Multiple limit cycles in a Leslie–Gower-type predator–prey model considering weak Allee effect on prey
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...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Vilnius University Press
2017-05-01
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| Series: | Nonlinear Analysis |
| Subjects: | |
| Online Access: | http://www.zurnalai.vu.lt/nonlinear-analysis/article/view/13391 |
| Summary: | 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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| ISSN: | 1392-5113 2335-8963 |