From discrete to continuous evolution models: a unifying approach to drift-diffusion and replicator dynamics

Abstract

We study the large population limit of the Moran process, assuming weak-selection, and for different scalings. Depending on the particular choice of scalings, we obtain a continuous model that may highlight the genetic-drift (neutral evolution) or natural selection; for one precise scaling, both effects are present. For the scalings that take the genetic-drift into account, the continuous model is given by a singular diffusion equation, together with two conservation laws that are already present at the discrete level. For scalings that take into account only natural selection, we obtain a hyperbolic singular equation that embeds the Replicator Dynamics and satisfies only one conservation law. The derivation is made in two steps: a formal one, where the candidate limit model is obtained, and a rigorous one, where convergence of the probability density is proved. Additional results on the fixation probabilities are also presented.

18 pages, 3 figures

References in corpus (4)

• Stochastic Dynamics of Invasion and Fixation
• The long-run behavior of the stochastic replicator dynamics
• Discrete versus continuous models in evolutionary dynamics: from simple to simpler -- and even simpler -- models
• The continuous limit of the Moran process and the diffusion of mutant genes in infinite populations

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• On the stochastic evolution of finite populations
• Fixation probabilities for the Moran process in evolutionary games with two strategies: graph shapes and large population asymptotics
• Fitness potentials and qualitative properties of the Wright-Fisher dynamics
• Predicting population extinction in lattice-based birth-death-movement models
• Well posedness and Maximum Entropy Approximation for the Dynamics of Quantitative Traits
• Mean extinction times in cyclic coevolutionary rock-paper-scissors dynamics
• Insuperable strategies in two-player and reducible multi-player games
• Finitary Process Evolution I: Information Geometry of Configuration Space and the Process-Replicator Dynamics