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Bifurcation and pattern formation in homogeneous diffusive predator-prey model
  • Hailong Yuan,
  • Yadi Wang
Hailong Yuan
Shaanxi University of Science and Technology

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Yadi Wang
Shaanxi University of Science and Technology Xi'an Campus
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Abstract

In this paper, we shall consider the existence of nonconstant solutions of a preadtor-prey model under homogeneous Neumann boundary conditions. In particular, the existence of the global bifurcations, the Hopf bifurcations and the steady state bifurcations of system are established, and the bifurcation direction of the bifurcating periodic solutions and the steady state bifurcations of system are investigated. Moreover, the stability of the Hopf bifurcation is also studied, and we show that the spatially homogeneous periodic solutions are asymptotically stable, while the spatially nonhomogeneous periodic solutions are unstable. Furthermore, the effect of cross-diffusion for existence of nonconstant solutions of system is studied, and we also show that cross-diffusion is not always helpful to create Turing instability. Our mathematical approach is based on the Leray-Schauder degree theorem, the bifurcation theorem, the technique of space decomposition and the implicit function theorem.