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

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.

💡 Research Summary

The paper presents a systematic tutorial on modeling coevolutionary dynamics using discrete‑time, discrete‑state stochastic processes and shows how, in the limit of infinite population size (N → ∞), these processes give rise to deterministic replicator equations at leading order and to stochastic Fokker‑Planck equations at first order in 1/√N. Starting from the foundations of evolutionary game theory, the author reviews basic concepts such as Nash equilibria, evolutionarily stable strategies (ESS), and the mapping from genotype to phenotype to fitness. The focus is on two‑strategy (2 × 2) games, with payoff matrix P =