## Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system.(English)Zbl 1203.35030

The authors investigate a diffusive predator-prey system with Holling type-II functional response and Neumann boundary conditions. For this system the analysis of Hopf bifurcations and steady state bifurcations is carried out in detail, including centre manifold reduction and normal form analysis.
It is shown that multiple spatially non-homogeneous period orbits exist and that there exist loops of periodic orbits and steady state solutions. This result can be regarded as evidence for the rich spatio-temporal behaviour observed in that class of systems.
The detailed analysis of the bifurcation equations, as it is carried out in this article, can also serve as a valuable resource for learning the applied methods.

### MSC:

 35B32 Bifurcations in context of PDEs 35K57 Reaction-diffusion equations 35B10 Periodic solutions to PDEs 92D25 Population dynamics (general) 35K51 Initial-boundary value problems for second-order parabolic systems
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