Stationary Pattern and Global Bifurcation for a Predator–Prey Model with Prey-Taxis and General Class of Functional Responses
Abstract
:1. Introduction
2. Stability and Pattern Formation
2.1. Linear Stability Analysis of the Local System (1)
2.2. Linear Stability Analysis of the Nonlocal System (2)
3. Global Bifurcation Analysis
- (1)
- for all .
- (2)
- The partial derivative exists and is continuous in near .
- (3)
- is closed for some , and .
- (4)
- where spans .Let be any complement of the space spanned by . Then, there exists an open interval and a continuously differentiable function such that AndMoreover, the entire solution set of near in V consists of the line and the curve .
- (5)
- In addition, if is a Fredholm operator for all , then the curve is contained in , which is a connected component of . Here, . Furthermore, either is not compact in V, or contains a point with
- (1)
- is unbounded in .
- (2)
- contains a point where
- (3)
- contains a point with .It is easy to see that the positive solution of System (29) must bifurcate from if and only if , so is excluded.
4. Numerical Analysis
5. Conclusions
- (1)
- By substituting the specific value proposed in numerical simulation, Figure 3 shows that is stable for the corresponding reaction–diffusion system without prey-taxis.
- (2)
- By simple calculation, we can discover that and . This means that remains locally asymptotically stable when System (1) possesses attractive () prey-taxis. can also be locally asymptotically stable when the system (1) possesses repulsive prey-taxis with (see Figure 5c,d). When , is unstable (see Figure 5a,b).
- (3)
- The numerical observations in Figure 1 are consistent with our theoretical results that the critical value of the necessary and sufficient conditions are almost similar when the study area is large enough (see Remark 2).
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Maimaiti, Y.; Zhang, W.; Muhammadhaji, A. Stationary Pattern and Global Bifurcation for a Predator–Prey Model with Prey-Taxis and General Class of Functional Responses. Mathematics 2023, 11, 4641. https://doi.org/10.3390/math11224641
Maimaiti Y, Zhang W, Muhammadhaji A. Stationary Pattern and Global Bifurcation for a Predator–Prey Model with Prey-Taxis and General Class of Functional Responses. Mathematics. 2023; 11(22):4641. https://doi.org/10.3390/math11224641
Chicago/Turabian StyleMaimaiti, Yimamu, Wang Zhang, and Ahmadjan Muhammadhaji. 2023. "Stationary Pattern and Global Bifurcation for a Predator–Prey Model with Prey-Taxis and General Class of Functional Responses" Mathematics 11, no. 22: 4641. https://doi.org/10.3390/math11224641
APA StyleMaimaiti, Y., Zhang, W., & Muhammadhaji, A. (2023). Stationary Pattern and Global Bifurcation for a Predator–Prey Model with Prey-Taxis and General Class of Functional Responses. Mathematics, 11(22), 4641. https://doi.org/10.3390/math11224641