Abstract

In the ecological literature, many models for the predator-prey interactions have been well formulated but partially analyzed. Assuming this analysis to be true and complete, some authors use that results to study a more complex relationship among species (food webs). Others employ more sophisticated mathematical tools for the analysis, without further questioning. The aim of this paper is to extend, complement and enhance the results established in an earlier article referred to a modified Leslie-Gower model. In that work, the authors proved only the boundedness of solutions, the existence of an attracting set, and the global stability of a single equilibrium point at the interior of the first quadrant. In this paper, new results for the same model are proven, establishing conditions in the parameter space for which up two positive equilibria exist. Assuming there exists a unique positive equilibrium point, we have proved, the existence of: i) a separatrix curve Sigma, dividing the trajectories in the phase plane, which can have different omega - limit, ii) a subset of the parameter space in which two concentric limit cycles exist, the innermost unstable and the outermost stable. Then, there exists the phenomenon of tri-stability, because simultaneously, it has: a local stable positive equilibrium point, a stable limit cycle, and an attractor equilibrium point over the vertical axis. Therefore, we warn the model studied have more rich and interesting properties that those shown that earlier papers. Numerical simulations and a bifurcation diagram are given to endorse the analytical results.