Hopf--Hopf bifurcation in a delayed diffusive predator-prey model with Beddington--DeAngelis functional response
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We study the spatiotemporal patterns around Hopf--Hopf bifurcation in a delayed diffusive predator-prey model with Beddington--DeAngelis functional response on one dimensional space. The locally and globally asymptotically stability conditions of positive equilibrium for nondelay system are obtained. Considering the maturity and gestation period of the prey, a delay is introduced and the existence of Hopf bifurcations and Hopf--Hopf bifurcations is discussed. The infinite-dimensional reaction-diffusion equation is transformed into an ordinary differential equation using the center manifold theorem and the normal form theory, leading to the derivation of the normal form of the Hopf--Hopf bifurcation. Near the bifurcation point we obtain the classification of dynamics and complex dynamics behaviors such as periodic oscillations, quasi-periodic oscillations of three torus and singular attractors.
Abstract
We study the spatiotemporal patterns around Hopf--Hopf bifurcation in a delayed diffusive predator-prey model with Beddington--DeAngelis functional response on one dimensional space. The locally and globally asymptotically stability conditions of positive equilibrium for nondelay system are obtained. Considering the maturity and gestation period of the prey, a delay is introduced and the existence of Hopf bifurcations and Hopf--Hopf bifurcations is discussed. The infinite-dimensional reaction-diffusion equation is transformed into an ordinary differential equation using the center manifold theorem and the normal form theory, leading to the derivation of the normal form of the Hopf--Hopf bifurcation. Near the bifurcation point we obtain the classification of dynamics and complex dynamics behaviors such as periodic oscillations, quasi-periodic oscillations of three torus and singular attractors.