Symmetries and exact solutions of a reaction-diffusion system arising in population dynamics

TL;DR

This paper analyzes a reaction-diffusion system from population dynamics, identifying symmetries and constructing new exact solutions, including those beyond classical symmetries, with applications and a general algorithm for finding Q-conditional symmetries.

Contribution

It introduces a comprehensive method for finding Lie and Q-conditional symmetries in nonlinear systems, leading to novel exact solutions and a practical algorithm for researchers.

Findings

Identified all Lie and Q-conditional symmetries for the system.

Constructed new exact solutions, including Lambert function solutions.

Presented a general algorithm for Q-conditional symmetry detection.

Abstract

A system of two cubic reaction-diffusion equations for two independent gene frequencies arising in population dynamics is studied. Depending on values of coefficients, all possible Lie and $Q$-conditional (nonclassical) symmetries are identified. A wide range of new exact solutions is constructed, including those expressible in terms of a Lambert function and not obtainable by Lie symmetries. An example of a new real-world application of the system is discussed. A general algorithm for finding Q-conditional symmetries of nonlinear evolution systems of the most general form is presented in a useful form for other researchers.