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Spatiotemporal patterns induced by Hopf bifurcations in a homogeneous diffusive predator-prey system. (English) Zbl 1435.92057

Summary: In this paper, we consider a diffusive predator-prey system where the prey exhibits the herd behavior in terms of the square root of the prey population. The model is supposed to impose on homogeneous Neumann boundary conditions in the bounded spatial domain. By using the abstract Hopf bifurcation theory in infinite dimensional dynamical system, we are capable of proving the existence of both spatial homogeneous and nonhomogeneous periodic solutions driven by Hopf bifurcations bifurcating from the positive constant steady state solutions. Our results allow for the clearer understanding of the mechanism of the spatiotemporal pattern formations of the predator-prey interactions in ecology.

MSC:

92D25 Population dynamics (general)
35B32 Bifurcations in context of PDEs
35K51 Initial-boundary value problems for second-order parabolic systems
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