Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect
We proposed and analyzed a predator–prey model with both the additive Allee effect and the fear effect in the prey. Firstly, we studied the existence and local stability of equilibria. Some sufficient conditions on the global stability of the positive equilibrium were established by applying the Dul...
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| Format: | Article |
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MDPI AG
2020-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/8/1280 |
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| author | Liyun Lai Zhenliang Zhu Fengde Chen |
| author_facet | Liyun Lai Zhenliang Zhu Fengde Chen |
| author_sort | Liyun Lai |
| collection | DOAJ |
| description | We proposed and analyzed a predator–prey model with both the additive Allee effect and the fear effect in the prey. Firstly, we studied the existence and local stability of equilibria. Some sufficient conditions on the global stability of the positive equilibrium were established by applying the Dulac theorem. Those results indicate that some bifurcations occur. We then confirmed the occurrence of saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Those theoretical results were demonstrated with numerical simulations. In the bifurcation analysis, we only considered the effect of the strong Allee effect. Finally, we found that the stronger the fear effect, the smaller the density of predator species. However, the fear effect has no influence on the final density of the prey. |
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| format | Article |
| id | doaj.art-f7eb4cb3d7ed4494a29829ed435d16e2 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T18:00:32Z |
| publishDate | 2020-08-01 |
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| series | Mathematics |
| spelling | doaj.art-f7eb4cb3d7ed4494a29829ed435d16e22023-11-20T08:57:05ZengMDPI AGMathematics2227-73902020-08-0188128010.3390/math8081280Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear EffectLiyun Lai0Zhenliang Zhu1Fengde Chen2College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, ChinaCollege of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, ChinaCollege of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, ChinaWe proposed and analyzed a predator–prey model with both the additive Allee effect and the fear effect in the prey. Firstly, we studied the existence and local stability of equilibria. Some sufficient conditions on the global stability of the positive equilibrium were established by applying the Dulac theorem. Those results indicate that some bifurcations occur. We then confirmed the occurrence of saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Those theoretical results were demonstrated with numerical simulations. In the bifurcation analysis, we only considered the effect of the strong Allee effect. Finally, we found that the stronger the fear effect, the smaller the density of predator species. However, the fear effect has no influence on the final density of the prey.https://www.mdpi.com/2227-7390/8/8/1280fear effectadditive allee effectsaddle-node bifurcationtranscritical bifurcationhopf bifucation |
| spellingShingle | Liyun Lai Zhenliang Zhu Fengde Chen Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect Mathematics fear effect additive allee effect saddle-node bifurcation transcritical bifurcation hopf bifucation |
| title | Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect |
| title_full | Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect |
| title_fullStr | Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect |
| title_full_unstemmed | Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect |
| title_short | Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect |
| title_sort | stability and bifurcation in a predator prey model with the additive allee effect and the fear effect |
| topic | fear effect additive allee effect saddle-node bifurcation transcritical bifurcation hopf bifucation |
| url | https://www.mdpi.com/2227-7390/8/8/1280 |
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