On a finite population variation of the Fisher-KPP equation

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.(C) 2023 Elsevier B.V. All rights reserved.

Automated summary

In this paper, we introduce a finite population variation of the Fisher-KPP equation by utilizing the replicator dynamic to generate the reaction term. Based on previous research, we demonstrate that the resulting system of partial differential equations possesses a travelling wave solution, which can be expressed in closed form. Surprisingly, this closed form solution is obtained by reversing the sign of the known closed form solution of the classic Fisher equation. Additionally, we develop an approximate closed form solution for the corresponding equilibrium problem on a finite interval with specific boundary conditions, and propose two conjectures on these equilibrium problems, which are then analyzed numerically.