# A modified May-Holling-Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator

## TL;DR

This paper analyzes a predator-prey model incorporating multiple Allee effects, an alternative food source, and bifurcation phenomena, revealing complex dynamics and stability conditions through analytical and numerical methods.

## Contribution

It introduces a modified predator-prey model with multiple Allee effects and an alternative food source, exploring its bifurcation structure and stability properties.

## Key findings

- Existence of at most two equilibrium points in the first quadrant.
- Identification of various bifurcations including saddle-node, Hopf, Bogadonov-Takens, and homoclinic bifurcations.
- The basin of attraction for the stable equilibrium increases when depensation effects are reduced.

## Abstract

We study a predator-prey model with Holling type I functional response, an alternative food source for the predator, and multiple Allee effects on the prey. We show that the model has at most two equilibrium points in the first quadrant, one is always a saddle point while the other can be a repeller or an attractor. Moreover, there is always a stable equilibrium point that corresponds to the persistence of the predator population and the extinction of the prey population. Additionally, we show that when the parameters are varied the model displays a wide range of different bifurcations, such as saddle-node bifurcations, Hopf bifurcations, Bogadonov-Takens bifurcations and homoclinic bifurcations. We use numerical simulations to illustrate the impact changing the predation rate, or the non-fertile prey population, and the proportion of alternative food source have on the basins of attraction of the stable equilibrium point in the first quadrant (when it exists). In particular, we also show that the basin of attraction of the stable positive equilibrium point in the first quadrant is bigger when we reduce the depensation in the model.