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

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.