On convergence of numerical algorithm of a class of the spatial segregation of reaction-diffusion system with two population densities

Recently, much interest has gained the numerical approximation of equations of the Spatial Segregation of Reaction-diffusion systems with m population densities. These problems are governed by a minimization problem subject to the 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 system with two population densities having disjoint support. We prove the convergence of the numerical algorithm for two competing populations with non-negative internal dynamics $f_i(x)\geq 0.$ At the end of the paper we present computational tests.