Abstract

In this work, a modified Leslie-Gower-type predator-prey model is analyzed, considering now that the prey population is affected by a weak Allee effect, complementing results obtained in previous papers in which the consequences of strong Allee effect for the same model were established. To simplify the calculations, a diffeomorphism is constructed to obtain a topological equivalent system for which we establish the boundedness of solutions, the nature of equilibrium points, the existence of a separatrix curve dividing the behavior of trajectories. Also, the existence of two concentric limit cycles surrounding a unique positive equilibrium point (generalized Hopf or Bautin bifurcation) is shown. Although the equilibrium point associated with the weak Allee effect lies in the second quadrant, the model has rich dynamics due to this phenomenon, such as it happens when a strong Allee effect is considered in the prey population. The model here analyzed has some similar behaviors with the model considering strong Allee effect, having both two limit cycles; nevertheless, they differ in the number of positive equilibrium points and the existence in our model of a non-infinitesimal limit cycle, which exists when the positive equilibrium is a repeller node. The main results obtained are reinforced by employing some numerical simulations.