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Stationary patterns caused by cross-diffusion for a three-species prey-predator model. (English) Zbl 1121.92069
Summary: We study a system arising from a three-component prey-predator model with prey- dependent and ratio-dependent functional responses, where cross-diffusion is included in such a way that the predator chases the prey and the prey runs away from the predator. We prove that cross-diffusion can generate stationary patterns (nonconstant positive steady states).

##### MSC:
 92D40 Ecology 35K57 Reaction-diffusion equations 35B35 Stability of solutions of PDE 35B40 Asymptotic behavior of solutions of PDE 35B32 Bifurcation (PDE) 35J55 Systems of elliptic equations, boundary value problems (MSC2000)
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