A numerical approach for a general class of the spatial segregation of reaction–diffusion systems arising in population dynamics

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 ≥ 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.2000 Mathematics Subject Classification. 35R35, 65N06, 65N22, 92D25.

Convergence of the finite difference scheme for a general class of the spatial segregation of reaction–diffusion systems

“…In this case the proposed algorithm is lack of deep analysis, especially for the case of three and more competing populations. In the recent work by the current author in collaboration [2] the existence and uniqueness of the scheme, which solves the system (2), have been proven, provided all f i (z, s) are nonnegative and nondecreasing with respect to s. It is noteworthy, that the difference schemes with the same spirit as the system (2), have been successfully applied in quadrature domains theory (see [12]) and in optimal partitions theory (see [10]). This makes us to strongly believe that the ideas behind the difference scheme (2) have great opportunities to be applied in different problems, where the segregated geometry arise.…”

“…. , u m h ) we define the vector, which solves the finite difference system (2). Then the following statements are true:…”

“…. , u m h ) to be the difference scheme, which solves the discrete system (2). For these two elements we set M h and R h as defined above.…”

On the Uniqueness of the Solution to a Strongly Competing System

“…In this section we prove the uniqueness for the limiting case as ε tends to zero. Our approach is motivated from the recent work related to the numerical analysis of a certain class of the spatial segregation of reaction-diffusion systems (see [1]). We heavily use the following notation:…”

Uniqueness of a solution to a general class of discrete system defined on connected graphs

“…In a recent series of papers the authors consider numerical approach to a certain class of spatial segregation problem (see [2,3,5,6]). It was developed certain finite difference scheme and proved its solution existence, uniqueness and convergence.…”