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Zbl 1197.35042
Zhang, Jia-Fang; Li, Wan-Tong; Yan, Xiang-Ping
Multiple bifurcations in a delayed predator-prey diffusion system with a functional response. (English)
[J] Nonlinear Anal., Real World Appl. 11, No. 4, 2708-2725 (2010). ISSN 1468-1218

Summary: The present paper is concerned with a delayed predator-prey diffusion system with a Beddington-DeAngelis functional response and homogeneous Neumann boundary conditions. If the positive constant steady state of the corresponding system without delay is stable, by choosing the delay as the bifurcation parameter, we can show that the increase of the delay can not only cause spatially homogeneous Hopf bifurcation at the positive constant steady state but also give rise to spatially heterogeneous ones. In particular, under appropriate conditions, we find that the system has a Bogdanov-Takens singularity at the positive constant steady state, whereas this singularity does not occur for the corresponding system without diffusion. In addition, by applying the normal form theory and center manifold theorem for partial functional differential equations, we give normal forms of Hopf bifurcation and Bogdanov-Takens bifurcation and the explicit formula for determining the properties of spatial Hopf bifurcations.

MSC 2000:: *35B32 Bifurcation (PDE)
35K51
35K58
92D25 Population dynamics

Keywords: discrete delay; diffusion effects; spatial Hopf bifurcation; Bogdanov-Takens bifurcation; one space dimension; Beddington-DeAngelis functional response; homogeneous Neumann boundary conditions; delay as bifurcation parameter; normal form theory; center manifold theorem

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