On a Finite Population Variation of the Fisher-KPP Equation

TL;DR

This paper introduces a finite population variant of the Fisher-KPP equation derived from replicator dynamics, providing new closed-form solutions for travelling waves and equilibrium problems with boundary conditions.

Contribution

It formulates a novel finite population Fisher-KPP model using game-theoretic reaction terms and derives explicit travelling wave solutions, extending classical results.

Findings

Closed-form travelling wave solutions derived from sign-reversal of classic Fisher solutions

Constructed equilibrium solutions with Dirichlet and Neumann boundary conditions

Presented and numerically analyzed two conjectures on equilibrium problems

Abstract

In this paper, we formulate a finite population variation of the Fisher-KPP equation using the fact that the reaction term can be generated from the replicator dynamic using a two-player two-strategy skew-symmetric game. We use prior results from Ablowitz and Zeppetella to show that the resulting system of partial differential equations admits a travelling wave solution, and that there are closed form solutions for this travelling wave. Interestingly, the closed form solution is constructed from a sign-reversal of the known closed form solution of the classic Fisher equation. We also construct a closed form solution approximation for the corresponding equilibrium problem on a finite interval with Dirichlet and Neumann boundary conditions. Two conjectures on these corresponding equilibrium problems are presented and analysed numerically.