A reaction-diffusion system with nonconstant diffusion coefficients: exact and numerical solutions

Abstract

A Lotka-Volterra type system with porous diffusion, which can be used as an alternative model to the classical Lotka-Volterra system, is under study. Multiparameter families of exact solutions of the system in question are constructed and their properties are established. It is shown that the solutions obtained can satisfy the zero Neumann conditions, which are typical conditions for mathematical models describing real-world processes. It is proved that the system possesses two stable steady-state points provided its coefficients are correctly-specified. In particular, this occurs when the system models the prey-predator interaction. The exact solutions are used for solving boundary-value problems. The analytical results are compared with numerical solutions of the same boundary-value problems but perturbed initial profiles. It is demonstrated that the numerical solutions coincide with the relevant exact solutions with high exactness in the case of sufficiently small perturbations of the initial profiles.