A Numerical Approach for a General Class of the Spatial Segregation of Reaction-Diffusion Systems Arising in Population Dynamics

TL;DR

This paper develops a numerical method for solving a broad class of reaction-diffusion systems modeling spatial segregation in population dynamics, proving theoretical properties and demonstrating computational effectiveness.

Contribution

It introduces a discrete multi-phase minimization framework and an iterative algorithm for these systems, including convergence proofs and special case analysis for two populations.

Findings

Proved existence and uniqueness of the finite difference scheme.

Established convergence of the scheme to viscosity solutions in the two-population case.

Validated the approach with computational tests on various models.

Abstract

In the current work we consider the numerical solutions of equations of stationary states for a general class of the spatial segregation of reaction-diffusion systems with $m\geq 2$ population densities. We introduce a discrete multi-phase minimization problem related to the segregation problem, which allows to prove the existence and uniqueness of the corresponding finite difference scheme. Based on that scheme, we suggest an iterative algorithm and show its consistency and stability. For the special case $m=2,$ we show that the problem gives rise to the generalized version of the so-called two-phase obstacle problem. In this particular case we introduce the notion of viscosity solutions and prove convergence of the difference scheme to the unique viscosity solution. At the end of the paper we present computational tests, for different internal dynamics, and discuss numerical results.