Impact of the fear and Allee effect on a Holling type II prey\u2013predator model

Binfeng Xie

doi:10.1186/s13662-021-03592-6

Abstract

In this paper, we propose and investigate a prey–predator model with Holling type II response function incorporating Allee and fear effect in the prey. First of all, we obtain all possible equilibria of the model and discuss their stability by analyzing the eigenvalues of Jacobian matrix around the equilibria. Secondly, it can be observed that the model undergoes Hopf bifurcation at the positive equilibrium by taking the level of fear as bifurcation parameter. Moreover, through the analysis of Allee and fear effect, we find that: (i) the fear effect can enhance the stability of the positive equilibrium of the system by excluding periodic solutions; (ii) increasing the level of fear and Allee can reduce the final number of predators; (iii) the Allee effect also has important influence on the permanence of the predator. Finally, numerical simulations are provided to check the validity of the theoretical results.

Highlights

Predator–prey model has always been a hot research topic in biological mathematics [1– 16]
The rest of the article is arranged as follows: In Sect. 2, we provide a qualitative analysis of the system, which include the stability of the equilibria and the sufficient condition for Hopf bifurcation at positive equilibrium and the corresponding biological interpretation
5 Conclusion In this article, we have studied the impact of the fear and Allee effect on a prey–predator model with Holling type II response function

Summary

Introduction

Predator–prey model has always been a hot research topic in biological mathematics [1– 16]. Zu et al [30] proposed a prey–predator system with Holling II type response function incorporating Allee effect in prey as follows:. The results show that the number of offspring of sparrow will be reduced by 40% only by adding fear effect to the prey, and the predation risk itself is enough to affect the change of wild animal population. 1 1+fy as the fear factor where f > 0 presents the level of fear induced by predators and y is the predator population density at time t, and studied the prey–predator system with Holling II type response function incorporating fear effect in prey as follows:. We obtain a Holling II type predator–prey model with Allee effect and fear effect in prey as follows:.

Equilibria and their existence condition

Without Allee effect

Without fear effect

Incorporate fear and Allee effect

Findings

Conclusion