The dynamics of a simple Laissez-Faire model with two predators
In this paper, we study the dynamics of a laissez-faire predator--prey model with both a specialist and a generalist predator. We analyze the stabilities of equilibria by performing linearized stability analyses. We then reexamine the stability of the equilibrium where the prey and predator coexist...
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| Format: | Article |
| Language: | English |
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AIMS Press
2008-11-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2009.6.145 |
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| _version_ | 1832590163374505984 |
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| author | Gunog Seo Mark Kot |
| author_facet | Gunog Seo Mark Kot |
| author_sort | Gunog Seo |
| collection | DOAJ |
| description | In this paper, we study the dynamics of a laissez-faire predator--prey model with both a specialist and a generalist predator. We analyze the stabilities of equilibria by performing linearized stability analyses. We then reexamine the stability of the equilibrium where the prey and predator coexist by constructing a Lyapunov function.If we hold the generalist predator population constant, treating it as a bifurcation parameter, we show that our model can possess multiple (up to three) limit cycles that surround an equilibrium in the interior of the first quadrant. Our model shows rich dynamics including fold, transcritical, pitchfork, Hopf, cyclic-fold, and Bautin bifurcations as well as heteroclinic connections.If we instead vary the generalist predator population slowly across bifurcations, the model exhibits bursting behavior as it alternates between a repetitive spiking phase and a quiescent phase. |
| format | Article |
| id | doaj-art-06d07e86eaae4567a95f69787c597c8e |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2008-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-06d07e86eaae4567a95f69787c597c8e2025-01-24T01:58:53ZengAIMS PressMathematical Biosciences and Engineering1551-00182008-11-016114517210.3934/mbe.2009.6.145The dynamics of a simple Laissez-Faire model with two predatorsGunog Seo0Mark Kot1University of Washington, Applied Mathematics, Box 352420, Seattle, WA 98195-2420University of Washington, Applied Mathematics, Box 352420, Seattle, WA 98195-2420In this paper, we study the dynamics of a laissez-faire predator--prey model with both a specialist and a generalist predator. We analyze the stabilities of equilibria by performing linearized stability analyses. We then reexamine the stability of the equilibrium where the prey and predator coexist by constructing a Lyapunov function.If we hold the generalist predator population constant, treating it as a bifurcation parameter, we show that our model can possess multiple (up to three) limit cycles that surround an equilibrium in the interior of the first quadrant. Our model shows rich dynamics including fold, transcritical, pitchfork, Hopf, cyclic-fold, and Bautin bifurcations as well as heteroclinic connections.If we instead vary the generalist predator population slowly across bifurcations, the model exhibits bursting behavior as it alternates between a repetitive spiking phase and a quiescent phase.https://www.aimspress.com/article/doi/10.3934/mbe.2009.6.145laissez-fairepredator--predator--prey modelburstingheteroclinicconnectionbautin bifurcation |
| spellingShingle | Gunog Seo Mark Kot The dynamics of a simple Laissez-Faire model with two predators Mathematical Biosciences and Engineering laissez-fairepredator--predator--prey model bursting heteroclinicconnection bautin bifurcation |
| title | The dynamics of a simple Laissez-Faire model with two predators |
| title_full | The dynamics of a simple Laissez-Faire model with two predators |
| title_fullStr | The dynamics of a simple Laissez-Faire model with two predators |
| title_full_unstemmed | The dynamics of a simple Laissez-Faire model with two predators |
| title_short | The dynamics of a simple Laissez-Faire model with two predators |
| title_sort | dynamics of a simple laissez faire model with two predators |
| topic | laissez-fairepredator--predator--prey model bursting heteroclinicconnection bautin bifurcation |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2009.6.145 |
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