Abstract
The paper is concerned with a delayed diffusive predator-prey system where the growth of prey population is governed by Allee effect and the predator population consumes the prey according to Beddington-DeAngelis type functional response. The situation of bi-stability and the existence of two coexisting equilibria for the proposed model system are addressed. The stability of the steady state together with its dependence on the magnitude of time delay has been obtained. The conditions that guarantee the occurrence of the Hopf bifurcation in presence of delay are demonstrated. Furthermore, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Finally, some numerical simulations have been carried out in order to validate the assumptions of the model.
Highlights
In this paper, we consider the following delayed diffusive predator-prey system with Beddington-DeAngelis functional response and strong Allee effect: ⎧ ⎪⎪⎪⎪⎪⎨ ∂u ∂t ∂v ∂t = = d d u + ru( u K)(u m) quv(t–τ ) a+bu+cv(t–τ
In Section, we investigate the direction and stability of Hopf bifurcation by applying the center manifold method and the normal form theory
We considered a delayed diffusive predator-prey system with Beddington
Summary
We consider the following delayed diffusive predator-prey system with Beddington-DeAngelis functional response and strong Allee effect:. If (H ) and (H ) hold, all roots of equation ( ) have negative real parts for all τ ≥. If (H ) and (H ) hold, equation ( ) has a pair of purely imaginary roots ±iωn ( ≤ n ≤ N∗) at τ τnj τn jπ ωn. If (H ) and (H ) hold, the following statements are true: (i) When τ ∈ [ , τ ), the coexisting equilibrium of system ( ) is locally asymptotically stable.
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