Dynamics Behavior of a Predator-Prey Diffusion Model Incorporating Hunting Cooperation and Predator-Taxis
Abstract
:1. Introduction
2. Global Existence of Classical Solution
2.1. Local Existence and Preliminary
- (i)
- There exists a positive constant such that model (2) admits a unique local-in-time classical solution , which satisfies;
- (ii)
- There exists a constant such that;
- (iii)
- There exists a constant such that;
- (iv)
- If , we have.
2.2. Global Existence of Solution
3. Stability/Instability Caused by Predator-Taxis
- (i)
- If , then for all , is always locally asymptotically stable;
- (ii)
- If and , then is locally asymptotically stable when and it is unstable when ;
- (iii)
- If , then is locally asymptotically stable when and it is unstable when or .
- (i)
- If and , it is evident that for any , then exhibits locally asymptotically stability;
- (ii)
- If , together with , one has for any . When , then for any , thus is locally asymptotically stability. When , we have for some , which implies is unstable;
- (iii)
- If , then is unstable for the corresponding reaction–diffusion model. From , it follows that for any . When , we observe that for any , and then is locally asymptotically stability. When and , we have for some , then is unstable.
- Next, we show that can reach its maximum on . Set
4. Numerical Simulations
5. Concluding Remarks
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Zhang, H. Dynamics Behavior of a Predator-Prey Diffusion Model Incorporating Hunting Cooperation and Predator-Taxis. Mathematics 2024, 12, 1474. https://doi.org/10.3390/math12101474
Zhang H. Dynamics Behavior of a Predator-Prey Diffusion Model Incorporating Hunting Cooperation and Predator-Taxis. Mathematics. 2024; 12(10):1474. https://doi.org/10.3390/math12101474
Chicago/Turabian StyleZhang, Huisen. 2024. "Dynamics Behavior of a Predator-Prey Diffusion Model Incorporating Hunting Cooperation and Predator-Taxis" Mathematics 12, no. 10: 1474. https://doi.org/10.3390/math12101474
APA StyleZhang, H. (2024). Dynamics Behavior of a Predator-Prey Diffusion Model Incorporating Hunting Cooperation and Predator-Taxis. Mathematics, 12(10), 1474. https://doi.org/10.3390/math12101474