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

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.