Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators
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, i...
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| Format: | Article |
| Language: | English |
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AIMS Press
2020-11-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2020392?viewType=HTML |
| _version_ | 1798027230652137472 |
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| author | Eduardo Gonzalez-Olivares Alejandro Rojas-Palma |
| author_facet | Eduardo Gonzalez-Olivares Alejandro Rojas-Palma |
| author_sort | Eduardo Gonzalez-Olivares |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-11T18:48:00Z |
| format | Article |
| id | doaj.art-e9ec7cbf1eb14cedb51f47790dc7f2a8 |
| institution | Directory Open Access Journal |
| issn | 1551-0018 |
| language | English |
| last_indexed | 2024-04-11T18:48:00Z |
| publishDate | 2020-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj.art-e9ec7cbf1eb14cedb51f47790dc7f2a82022-12-22T04:08:34ZengAIMS PressMathematical Biosciences and Engineering1551-00182020-11-011767708773110.3934/mbe.2020392Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predatorsEduardo Gonzalez-Olivares0Alejandro Rojas-Palma11. Pontificia Universidad Católica de Valparaíso, Chile2. Departamento de Matemáticas, Física y Estadística, Facultad de Ciencias Básicas, Universidad Católica del Maule, Talca, ChileIn 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.https://www.aimspress.com/article/doi/10.3934/mbe.2020392?viewType=HTMLpredator-prey modelfunctional responsebifurcationlimit cycleseparatrix curvestability |
| spellingShingle | Eduardo Gonzalez-Olivares Alejandro Rojas-Palma Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators Mathematical Biosciences and Engineering predator-prey model functional response bifurcation limit cycle separatrix curve stability |
| title | Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators |
| title_full | Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators |
| title_fullStr | Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators |
| title_full_unstemmed | Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators |
| title_short | Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators |
| title_sort | global stability in a modified leslie gower type predation model assuming mutual interference among generalist predators |
| topic | predator-prey model functional response bifurcation limit cycle separatrix curve stability |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2020392?viewType=HTML |
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