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Diffusion, cross-diffusion and competitive interaction. (English) Zbl 1113.92064
Summary: The cross-diffusion competition systems were introduced by N. Shigesada et al. [J. Theor. Biol. 79, 83–99 (1979); see also Lect. Notes Biomath. 54, 478–491 (1984; Zbl 0537.92028); J. Math. Biol. 9, 85–96 (1980; Zbl 0427.92015)] to describe the population pressure by other species. In this paper, introducing the densities of the active individuals and the less active ones, we show that the cross-diffusion competition system can be approximated by a reaction-diffusion system which only includes a linear diffusion. The linearized stability around the constant equilibrium solution is also studied, which implies that the cross-diffusion induced instability can be regarded as Turing instability of the corresponding reaction-diffusion system.

35K57Reaction-diffusion equations
35B25Singular perturbations (PDE)
35B35Stability of solutions of PDE
35K55Nonlinear parabolic equations
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