Journal
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
Volume 77, Issue -, Pages 141-167Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/j.cnsns.2019.04.024
Keywords
Delayed predator-prey model; Turing-Hopf bifurcation; Multiple time scale
Categories
Funding
- National Natural Science Foundation of China [11571016, 61403115]
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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. (C) 2019 Elsevier B.V. All rights reserved.
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