Dynamics of a modified Leslie-Gower predation model considering a generalist predator and the hyperbolic functional response

Eduardo González-Olivares; Claudio Arancibia-Ibarra; Alejandro Rojas-Palma; Betsabé González-Yañez; Eduardo González-Olivares; Claudio Arancibia-Ibarra; Alejandro Rojas-Palma; Betsabé González-Yañez

doi:10.3934/mbe.2019403

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 7995-8024. doi: 10.3934/mbe.2019403

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Dynamics of a modified Leslie-Gower predation model considering a generalist predator and the hyperbolic functional response

1.
Pontificia Universidad Católica de Valparaíso, Chile;
2.
Facultad de Educación, Universidad de Las Américas, Chile;
3.
Departamento de Matemática, Física y Estadítica, Facultad de Ciencias Básicas, Universidad Catolica del Maule, Talca, Chile

Received: 30 November 2018 Accepted: 25 August 2019 Published: 03 September 2019

In the ecological literature, many models for the predator-prey interactions have been well formulated but partially analyzed.Assuming this analysis to be true and complete, some authors use that results to study a more complex relationship among species (food webs).Others employ more sophisticated mathematical tools for the analysis, without further questioning.The aim of this paper is to extend, complement and enhance the results established in an earlier article referred to a modified Leslie-Gower model.In that work, the authors proved only the boundedness of solutions, the existence of an attracting set, and the global stability of a single equilibrium point at the interior of the first quadrant.In this paper, new results for the same model are proven, establishing conditions in the parameter space for which up two positive equilibria exist.Assuming there exists a unique positive equilibrium point, we have proved, the existence of:ⅰ) a separatrix curve Σ, dividing the trajectories in the phase plane, which can have different ω-limit, ⅱ) a subset of the parameter space in which two concentric limit cycles exist, the innermost unstable and the outermost stable.Then, there exists the phenomenon of tri-stability, because simultaneously, it has:a local stable positive equilibrium point, a stable limit cycle, and an attractor equilibrium point over the vertical axis.Therefore, we warn the model studied have more rich and interesting properties that those shown that earlier papers.Numerical simulations and a bifurcation diagram are given to endorse the analytical results.
- predator-prey model,
- bifurcation, limit cycle,
- separatrix curve, stability
Citation: Eduardo González-Olivares, Claudio Arancibia-Ibarra, Alejandro Rojas-Palma, Betsabé González-Yañez. Dynamics of a modified Leslie-Gower predation model considering a generalist predator and the hyperbolic functional response[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7995-8024. doi: 10.3934/mbe.2019403

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Abstract

In the ecological literature, many models for the predator-prey interactions have been well formulated but partially analyzed.Assuming this analysis to be true and complete, some authors use that results to study a more complex relationship among species (food webs).Others employ more sophisticated mathematical tools for the analysis, without further questioning.The aim of this paper is to extend, complement and enhance the results established in an earlier article referred to a modified Leslie-Gower model.In that work, the authors proved only the boundedness of solutions, the existence of an attracting set, and the global stability of a single equilibrium point at the interior of the first quadrant.In this paper, new results for the same model are proven, establishing conditions in the parameter space for which up two positive equilibria exist.Assuming there exists a unique positive equilibrium point, we have proved, the existence of:ⅰ) a separatrix curve Σ, dividing the trajectories in the phase plane, which can have different ω-limit, ⅱ) a subset of the parameter space in which two concentric limit cycles exist, the innermost unstable and the outermost stable.Then, there exists the phenomenon of tri-stability, because simultaneously, it has:a local stable positive equilibrium point, a stable limit cycle, and an attractor equilibrium point over the vertical axis.Therefore, we warn the model studied have more rich and interesting properties that those shown that earlier papers.Numerical simulations and a bifurcation diagram are given to endorse the analytical results.

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