Bifurcations in a discontinuous Leslie-Gower model with harvesting and alternative food for predators and constant prey refuge at low density

Christian Cortés García; Christian Cortés García

doi:10.3934/mbe.2022653

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 12: 14029-14055. doi: 10.3934/mbe.2022653

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Research article

Bifurcations in a discontinuous Leslie-Gower model with harvesting and alternative food for predators and constant prey refuge at low density

Christian Cortés García ^{1,2
,
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1.
Department of Mathematics, Universidad Carlos Ⅲ de Madrid, 30 University Avenue, Madrid, Spain
2.
Department of Systems Biology, Centro Nacional de Biotecnología, 3 Darwin Street, Madrid, Spain

Academic Editor: Haiyan Wang

Received: 01 August 2022 Revised: 08 September 2022 Accepted: 12 September 2022 Published: 22 September 2022

Since environmental studies have shown that a constant quantity of prey become refuges from the predator at low densities and become accessible again for consumption when they reach a higher density, in this work we propose a discontinuous mathematical model, Lesli-Gower type, which describes the dynamics between prey and predators, interacting under the same environment, and whose predator functional response, of linear type, is altered by a refuge constant in the prey when below a critical value. Assuming that predators can be captured and have alternative food, the qualitative analysis of the proposed discontinuous model is performed by analyzing each of the vector fields that compose it, which serves as the basis for the calculation of the bifurcation curves of the discontinuous model, with respect to the threshold value of the prey and the harvest rate of predators. It is concluded that the perturbations of the parameters of the model leads either to the extinction of the predators or to a stabilization in the growth of both species, regardless of their initial conditions.
- filippov systems,
- threshold value,
- bifurcation analysis,
- logistic growth,
- pseudo-equilibrium
Citation: Christian Cortés García. Bifurcations in a discontinuous Leslie-Gower model with harvesting and alternative food for predators and constant prey refuge at low density[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 14029-14055. doi: 10.3934/mbe.2022653

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Abstract

Since environmental studies have shown that a constant quantity of prey become refuges from the predator at low densities and become accessible again for consumption when they reach a higher density, in this work we propose a discontinuous mathematical model, Lesli-Gower type, which describes the dynamics between prey and predators, interacting under the same environment, and whose predator functional response, of linear type, is altered by a refuge constant in the prey when below a critical value. Assuming that predators can be captured and have alternative food, the qualitative analysis of the proposed discontinuous model is performed by analyzing each of the vector fields that compose it, which serves as the basis for the calculation of the bifurcation curves of the discontinuous model, with respect to the threshold value of the prey and the harvest rate of predators. It is concluded that the perturbations of the parameters of the model leads either to the extinction of the predators or to a stabilization in the growth of both species, regardless of their initial conditions.

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