Abstract
The dynamics of a delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis functional response is investigated. The main results are given in terms of local stability and local Hopf bifurcation. By regarding the possible combination of the feedback delays of the prey and the predator as a bifurcation parameter, sufficient conditions for the local stability and existence of the local Hopf bifurcation of the system are obtained. In particular, the properties of the local Hopf bifurcation such as direction and stability are determined by using the normal form method and center manifold theorem. Finally, numerical simulations are carried out to illustrate the main theoretical results.
Highlights
As we all know, one of the dominant themes in mathematical ecology is the dynamic relationship between predators and their prey
Stimulated by this, in this paper we investigate the Hopf bifurcation of the following delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis functional response: dx(t) dt αy(t) +bx(t)+cy(t)
System ( ) undergoes a Hopf bifurcation at the positive equilibrium E∗(x∗, y∗) of system ( ) when τ = τ and a family of periodic solutions bifurcate from the positive equilibrium E∗(x∗, y∗) of system ( )
Summary
One of the dominant themes in mathematical ecology is the dynamic relationship between predators and their prey. Predator-prey systems with a Beddington-DeAngelis functional response have been studied by many authors [ – ]. Local stability, and global stability of the following modified Leslie-Gower predator-prey system with Beddington-DeAngelis functional. Li and Wang studied the stability and Hopf bifurcation of a delayed three-level food chain model with Beddington-DeAngelis functional response [ ]. In [ ], Bianca et al further studied an economic growth model with two delays and they investigated the existence and properties of Hopf bifurcation of the model by regarding the possible combination of the two delays as the bifurcation parameter. To the best of our knowledge, seldom did authors consider the Hopf bifurcation of the delayed predatorprey system with modified Leslie-Gower and Beddington-DeAngelis functional response. Stimulated by this, in this paper we investigate the Hopf bifurcation of the following delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis functional response: px(t τ ) –.
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