Abstract
We consider a time delay predator-prey model with Holling type-IV functional response and stage-structured for the prey. Our aim is to observe the dynamics of this model under the influence of gestation delay of the predator. We obtain sufficient conditions for the local stability of each of feasible equilibria of the system and the existence of a Hopf bifurcation at the coexistence equilibrium. By using the normal form theory and center manifold theory we also derive some explicit formulae determining the bifurcation direction and the stability of the bifurcated periodic solutions. Finally, numerical simulations are given to explain the theoretical results.
Highlights
The dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance
One of the familiar factors affecting the dynamics of predator-prey models is the functional response, which relates the single predator’s prey consumption rate to the prey population density
It is well known that delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay can induce various oscillations and periodic solutions
Summary
The dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. One of the familiar factors affecting the dynamics of predator-prey models is the functional response, which relates the single predator’s prey consumption rate to the prey population density. The single-species model with stage structure was studied by Aiello and Freedman [ ]. Xu et al [ ] considered a ratiodependent predator-prey model with stage structure for the prey. By constructing appropriate Lyapunov functions sufficient conditions are obtained for the global asymptotic stability of nonnegative equilibria of the model. Motivated by the works of Chen and Jing [ ], Wangersky and Cunningham [ ], and Xu et al [ , ], we consider the following predator-prey model with Holling type-IV func-. In order to illustrate the validity of the theoretical result, some numerical simulations are included
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