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Spatiotemporal dynamics in a ratio-dependent predator-prey model with time delay near the Turing-Hopf bifurcation point. (English) Zbl 1509.35034

Summary: Spatiotemporal dynamics of a ratio-dependent predator-prey model with time delay and the Neumann boundary condition are considered. The existence of the codimension-two Turing-Hopf bifurcation point is proved, so that both the Turing and Hopf bifurcations will occur simultaneously. Then, through the global Hopf bifurcation results, the global Hopf bifurcation of the ratio-dependent predator-prey model is investigated when diffusion is absent. Further, to identify the spatiotemporal dynamics during the Turing-Hopf bifurcation, the method of the multiple time scale analysis is employed to derive the amplitude equations near the codimension-two bifurcation point, instead of employing the center manifold theory and the normal form reduction. By analyzing the amplitude equations, it is found that the original ratio-dependent predator-prey model may exhibit the spatial, temporal or the spatiotemporal patterns, such as the nonconstant steady state, spatially homogeneous periodic solutions as well as the spatially inhomogeneous periodic solutions. Numerical simulations are carried out to validate the theoretical results.

MSC:

35B36 Pattern formations in context of PDEs
35B32 Bifurcations in context of PDEs
35K51 Initial-boundary value problems for second-order parabolic systems
35K57 Reaction-diffusion equations
92D25 Population dynamics (general)
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