Multiple limit cycles in a Leslie–Gower-type predator–prey model considering weak Allee effect on prey
Articles
Eduardo González-Olivares
Pontificia Universidad Católica de Valparaíso, Chile
Alejandro Rojas-Palma
Pontificia Universidad Católica de Valparaíso, Chile
Betsabé González-Yañez
Pontificia Universidad Católica de Valparaíso, Chile
Published 2017-05-10
https://doi.org/10.15388/NA.2017.3.5
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Keywords

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

How to Cite

González-Olivares E., Rojas-Palma A. and González-Yañez B. (2017) “Multiple limit cycles in a Leslie–Gower-type predator–prey model considering weak Allee effect on prey”, Nonlinear Analysis: Modelling and Control, 22(3), pp. 347-365. doi: 10.15388/NA.2017.3.5.

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.

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