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Volume 9, Number 3, 2019, Pages 1132-1164                                                                DOI:10.11948/2156-907X.20190015
Turing-Hopf bifurcation in the reaction-diffusion system with delay and application to a diffusive predator-prey model
Yongli Song,Heping Jiang,Yuan Yuan
Keywords:Diffusion, delay, Turing-Hopf bifurcation, normal form, predator-prey model.
      The interactions of diffusion-driven Turing instability and delay-induced Hopf bifurcation always give rise to rich spatiotemporal dynamics. In this paper, we first derive the algorithm for the normal forms associated with the Turing-Hopf bifurcation in the reaction-diffusion system with delay, which can be used to investigate the spatiotemporal dynamical classification near the Turing-Hopf bifurcation point in the parameter plane. Then, we consider a diffusive predator-prey model with weak Allee effect and delay. Through investigating the dynamics of the corresponding normal form of Turing-Hopf bifurcation induced by diffusion and delay, the spatiotemporal dynamics near this bifurcation point can be divided into six categories. Especially, stable spatially homogeneous/inhomogeneous periodic solutions and steady states, coexistence of two stable spatially inhomogeneous periodic solutions, coexistence of two stable spaially inhomogeneous steady states and the transition from one kind of spatiotemporal patterns to another are found.
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