On the Local Stability of Constant Equilibria and a Meshless Numerical Method for a System with Intraspecific and Interspecific Competition

In this paper we study a novel mathematical model with intraspecific and interspecific competition between two species consisting of a non-linear parabolicODE-parabolic system. It describes the evolution of two populations in competition for a resource, one of which is subject to chemotaxis. We analyze the local stability of the constant equilibrium solutions and we obtain the periodic behavior of the solution for certain data of the problem. For this purpose, we apply the meshless numerical method of Generalized Finite Differences (GFDM) and we prove the conditional convergence of the discrete solution to the analytical one. The conditional convergence of the numerical method is demostrated and, thought its implementation, we obtain numerical solutions whose asymptotic behavior agrees with the analytically one expected. We give several numerical examples on the applications of this meshless method over regular and irregular domains to illustrate its potential.