On study of transition fronts of Fisher-KPP type reaction-diffusion PDEs by non-linear transformations into exactly solvable class

TL;DR

This paper introduces a nonlinear transformation method that converts Fisher-KPP reaction-diffusion PDEs into an exactly solvable class, enabling precise analysis of wave propagation and population dynamics.

Contribution

The authors develop a novel nonlinear transformation technique that preserves solution behavior and allows exact solutions for Fisher-KPP PDEs, advancing analytical understanding of reaction-diffusion systems.

Findings

Derived exact solutions for Fisher-KPP PDEs using the transformation.

Calculated wave front velocity and shape for various initial conditions.

Verified conjectures and analyzed relaxation behavior of solutions.

Abstract

Spatio-temporal dynamics of the evolution of population involving growth and diffusion processes can be modeled by class of partial diffusion equations (PDEs) known as reaction-diffusion systems. In this work, we developed a nonlinear transformations method that converts the original nonlinear Fisher-KPP class of PDEs into an exactly solvable class. We then demonstrated that the proposed nonlinear transformation method intrinsically preserves the relaxation behavior of the solutions to asymptotic values of the non-linear dynamical system. We also show that these particular transforms are very amenable to yield an exact closed form solution in terms of the heat kernel and analytical approximations through the two variable Hermite polynomials. With this proposed method, we calculated the front velocity and shape of the propagating wave and showed how the non-linear transformation affects these parameters for both short and long epochs. As applications, we focus on solving pertinent cases of the Fisher-KPP type of PDEs relating to the evolutionary dynamics by assigning fitness to the mutant gene according to zygosity conditions. We calculated the relaxation of velocity with the parameters of the initial conditions in the following cases, namely, the Fisher, the heterozygote inferior fitness, the heterozygote superior fitness, and finally a general nonlinearity case. We also verified previous conjectures through the exact solutions computed using the proposed method.