A Velocity-diffusion Method For A Lotka-volterra System With Nonlinear Cross And Self-diffusion

摘要

The aim of this paper is to introduce a deterministic particle method for the solution of two strongly coupled reaction-diffusion equations. In these equations the diffusion is nonlinear because we consider the cross and self-diffusion effects. The reaction terms on which we focus are of the Lotka-Volterra type.rnOur treatment of the diffusion terms is a generalization of the idea, introduced in [P. Degond, F.-J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. Sci. Stat. Comput. 11 (1990) 293-310] for the linear diffusion, of interpreting Fick's law in a deterministic way as a prescription on the particle velocity. Time discretization is based on the Peaceman-Rachford operator splitting scheme.rnNumerical experiments show good agreement with the previously appeared results. We also observe travelling front solutions, the phenomenon of pattern formation and the possibility of survival for a dominated species due to a segregation effect.