Convergence and numerical simulations of prey-predator interactions via a meshless method

Abstract

We study two mathematical models consisting of nonlinear systems of partial differential equations, a predator prey and a competitive two-species chemotaxis systems with two chemicals satisfying their corresponding elliptic equations in a smooth bounded domain. By introducing global factors, for different ranges of parameters and by deriving a discretization of the system by means of the Generalized Finite Difference Method (GFDM) we prove that any positive and bounded discrete solution converges to the analytical one, i.e., a spatially homogeneous state. We apply the meshless method over regular and irregular domains where we simulate the behavior of the solution with the tools of several numerical examples.