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Discrete stochastic processes, replicator and Fokker-Planck equations of coevolutionary dynamics in finite and infinite populations

Jens Christian Claussen (2008)

Banach Center Publications

Finite-size fluctuations in coevolutionary dynamics arise in models of biological as well as of social and economic systems. This brief tutorial review surveys a systematic approach starting from a stochastic process discrete both in time and state. The limit N → ∞ of an infinite population can be considered explicitly, generally leading to a replicator-type equation in zero order, and to a Fokker-Planck-type equation in first order in 1/√N. Consequences and relations to some previous approaches...

Equilibrium transitions in finite populations of players

J. Miękisz (2006)

Banach Center Publications

We discuss stochastic dynamics of finite populations of individuals playing symmetric games. We review recent results concerning the dependence of the long-run behavior of such systems on the number of players and the noise level. In the case of two-player games with two symmetric Nash equilibria, when the number of players increases, the population undergoes multiple transitions between its equilibria.

Evolutionary Games in Space

N. Kronik, Y. Cohen (2009)

Mathematical Modelling of Natural Phenomena

The G-function formalism has been widely used in the context of evolutionary games for identifying evolutionarily stable strategies (ESS). This formalism was developed for and applied to point processes. Here, we examine the G-function formalism in the settings of spatial evolutionary games and strategy dynamics, based on reaction-diffusion models. We start by extending the point process maximum principle to reaction-diffusion models with homogeneous, locally stable surfaces. We then develop...

Problems with classical models of sex-ratio evolution

Krzysztof Argasiński (2008)

Banach Center Publications

The classical theory of the sex-ratio evolution, known as the sex-ratio game, is based on the maximization of the number of grandchildren, treated as a fitness measure of a female producing offspring of the sex ratio that is coded in her genes. The theory predicts that it is more profitable to produce offspring with less numerous sex. We can find in the literature mutually exclusive conclusions based on this prediction: some textbooks say that populations with the equal number of sons and daughters...

Spreading mechanisms of cooperation for the evolutionary Prisoner's Dilemma games

György Szabó (2008)

Banach Center Publications

We survey several mechanisms supporting the maintenance of cooperation for evolutionary Prisoner's Dilemma games. In these models players are located on the sites of a lattice or graph and they can follow one of the pure strategies: cooperation (C) or defection (D). Their total income comes from Prisoner's Dilemma games with their neighbors. We discuss the consequences of different evolutionary rules determining the time-dependence of the strategy distribution and compare the results of spreading...

Stochastic stability in spatial games

Jacek Miękisz (2008)

Banach Center Publications

We compare two concepts of stochastic stability in spatial games. The classical approach to stochastic stability, introduced by Foster and Young [8], involves single configurations in the zero-noise limit. Ensemble stability discussed in [17] refers to ensembles of configurations in the limit of an infinite number of players. The above two limits may not commute. We will discuss reasons of such behaviour. We review some results concerning the effect of the number of players and the noise level on...

Theorem on signatures

Władysław Kulpa, Andrzej Szymański (2007)

Acta Universitatis Carolinae. Mathematica et Physica

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