Dynamical Behavior and Stability Analysis in a Stage-Structured Prey Predator Model with Discrete Delay and Distributed Delay
We propose a prey predator model with stage structure for prey. A discrete delay and a distributed delay for predator described by an integral with a strong delay kernel are also considered. Existence of two feasible boundary equilibria and a unique interior equilibrium are analytically investigated...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/184174 |
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| _version_ | 1832558694025396224 |
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| author | Chao Liu Qingling Zhang |
| author_facet | Chao Liu Qingling Zhang |
| author_sort | Chao Liu |
| collection | DOAJ |
| description | We propose a prey predator model with stage structure for prey. A discrete delay and a distributed delay for predator described by an integral with a strong delay kernel are also considered. Existence of two feasible boundary equilibria and a unique interior equilibrium are analytically investigated. By analyzing associated characteristic equation, local stability analysis of boundary equilibrium and interior equilibrium is discussed, respectively. It reveals that interior equilibrium is locally stable when discrete delay is less than a critical value. According to Hopf bifurcation theorem for functional differential equations, it can be found that model undergoes Hopf bifurcation around the interior equilibrium when local stability switch occurs and corresponding stable limit cycle is observed. Furthermore, directions of Hopf bifurcation and stability of the bifurcating periodic solutions are studied based on normal form theory and center manifold theorem. Numerical simulations are carried out to show consistency with theoretical analysis. |
| format | Article |
| id | doaj-art-e88c59a94db74b5cad13b1755f9dee66 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-e88c59a94db74b5cad13b1755f9dee662025-02-03T01:31:51ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/184174184174Dynamical Behavior and Stability Analysis in a Stage-Structured Prey Predator Model with Discrete Delay and Distributed DelayChao Liu0Qingling Zhang1Institute of Systems Science, Northeastern University, Shenyang 110004, ChinaInstitute of Systems Science, Northeastern University, Shenyang 110004, ChinaWe propose a prey predator model with stage structure for prey. A discrete delay and a distributed delay for predator described by an integral with a strong delay kernel are also considered. Existence of two feasible boundary equilibria and a unique interior equilibrium are analytically investigated. By analyzing associated characteristic equation, local stability analysis of boundary equilibrium and interior equilibrium is discussed, respectively. It reveals that interior equilibrium is locally stable when discrete delay is less than a critical value. According to Hopf bifurcation theorem for functional differential equations, it can be found that model undergoes Hopf bifurcation around the interior equilibrium when local stability switch occurs and corresponding stable limit cycle is observed. Furthermore, directions of Hopf bifurcation and stability of the bifurcating periodic solutions are studied based on normal form theory and center manifold theorem. Numerical simulations are carried out to show consistency with theoretical analysis.http://dx.doi.org/10.1155/2014/184174 |
| spellingShingle | Chao Liu Qingling Zhang Dynamical Behavior and Stability Analysis in a Stage-Structured Prey Predator Model with Discrete Delay and Distributed Delay Abstract and Applied Analysis |
| title | Dynamical Behavior and Stability Analysis in a Stage-Structured Prey Predator Model with Discrete Delay and Distributed Delay |
| title_full | Dynamical Behavior and Stability Analysis in a Stage-Structured Prey Predator Model with Discrete Delay and Distributed Delay |
| title_fullStr | Dynamical Behavior and Stability Analysis in a Stage-Structured Prey Predator Model with Discrete Delay and Distributed Delay |
| title_full_unstemmed | Dynamical Behavior and Stability Analysis in a Stage-Structured Prey Predator Model with Discrete Delay and Distributed Delay |
| title_short | Dynamical Behavior and Stability Analysis in a Stage-Structured Prey Predator Model with Discrete Delay and Distributed Delay |
| title_sort | dynamical behavior and stability analysis in a stage structured prey predator model with discrete delay and distributed delay |
| url | http://dx.doi.org/10.1155/2014/184174 |
| work_keys_str_mv | AT chaoliu dynamicalbehaviorandstabilityanalysisinastagestructuredpreypredatormodelwithdiscretedelayanddistributeddelay AT qinglingzhang dynamicalbehaviorandstabilityanalysisinastagestructuredpreypredatormodelwithdiscretedelayanddistributeddelay |