Global dynamics of a predator-prey system with immigration in both species

Érika Diz-Pita; Érika Diz-Pita

doi:10.3934/era.2024036

Electronic Research Archive

2024, Volume 32, Issue 2: 762-778. doi: 10.3934/era.2024036

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Global dynamics of a predator-prey system with immigration in both species

Érika Diz-Pita ^,

Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela

Received: 13 November 2023 Revised: 26 December 2023 Accepted: 03 January 2024 Published: 10 January 2024

In nature, the vast majority of species live in ecosystems that are not isolated, and the same is true for predator-prey ecological systems. With this work, we extend a predator-prey model by considering the inclusion of an immigration term in both species. From a biological point of view, that allows us to achieve a more realistic model. We consider a system with a Holling type Ⅰ functional response and study its global dynamics, which allows to not only determine the behavior in a region of the plane $ \mathbb{R}^2 $, but also to control the orbits that either go or come to infinity. First, we study the local dynamics of the system, by analyzing the singular points and their stability, as well as the possible behavior of the limit cycles when they exist. By using the Poincaré compactification, we determine the global dynamics by studying the global phase portraits in the positive quadrant of the Poincaré disk, which is the region where the system is of interest from a biological point of view.
- predator-prey,
- immigration,
- stability,
- global dynamics,
- phase portrait
Citation: Érika Diz-Pita. Global dynamics of a predator-prey system with immigration in both species[J]. Electronic Research Archive, 2024, 32(2): 762-778. doi: 10.3934/era.2024036

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Abstract

In nature, the vast majority of species live in ecosystems that are not isolated, and the same is true for predator-prey ecological systems. With this work, we extend a predator-prey model by considering the inclusion of an immigration term in both species. From a biological point of view, that allows us to achieve a more realistic model. We consider a system with a Holling type Ⅰ functional response and study its global dynamics, which allows to not only determine the behavior in a region of the plane $ \mathbb{R}^2 $, but also to control the orbits that either go or come to infinity. First, we study the local dynamics of the system, by analyzing the singular points and their stability, as well as the possible behavior of the limit cycles when they exist. By using the Poincaré compactification, we determine the global dynamics by studying the global phase portraits in the positive quadrant of the Poincaré disk, which is the region where the system is of interest from a biological point of view.

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