Turing–Hopf Bifurcation Analysis in a Diffusive Ratio-Dependent Predator–Prey Model with Allee Effect and Predator Harvesting
Abstract
:1. Introduction
2. Dynamics of the ODE Model
- (i)
- is a stable node;
- (ii)
- If , then is a source, and is a saddle;
- (iii)
- If , then is a saddle, and is a stable node.
- (i)
- is unstable;
- (ii)
- If holds, then there exists a unique such that is asymptotically stable when , while is unstable when , where is the positive zero of (11).
- (i)
- If , the bifurcating periodic solutions are orbitally asymptotically stable, and periodic solutions occur as h decreases and passes .
- (ii)
- If , the bifurcating periodic solutions are unstable, and periodic solutions occur as h decreases and passes .
3. Turing Instability Induced by Diffusion and Turing–Hopf Bifurcation
- (i)
- (ii)
- The Turing–Hopf bifurcation occurs at when , where
4. Normal Forms for Turing–Hopf Bifurcation
5. Numerical Simulations
- ,
- , for ,
- , for ,
- ,
6. Conclusions and Discussion
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Region | Steady States | Stability of the Steady States |
---|---|---|
, | is locally asymptotically stable. | |
, , | is locally asymptotically stable; is unstable. | |
, , | are locally asymptotically stable; , are unstable. | |
, , , | are locally asymptotically stable; and are unstable. | |
, , | are locally asymptotically stable; are unstable. | |
, | are locally asymptotically stable; is unstable. |
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Chen, M.; Xu, Y.; Zhao, J.; Wei, X. Turing–Hopf Bifurcation Analysis in a Diffusive Ratio-Dependent Predator–Prey Model with Allee Effect and Predator Harvesting. Entropy 2024, 26, 18. https://doi.org/10.3390/e26010018
Chen M, Xu Y, Zhao J, Wei X. Turing–Hopf Bifurcation Analysis in a Diffusive Ratio-Dependent Predator–Prey Model with Allee Effect and Predator Harvesting. Entropy. 2024; 26(1):18. https://doi.org/10.3390/e26010018
Chicago/Turabian StyleChen, Meiyao, Yingting Xu, Jiantao Zhao, and Xin Wei. 2024. "Turing–Hopf Bifurcation Analysis in a Diffusive Ratio-Dependent Predator–Prey Model with Allee Effect and Predator Harvesting" Entropy 26, no. 1: 18. https://doi.org/10.3390/e26010018
APA StyleChen, M., Xu, Y., Zhao, J., & Wei, X. (2024). Turing–Hopf Bifurcation Analysis in a Diffusive Ratio-Dependent Predator–Prey Model with Allee Effect and Predator Harvesting. Entropy, 26(1), 18. https://doi.org/10.3390/e26010018