Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations

Abstract

<p style='text-indent:20px;'>We investigate spatiotemporal patterns near the Turing-Hopf and double Hopf bifurcations in a diffusive Holling-Tanner model on a one- dimensional spatial domain. Local and global stability of the positive constant steady state for the non-delayed system is studied. Introducing the generation time delay in prey growth, we discuss the existence of Turing-Hopf and double Hopf bifurcations and give the explicit dynamical classification near these bifurcation points. Finally, we obtain the complicated dynamics, including periodic oscillations, quasi-periodic oscillations on a three-dimensional torus, the coexistence of two stable nonconstant steady states, the coexistence of two spatially inhomogeneous periodic solutions, and strange attractors.</p>

Hopf-Hopf bifurcation and spatially staggered time-periodic orbits induced by the joint effects of predator-taxis and delay

“…Let A be the infinitesimal generator associated with the semiflow of the linearized equation (6), then…”

“…In [5], the normal form formulae near the codimension-two Hopf-Hopf bifurcation point of a reactiondiffusion system with time delay have been established. Subsequently, Duan et al [6] have investigated a delayed predator-prey model with fear effect. By using the formulae in [5], it has been shown the existence of quasi-periodic orbits on three-torus near a Hopf-Hopf bifurcation point, leading to a strange attractor when further varying the parameter.…”

Spatiotemporal patterns induced by Turing and Turing-Hopf bifurcations in a predator-prey system

“…In particular, Song et al [19] and Jiang et al [20] derived the normal form of the Turing-Hopf bifurcation of partial differential equations (PDEs) and partial functional differential equations (PFDEs), respectively. Following the method proposed, there are many subsequent works on normal forms of the Turing-Hopf bifurcation [21][22][23][24].…”