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

TL;DR

This paper reviews a systematic approach to modeling coevolutionary dynamics using stochastic processes, deriving replicator and Fokker-Planck equations for finite and infinite populations, with implications for biological, social, and economic systems.

Contribution

It introduces a discrete stochastic process framework for coevolutionary dynamics, connecting finite and infinite population models through explicit derivations of replicator and Fokker-Planck equations.

Findings

Derivation of replicator equations from stochastic processes

Derivation of Fokker-Planck equations as first-order corrections

Clarification of the relation between finite and infinite population models

Abstract

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 $N\to \infty$ 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 $1/\sqrt{N}$. Consequences and relations to some previous approaches are outlined.