Multiple limit cycles in a Leslie–Gower-type predator–prey model considering weak Allee effect on prey - González-Olivares, Eduardo; Rojas-Palma, Alejandro; González-Yañez, Betsabé (Nonlinear Analysis: Modelling and Control, 2017)
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