Abstract
In this paper, we consider a Holling type II predator–prey model incorporating time delay and Allee effect in prey. We discuss the influence of Allee effect on the logistic equation. By analyzing the characteristic equation of the corresponding linearized system, we give the threshold condition for the local asymptotic stability of the system according to the change of birth rate or Allee effect in prey. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. In addition, we show that if the Allee effect is large or the birth rate is small, then both predators and prey are extinct. The Allee effect can influence the stability of the system.
Highlights
The predator–prey model is one of the basic models between different species in nature which has been widely researched [1,2,3,4,5,6]
Scholars have obtained a large number of interesting results about predator–prey models associated with Holling type II functional response [9,10,11]
5 Conclusion In this paper, we investigate the stability and Hopf bifurcation of a delayed Holling type II predator–prey model when the prey is subject to Allee effect
Summary
The predator–prey model is one of the basic models between different species in nature which has been widely researched [1,2,3,4,5,6]. Remark 3.1 When a ≤ d, it follows from Theorem 3.2 that E0 of system (4) is globally asymptotically stable for any value of the Allee effect. (i) If a = 5, that is, a < a1, it follows from Theorem 3.2 that E0 is globally asymptotically stable Both prey and predators are extinct (see Fig. 3). According to Theorem 3.6, E∗ is locally asymptotically stable if 0 < τ < τ0 and unstable if τ > τ0, and system (4) undergoes a Hopf bifurcation at E∗ when τ = τ0, where τ = 0.07 is shown in Fig. and τ = 0.15 is shown in Fig.
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