Control optimization and homoclinic bifurcation of a prey–predator model with ratio-dependent
Abstract In this paper, a predator–prey model with ratio-dependent and impulsive state feedback control is constructed, where the pest growth rate is related to an Allee effect. Firstly, the existence condition of the homoclinic cycle is obtained by analyzing the control parameter q. The existence,...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2019-01-01
|
| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13662-018-1933-z |
| _version_ | 1828399747693543424 |
|---|---|
| author | Zhenzhen Shi Jianmei Wang Qingjian Li Huidong Cheng |
| author_facet | Zhenzhen Shi Jianmei Wang Qingjian Li Huidong Cheng |
| author_sort | Zhenzhen Shi |
| collection | DOAJ |
| description | Abstract In this paper, a predator–prey model with ratio-dependent and impulsive state feedback control is constructed, where the pest growth rate is related to an Allee effect. Firstly, the existence condition of the homoclinic cycle is obtained by analyzing the control parameter q. The existence, uniqueness and asymptotic stability of the periodic orbit are discussed by using the geometric theory of the differential equations, the method of successor functions and analog of the Poincaré criterion. Secondly, we formulate a control optimization with a minimal total cost in pest management, and we obtain an optimal economic threshold. Finally, we verify the main results by numerical simulation. |
| first_indexed | 2024-12-10T09:21:55Z |
| format | Article |
| id | doaj.art-1712738c1d7c4edea90567b8096f276b |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-10T09:21:55Z |
| publishDate | 2019-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-1712738c1d7c4edea90567b8096f276b2022-12-22T01:54:40ZengSpringerOpenAdvances in Difference Equations1687-18472019-01-012019111410.1186/s13662-018-1933-zControl optimization and homoclinic bifurcation of a prey–predator model with ratio-dependentZhenzhen Shi0Jianmei Wang1Qingjian Li2Huidong Cheng3College of Mathematics and Systems Science, Shandong University of Science and TechnologyCollege of Mathematics and Systems Science, Shandong University of Science and TechnologyCollege of Foreign Languages, Shandong University of Science and TechnologyCollege of Mathematics and Systems Science, Shandong University of Science and TechnologyAbstract In this paper, a predator–prey model with ratio-dependent and impulsive state feedback control is constructed, where the pest growth rate is related to an Allee effect. Firstly, the existence condition of the homoclinic cycle is obtained by analyzing the control parameter q. The existence, uniqueness and asymptotic stability of the periodic orbit are discussed by using the geometric theory of the differential equations, the method of successor functions and analog of the Poincaré criterion. Secondly, we formulate a control optimization with a minimal total cost in pest management, and we obtain an optimal economic threshold. Finally, we verify the main results by numerical simulation.http://link.springer.com/article/10.1186/s13662-018-1933-zSemi-continuous dynamic systemsOrder one periodic orbitHomoclinic cycleSubsequence functionsOptimization |
| spellingShingle | Zhenzhen Shi Jianmei Wang Qingjian Li Huidong Cheng Control optimization and homoclinic bifurcation of a prey–predator model with ratio-dependent Advances in Difference Equations Semi-continuous dynamic systems Order one periodic orbit Homoclinic cycle Subsequence functions Optimization |
| title | Control optimization and homoclinic bifurcation of a prey–predator model with ratio-dependent |
| title_full | Control optimization and homoclinic bifurcation of a prey–predator model with ratio-dependent |
| title_fullStr | Control optimization and homoclinic bifurcation of a prey–predator model with ratio-dependent |
| title_full_unstemmed | Control optimization and homoclinic bifurcation of a prey–predator model with ratio-dependent |
| title_short | Control optimization and homoclinic bifurcation of a prey–predator model with ratio-dependent |
| title_sort | control optimization and homoclinic bifurcation of a prey predator model with ratio dependent |
| topic | Semi-continuous dynamic systems Order one periodic orbit Homoclinic cycle Subsequence functions Optimization |
| url | http://link.springer.com/article/10.1186/s13662-018-1933-z |
| work_keys_str_mv | AT zhenzhenshi controloptimizationandhomoclinicbifurcationofapreypredatormodelwithratiodependent AT jianmeiwang controloptimizationandhomoclinicbifurcationofapreypredatormodelwithratiodependent AT qingjianli controloptimizationandhomoclinicbifurcationofapreypredatormodelwithratiodependent AT huidongcheng controloptimizationandhomoclinicbifurcationofapreypredatormodelwithratiodependent |