Multiple limit cycles in a Leslie–Gower-type predator–prey model considering weak Allee effect on prey

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. In order 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 to the weak Allee effect lies in the second quadrant, the model has a rich dynamics due to this phenomenon, such as it happens when a strong Allee effect is considered in 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 amount 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 means of some numerical simulations.

Keywords

• bifurcations
• limit cycle
• separatrix curve
• Allee effect
• Leslie–Gower predator–prey models