TY - JOUR

T1 - On a finite population variation of the Fisher–KPP equation

AU - Griffin, Christopher

N1 - Publisher Copyright:
© 2023 Elsevier B.V.

PY - 2023/10

Y1 - 2023/10

N2 - 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.

AB - 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.

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U2 - 10.1016/j.cnsns.2023.107369

DO - 10.1016/j.cnsns.2023.107369

M3 - Article

AN - SCOPUS:85163866957

SN - 1007-5704

VL - 125

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

M1 - 107369

ER -