Numerical Algorithms for a Variational Problem of the Spatial Segregation of Reaction-Diffusion Systems

TL;DR

This paper develops and analyzes a numerical scheme for approximating stationary states in reaction-diffusion systems with segregated supports, demonstrating convergence and applying it to ecological models with complex boundary conditions.

Contribution

The paper introduces a new numerical method for reaction-diffusion segregation problems, with convergence proofs and applications to Lotka-Volterra models under high competition.

Findings

Numerical scheme converges in specific cases.

Effective simulation of spatial segregation in ecological models.

Demonstrated applicability to high competition scenarios.

Abstract

In this paper, we study a numerical approximation for a class of stationary states for reaction-diffusion system with m densities having disjoint support, which are governed by a minimization problem. We use quantitative properties of both solutions and free boundaries to derive our scheme. Furthermore, the proof of convergence of the numerical method is given in some particular cases. We also apply our numerical simulations for the spatial segregation limit of diffusive Lotka-Volterra models in presence of high competition and inhomogeneous Dirichlet boundary conditions. We discuss numerical implementations of the resulting approach and present computational tests.