Dynamical Analysis of a Predator-Prey Model with Additive Allee Effect and Prey Group Defense

TL;DR

This paper analyzes a predator-prey model incorporating Allee effects and prey group defense, revealing how different Allee effect strengths influence population stability, bifurcations, and potential extinction scenarios through analytical and numerical methods.

Contribution

It introduces a predator-prey model with Allee effect and prey group defense, analyzing stability and bifurcations under varying Allee effect strengths for the first time.

Findings

Strong Allee effect can cause prey and predator extinction.

Weak Allee effect leads to bifurcations at predator extinction point.

Bistability occurs under strong Allee effect conditions.

Abstract

In this article, we develop a predator-prey model with Allee effect and prey group defense. The model has three equilibrium points i.e. the trivial point, the predator extinction point, and the coexistence point. All equilibrium points are locally asymptotically stable under certain conditions. The Allee effect in this model influences the stability of the equilibrium point. If the Allee effect is weak, then the trivial equilibrium point is unstable. Meanwhile, if the Allee effect is strong, then the trivial equilibrium point is locally asymptotically stable. Those mean that a strong Allee effect can lead to the extinction of both populations. Moreover, under weak Allee condition, forward bifurcation and Hopf bifurcation occur at the predator extinction equilibrium point. Meanwhile, a strong Allee effect may induce bistability at both the trivial equilibrium point and the predator extinction equilibrium point. Those mean that prey can survive without the presence of predators, but a strong Allee effect can lead to prey extinction if the population size is very small. To support our analytical findings, we perform some numerical simulations in the final section.