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Zbl 1113.92064
Iida, Masato; Mimura, Masayasu; Ninomiya, Hirokazu
Diffusion, cross-diffusion and competitive interaction. (English)
[J] J. Math. Biol. 53, No. 4, 617-641 (2006). ISSN 0303-6812; ISSN 1432-1416/e

Summary: The cross-diffusion competition systems were introduced by {\it 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.

MSC 2000:: *92D40 Ecology
35K57 Reaction-diffusion equations
35B25 Singular perturbations (PDE)
35B35 Stability of solutions of PDE
35K55 Nonlinear parabolic equations

Keywords: Reaction-diffusion systems; Cross-diffusion systems; Turing's instability

Citations: Zbl 0537.92028; Zbl 0427.92015

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