Impact of alternative food on predator diet in a Leslie-Gower model with prey refuge and Holling Ⅱ functional response

Christian Cortés García; Jasmidt Vera Cuenca; Christian Cortés García; Jasmidt Vera Cuenca

doi:10.3934/mbe.2023610

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 8: 13681-13703. doi: 10.3934/mbe.2023610

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Research article

Impact of alternative food on predator diet in a Leslie-Gower model with prey refuge and Holling Ⅱ functional response

Christian Cortés García ^{1,2,3,†
,
,},
Jasmidt Vera Cuenca ^{4,†
,
,}

1.
Departamento de Matemáticas, Universidad Carlos Ⅲ de Madrid, Avenida de la Universidad 30, Madrid, Spain
2.
Departamento de Biología de Sistemas, Centro Nacional de Biotecnología, Calle Darwin 3, Madrid, Spain
3.
Grupo Interdisciplinar de Sistemas Complejos (GISC), Madrid, Spain
4.
Facultad de Ciencias Exactas y Naturales, Departamento de Matemáticas y Estadística, Universidad Surcolombiana, Avenida Pastrana Borrero - Carrera 1, Neiva, Colombia

† The authors contributed equally to this work.
Academic Editor: Yang Kuang

Received: 08 April 2023 Revised: 26 May 2023 Accepted: 09 June 2023 Published: 15 June 2023

Since certain prey hide from predators to protect themselves within their habitats, predators are forced to change their diet due to a lack of prey for consumption, or on the contrary, subsist only with alternative food provided by the environment. Therefore, in this paper, we propose and mathematically contrast a predator-prey, where alternative food for predators is either considered or not when the prey population size is above the refuge threshold size. Since the model with no alternative food for predators has a Hopf bifurcation and a transcritical bifurcation, in addition to a stable limit cycle surrounding the unique interior equilibrium, such bifurcation cases are transferred to the model when considering alternative food for predators when the prey size is above the refuge. However, such a model has two saddle-node bifurcations and a homoclinic bifurcation, characterized by a homoclinic curve surrounding one of the three interior equilibrium points of the model.
- filippov systems,
- crossing region,
- critical threshold,
- harvesting,
- bifurcation theory
Citation: Christian Cortés García, Jasmidt Vera Cuenca. Impact of alternative food on predator diet in a Leslie-Gower model with prey refuge and Holling Ⅱ functional response[J]. Mathematical Biosciences and Engineering, 2023, 20(8): 13681-13703. doi: 10.3934/mbe.2023610

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Abstract

Since certain prey hide from predators to protect themselves within their habitats, predators are forced to change their diet due to a lack of prey for consumption, or on the contrary, subsist only with alternative food provided by the environment. Therefore, in this paper, we propose and mathematically contrast a predator-prey, where alternative food for predators is either considered or not when the prey population size is above the refuge threshold size. Since the model with no alternative food for predators has a Hopf bifurcation and a transcritical bifurcation, in addition to a stable limit cycle surrounding the unique interior equilibrium, such bifurcation cases are transferred to the model when considering alternative food for predators when the prey size is above the refuge. However, such a model has two saddle-node bifurcations and a homoclinic bifurcation, characterized by a homoclinic curve surrounding one of the three interior equilibrium points of the model.

References

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