Global dynamics and integrability of a Leslie-Gower predator-prey model with linear functional response and generalist predator

TL;DR

This paper analyzes a Leslie-Gower predator-prey model with a linear functional response, proving it is not Liouvillian integrable and characterizing its global dynamics through phase portraits.

Contribution

It demonstrates the non-integrability of the model and provides a complete topological classification of its phase portraits using Poincaré compactification.

Findings

The system is not Liouvillian integrable.

Two distinct topological phase portraits are identified.

Global dynamics are characterized in the Poincaré disc.

Abstract

We deal with a Leslie-Gower predator-prey model with a generalist or alternating food for predator and linear functional response. Using a topological equivalent polynomial system we prove that the system is not Liouvillian (hence also not Darboux) integrable. In order to study the global dynamics of this model, we use the Poincar\'e compactification of $\mathbb{R}^2$ to characterize all phase portraits in the Poincar\'e disc, obtaining two different topological phase portraits.