Uniqueness of limit cycles and multiple attractors in a Gause-typepredator-prey model with nonmonotonic functional response and Allee effecton prey
The main purpose of this work is to analyze a Gause type predator-prey modelin which two ecological phenomena are considered: the Allee effect affectingthe prey growth function and the formation of group defence by prey in orderto avoid the predation. We prove the existence of a...
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AIMS Press
2012-12-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.345 |
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| author | Eduardo González-Olivares Betsabé González-Yañez Jaime Mena-Lorca José D. Flores |
| author_facet | Eduardo González-Olivares Betsabé González-Yañez Jaime Mena-Lorca José D. Flores |
| author_sort | Eduardo González-Olivares |
| collection | DOAJ |
| description | The main purpose of this work is to analyze a Gause type predator-prey modelin which two ecological phenomena are considered: the Allee effect affectingthe prey growth function and the formation of group defence by prey in orderto avoid the predation. We prove the existence of a separatrix curves in the phase plane, determinedby the stable manifold of the equilibrium point associated to the Alleeeffect, implying that the solutions are highly sensitive to the initialconditions. Trajectories starting at one side of this separatrix curve have theequilibrium point $(0,0)$ as their $\omega $-limit, while trajectoriesstarting at the other side will approach to one of the following threeattractors: a stable limit cycle, a stable coexistence point or the stableequilibrium point $(K,0)$ in which the predators disappear andprey attains their carrying capacity. We obtain conditions on the parameter values for the existence of one or twopositive hyperbolic equilibrium points and the existence of a limit cyclesurrounding one of them. Both ecological processes under study, namely thenonmonotonic functional response and the Allee effect on prey, exert astrong influence on the system dynamics, resulting in multiple domains ofattraction. Using Liapunov quantities we demonstrate the uniqueness of limit cycle, whichconstitutes one of the main differences with the model where the Alleeeffect is not considered. Computer simulations are also given in support ofthe conclusions. |
| format | Article |
| id | doaj-art-1f989f367eb749c18a3857103cde0365 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2012-12-01 |
| publisher | AIMS Press |
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| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-1f989f367eb749c18a3857103cde03652025-01-24T02:25:53ZengAIMS PressMathematical Biosciences and Engineering1551-00182012-12-0110234536710.3934/mbe.2013.10.345Uniqueness of limit cycles and multiple attractors in a Gause-typepredator-prey model with nonmonotonic functional response and Allee effecton preyEduardo González-Olivares0Betsabé González-Yañez1Jaime Mena-Lorca2José D. Flores3Grupo de Ecología Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, ValparaísoGrupo de Ecología Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, ValparaísoGrupo de Ecología Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, ValparaísoGrupo de Ecología Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, ValparaísoThe main purpose of this work is to analyze a Gause type predator-prey modelin which two ecological phenomena are considered: the Allee effect affectingthe prey growth function and the formation of group defence by prey in orderto avoid the predation. We prove the existence of a separatrix curves in the phase plane, determinedby the stable manifold of the equilibrium point associated to the Alleeeffect, implying that the solutions are highly sensitive to the initialconditions. Trajectories starting at one side of this separatrix curve have theequilibrium point $(0,0)$ as their $\omega $-limit, while trajectoriesstarting at the other side will approach to one of the following threeattractors: a stable limit cycle, a stable coexistence point or the stableequilibrium point $(K,0)$ in which the predators disappear andprey attains their carrying capacity. We obtain conditions on the parameter values for the existence of one or twopositive hyperbolic equilibrium points and the existence of a limit cyclesurrounding one of them. Both ecological processes under study, namely thenonmonotonic functional response and the Allee effect on prey, exert astrong influence on the system dynamics, resulting in multiple domains ofattraction. Using Liapunov quantities we demonstrate the uniqueness of limit cycle, whichconstitutes one of the main differences with the model where the Alleeeffect is not considered. Computer simulations are also given in support ofthe conclusions.https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.345stabilityallee effectnon-monotonic functional responsepredator-prey modelbifurcationlimit cycle. |
| spellingShingle | Eduardo González-Olivares Betsabé González-Yañez Jaime Mena-Lorca José D. Flores Uniqueness of limit cycles and multiple attractors in a Gause-typepredator-prey model with nonmonotonic functional response and Allee effecton prey Mathematical Biosciences and Engineering stability allee effect non-monotonic functional response predator-prey model bifurcation limit cycle. |
| title | Uniqueness of limit cycles and multiple attractors in a Gause-typepredator-prey model with nonmonotonic functional response and Allee effecton prey |
| title_full | Uniqueness of limit cycles and multiple attractors in a Gause-typepredator-prey model with nonmonotonic functional response and Allee effecton prey |
| title_fullStr | Uniqueness of limit cycles and multiple attractors in a Gause-typepredator-prey model with nonmonotonic functional response and Allee effecton prey |
| title_full_unstemmed | Uniqueness of limit cycles and multiple attractors in a Gause-typepredator-prey model with nonmonotonic functional response and Allee effecton prey |
| title_short | Uniqueness of limit cycles and multiple attractors in a Gause-typepredator-prey model with nonmonotonic functional response and Allee effecton prey |
| title_sort | uniqueness of limit cycles and multiple attractors in a gause typepredator prey model with nonmonotonic functional response and allee effecton prey |
| topic | stability allee effect non-monotonic functional response predator-prey model bifurcation limit cycle. |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.345 |
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