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Abstract: In this work we investigate the process of pattern formation in a two dimensional domain for a reaction–diffusion system with nonlinear diffusion terms and the competitive Lotka–Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, and hexagonal patterns. [Copyright &y& Elsevier]

Abstract: 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. Our 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. Numerical 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. [Copyright &y& Elsevier]