Bifurcation on diffusive Holling–Tanner predator–prey model with stoichiometric density dependence

This paper studies a diffusive Holling–Tanner predator–prey system with stoichiometric density dependence. The local stability of positive equilibrium, the existence of Hopf bifurcation and stability of bifurcating periodic solutions have been obtained in the absence of diffusion. We also study the spatially homogeneous and nonhomogeneous periodic solutions through all parameters of the system, which are spatially homogeneous. In order to verify our theoretical results, some numerical simulations are carried out.&nbsp

Existence and stability of periodic solutions for a delayed prey–predator model with diffusion effects

Existence and stability of spatially periodic solutions for a delay prey-predator diffusion system are concerned in this work. We obtain that the system can generate the spatially nonhomogeneous periodic solutions when the diffusive rates are suitably small. This result demonstrates that the diffusion plays an important role on deriving the complex spatiotemporal dynamics. Meanwhile, the stability of the spatially periodic solutions is also studied. Finally, in order to verify our theoretical results, some numerical simulations are also included

Stability and Hopf bifurcation of a diffusive Gompertz population model with nonlocal delay effect

In this paper, we investigate the dynamics of a diffusive Gompertz population model with nonlocal delay effect and Dirichlet boundary condition. The stability of the positive spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcations with the change of the time delay are discussed by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. Then we derive the stability and bifurcation direction of Hopf bifurcating periodic orbits by using the normal form theory and the center manifold reduction. Finally, we give some numerical simulations

Bifurcation analysis of a reaction-diffusion-advection predator-prey system with delay

A diffusive predator-prey system with advection and time delay is considered. Choosing the conversion delay $\tau$ as a bifurcation parameter, we find that as $\tau$ varies, the system will generate Hopf bifurcation. Then, for the reaction diffusion model proposed in this paper, we use an improved center manifold reduction method and normal form theory to derive an algorithm for determining the direction and stability of Hopf bifurcation. Finally, we provide simulations to illustrate the effects of time delay $\tau$ and advection $\alpha$ on system behaviors

Multi-stable and spatiotemporal staggered patterns in a predator-prey model with predator-taxis and delay

The effects of predator-taxis and conversion time delay on formations of spatiotemporal patterns in a predator-prey model are explored. First, the well-posedness, which implies global existence of classical solutions, is proved. Then, we establish critical conditions for the destabilization of the coexistence equilibrium via Turing/Turing-Turing bifurcations by describing the first Turing bifurcation curve; we also theoretically predict possible bistable/multi-stable spatially heterogeneous patterns. Next, we demonstrate that the coexistence equilibrium can also be destabilized via Hopf, Hopf-Hopf and Turing-Hopf bifurcations; also possible stable/bistable spatially inhomogeneous staggered periodic patterns and bistable spatially inhomogeneous synchronous periodic patterns are theoretically predicted. Finally, numerical experiments also support theoretical predictions and partially extend them. In a word, theoretical analyses indicate that, on the one hand, strong predator-taxis can eliminate spatial patterns caused by self-diffusion; on the other hand, the joint effects of predator-taxis and conversion time delay can induce complex survival patterns, e.g., bistable spatially heterogeneous staggered/synchronous periodic patterns, thus diversifying populations' survival patterns

Local Bifurcations for a Delay Differential Model of Plankton Allelopathy

Abstract: This paper is concerned with a two-species competitive system of plankton allelopathy with delay. A modified differential equation model of plankton allelopathy having stimulatory effects on each other is investigated in this paper. By regarding the delay τ as the bifurcation parameter, firstly, the stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results