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A ratio-dependent predator-prey model with diffusion. (English) Zbl 1124.35027

The existence and nonexistence of nonconstant positive steady-state of a system of strongly coupled parabolic equations arising in mathematical biology as a ratio-dependent predator-prey model (with diffusion) of two species, which co-interact and migrate in the same habitat is investigated. The boundary conditions are homogeneous of Neumann type. Conditions under which diffusion can or cannot create a nonconstant positive steady-state solution are obtained. The Turing stability is analyzed and the a priori estimates of positive solutions obtained. The local analysis at the constant positive steady-states is also studied by linearization and the application of topological degree.

MSC:

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D25 Population dynamics (general)
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
47H11 Degree theory for nonlinear operators
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