Exact solutions for logistic reaction-diffusion in biology

TL;DR

This paper introduces a solution technique for nonlinear reaction-diffusion equations in multiple dimensions, revealing exponential time behavior and linear spatial solutions, with applications to biological modeling.

Contribution

It develops a nonclassical symmetry method to find exact solutions for nonlinear reaction-diffusion equations in N-dimensions, linking diffusion and reaction terms.

Findings

Solutions are exponential in time, either growth or decay.

Spatial solutions satisfy the linear Helmholtz equation.

Examples provided for quadratic and cubic reactions in two dimensions.

Abstract

Reaction-diffusion equations with a nonlinear source have been widely used to model various systems, with particular application to biology. Here, we provide a solution technique for these types of equations in $N$-dimensions. The nonclassical symmetry method leads to a single relationship between the nonlinear diffusion coefficient and the nonlinear reaction term; the subsequent solutions for the Kirchhoff variable are exponential in time (either growth or decay) and satisfy the linear Helmholtz equation in space. Example solutions are given in two dimensions for particular parameter sets for both quadratic and cubic reaction terms.