Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators

Eduardo Gonzalez-Olivares; Alejandro Rojas-Palma; Eduardo Gonzalez-Olivares; Alejandro Rojas-Palma

doi:10.3934/mbe.2020392

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 7708-7731. doi: 10.3934/mbe.2020392

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Research article

Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators

Eduardo Gonzalez-Olivares ^{1
,
,},
Alejandro Rojas-Palma ²

1.
Pontificia Universidad Católica de Valparaíso, Chile
2.
Departamento de Matemáticas, Física y Estadística, Facultad de Ciencias Básicas, Universidad Católica del Maule, Talca, Chile

Received: 25 June 2020 Accepted: 25 October 2020 Published: 05 November 2020

Abstract
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In the ecological literature, mutual interference (self-interference) or competition among predators (CAP) to effect the harvesting of their prey has been modeled through different mathematical formulations. In this work, the dynamical properties of a Leslie-Gower type predation model is analyzed, incorporating one of these forms, which is described by the function $g\left(y\right) = y^{\beta }$, with $0 < \beta < 1$. This function $g$ is not differentiable for $y = 0$, and neither the Jacobian matrix of the system is not defined in the equilibrium points over the horizontal axis ($x-axis$). To determine the nature of these points, we had to use a non-standard methodology. Previously, we have shown the fundamental properties of the Leslie-Gower type model with generalist predators, to carry out an adequate comparative analysis with the model where the competition among predators (CAP) is incorporated. The main obtained outcomes in both systems are: (ⅰ) The unique positive equilibrium point, when exists, is globally asymptotically stable (g.a.s), which is proven using a suitable Lyapunov function. (ⅱ) There not exist periodic orbits, which was proved constructing an adequate Dulac function.
- predator-prey model,
- functional response,
- bifurcation,
- limit cycle,
- separatrix curve,
- stability
Citation: Eduardo Gonzalez-Olivares, Alejandro Rojas-Palma. Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7708-7731. doi: 10.3934/mbe.2020392

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Abstract

In the ecological literature, mutual interference (self-interference) or competition among predators (CAP) to effect the harvesting of their prey has been modeled through different mathematical formulations. In this work, the dynamical properties of a Leslie-Gower type predation model is analyzed, incorporating one of these forms, which is described by the function $g\left(y\right) = y^{\beta }$, with $0 < \beta < 1$. This function $g$ is not differentiable for $y = 0$, and neither the Jacobian matrix of the system is not defined in the equilibrium points over the horizontal axis ($x-axis$). To determine the nature of these points, we had to use a non-standard methodology. Previously, we have shown the fundamental properties of the Leslie-Gower type model with generalist predators, to carry out an adequate comparative analysis with the model where the competition among predators (CAP) is incorporated. The main obtained outcomes in both systems are: (ⅰ) The unique positive equilibrium point, when exists, is globally asymptotically stable (g.a.s), which is proven using a suitable Lyapunov function. (ⅱ) There not exist periodic orbits, which was proved constructing an adequate Dulac function.

References

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