Stability and bifurcation in a Holling type II predator–prey model with Allee effect and time delay

Xiao, Zaowang; Xie, Xue‐Jun; Xue, Yalong

doi:10.1186/s13662-018-1742-4

Cited by 19 publications

(7 citation statements)

References 34 publications

(48 reference statements)

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“…They showed that an alternative resource has the ability to prohibit top predator annihilation. Xiao et al [10] proposed and investigated a predator-prey model that associates time delay with a Holling type II and Allee impact in prey. They observed that if the Allee impact is high or the birth rate is low, then both species of the system are fading and hence the stability of the system is affected.…”

Section: Jaber and Bahloolmentioning

confidence: 99%

The Dynamics of a Food Web System: Role of a Prey Refuge Depending on Both Species

Jabr

Bahlool

2021

eijs

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This paper aims to study the role of a prey refuge that depends on both prey and predator species on the dynamics of a food web model. It is assumed that the food transfer among the web levels occurs according to Lotka-Volterra functional response. The solution properties, such as existence, uniqueness, and uniform boundedness, are discussed. The local, as well as the global, stabilities of the solution of the system are investigated. The persistence of the system is studied with the assistance of average Lyapunov function. The local bifurcation conditions that may occur near the equilibrium points are established. Finally, numerical simulation is used to confirm our obtained results. It is observed that the system has only one type of attractors that is a stable point, while periodic dynamics do not exist even on the boundary planes.

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Section: Jaber and Bahloolmentioning

confidence: 99%

The Dynamics of a Food Web System: Role of a Prey Refuge Depending on Both Species

Jabr

Bahlool

2021

eijs

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“…Our main objective is to investigate the effects of refuge, Allee and intra-specific competition among the predators on the equilibria, stability and bifurcation, which may be effective for the conservation of biodiversity. We have already investigated our previous study [20] with additive Allee on the prey species and incorporating prey refuge depending on both prey and predator species but our present study deals with a predator-prey dynamical model considering the followings: (i) Allee effect on the predator population; (ii) Holling type II [12,13,42] response function; (iii) nonlinear prey refuge proportional to the direct interaction on both the populations; and (iv) incorporating intra-specific competition among the predators. We have organized the rest of our study as follows.…”

Section: Introductionmentioning

confidence: 99%

Predator-prey dynamics with Allee effect on predator species subject to intra-specific competition and nonlinear prey refuge

Molla¹,

Sarwardi²,

Sajid³

2021

J. Math. Computer Sci.

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A modified version of our previously analyzed prey-predator refuge model is presented in this article by introducing Allee effect on the predator species and mutual interference among the predators. Possible number of coexistence equilibrium points are investigated with the help of prey and predator nullcline. The local stability and Hopf-bifurcation conditions are established around the coexistence equilibria. We have also discussed the nature of Hopf-bifurcation around the unique coexistence equilibrium point of the system as well. Finally, a comprehensive numerical simulation is carried out to justify our obtained analytical findings.

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“…The Allee effect was also researched in predator-prey systems with delay, impulse or diffusion. Xiao et al [19]…”

Section: Introductionmentioning

confidence: 99%

The Dynamical Behavior of a Predator-Prey System with Holling Type II Functional Response and Allee Effect

Wang¹,

Yu²,

Dai³

et al. 2020

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In this paper, we mainly considered the dynamical behavior of a predator-prey system with Holling type II functional response and Allee-like effect on predator, including stability analysis of equilibria and Hopf bifurcation. Firstly, we gave some sufficient conditions to guarantee the existence, the local and global stability of equilibria as well as non-existence of limit cycles. By using the cobweb model, some cases about the existence of interior equilibrium are also illustrated with numerical outcomes. These existence and stability conclusions of interior equilibrium are also suitable in corresponding homogeneous reaction-diffusion system subject to the Neumann boundary conditions. Secondly, we theoretically deduced that our system has saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation under certain conditions. Finally, for the Hopf bifurcation, we choose d as the bifurcation parameter and presented some numerical simulations to verify feasibility and effectiveness of the theoretical derivation corresponding to the existence of k y , respectively. The Hopf bifurcations are supercritical and limit cycles generated by the critical points are stable.

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Stability and bifurcation in a Holling type II predator–prey model with Allee effect and time delay

Cited by 19 publications

References 34 publications

The Dynamics of a Food Web System: Role of a Prey Refuge Depending on Both Species

The Dynamics of a Food Web System: Role of a Prey Refuge Depending on Both Species

Predator-prey dynamics with Allee effect on predator species subject to intra-specific competition and nonlinear prey refuge

The Dynamical Behavior of a Predator-Prey System with Holling Type II Functional Response and Allee Effect

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