamics for population games are studied and their asymptotic properties analyzed. In the considered class of imitation dynamics - that encompass the replicator equation as well as other models previously considered in evolutionary biology - players have no global information about the game structure, and all they know is their own current utility and the one of fellow players contacted through pairwise interactions. For potential population games, global asymptoti...

We propose a game-theoretic dynamics of a population of replicating individuals. It consists of two parts: the standard replicator one and a migration between two different habitats. We consider symmetric two-player games with two evolutionarily stable strategies: the efficient one in which the population is in a state with a maximal payoff and the risk-dominant one where players are averse to risk. We show that for a large range of parameters of our dynamics, even if the ini...