Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect†
College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Mathematics 2020, 8(8), 1280; https://doi.org/10.3390/math8081280
Received: 12 July 2020 / Revised: 26 July 2020 / Accepted: 31 July 2020 / Published: 3 August 2020
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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Keywords:
fear effect; additive allee effect; saddle-node bifurcation; transcritical bifurcation; hopf bifucation
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MDPI and ACS Style
Lai, L.; Zhu, Z.; Chen, F. Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect. Mathematics 2020, 8, 1280. https://doi.org/10.3390/math8081280
AMA Style
Lai L, Zhu Z, Chen F. Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect. Mathematics. 2020; 8(8):1280. https://doi.org/10.3390/math8081280
Chicago/Turabian StyleLai, Liyun; Zhu, Zhenliang; Chen, Fengde. 2020. "Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect" Mathematics 8, no. 8: 1280. https://doi.org/10.3390/math8081280
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