Convergence Analysis of a Numerical Algorithm for Spatial Segregation of Two Population Densities

Abstract

Over the last few decades, the numerical approximation of spatial segregation equations for reaction-diffusion systems with $m$ population densities has gained significant interest. These problems are governed by a minimization problem subject to a closed but non-convex set. In the present work, we deal with the numerical approximation of equations of stationary states for a certain class of the spatial segregation of reaction-diffusion systems with two population densities having disjoint support. We prove the convergence of the numerical algorithm for two competing populations with non-negative internal dynamics $f_i(x)\ge 0$. At the end of the paper, we present computational tests.

arXiv comment16 pages, 8 figures, Keywords: Free boundary, Two-phase membrane problem, Reaction-diffusion systems, Finite difference