Spatiotemporal Dynamics of a Predator–Prey Model with Harvest and Disease in Prey
Abstract
1. Introduction
2. Dynamics of the Temporal Model
2.1. Boundedness and Equilibria
- (i)
- The trivial equilibrium .
- (ii)
- The boundary equilibrium , which is feasible since in a biological sense.
- (iii)
- The infection prey free equilibrium ; where and is a positive root of cubic equation , the equilibrium is feasible for .
- (iv)
- The predator free equilibrium ; where , the equilibrium is feasible for .
- (v)
- The coexistence equilibrium , where , and is a positive root of equation
2.2. Local Asymptotic Stability and Hopf Bifurcation
- (i)
- is always unstable.
- (ii)
- is locally stable when , unstable when .
- (iii)
- is stable when unstable when or
- (iv)
- is locally stable when , unstable when .
3. Dynamics of the Spatiotemporal Model (3)
3.1. A Prior Estimates of Positive Solutions
- (i)
- If satisfies in on and , then .
- (ii)
- If satisfies in on and , then .
3.2. Non-Existence of Non-Constant Positive Steady State
- (i)
- are the eigenvalues of operator on under the homogeneous Neumann boundary condition.
- (ii)
- is the eigenspace corresponding to the eigenvalue .
- (iii)
- , where are the orthonormal basis of for .
- (iv)
- , and so
3.3. Existence of Non-Constant Positive Steady State
4. Discussion and Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Yang, J.; Zhao, Z.; Kong, Y.; Xu, J. Spatiotemporal Dynamics of a Predator–Prey Model with Harvest and Disease in Prey. Mathematics 2025, 13, 2474. https://doi.org/10.3390/math13152474
Yang J, Zhao Z, Kong Y, Xu J. Spatiotemporal Dynamics of a Predator–Prey Model with Harvest and Disease in Prey. Mathematics. 2025; 13(15):2474. https://doi.org/10.3390/math13152474
Chicago/Turabian StyleYang, Jingen, Zhong Zhao, Yingying Kong, and Jing Xu. 2025. "Spatiotemporal Dynamics of a Predator–Prey Model with Harvest and Disease in Prey" Mathematics 13, no. 15: 2474. https://doi.org/10.3390/math13152474
APA StyleYang, J., Zhao, Z., Kong, Y., & Xu, J. (2025). Spatiotemporal Dynamics of a Predator–Prey Model with Harvest and Disease in Prey. Mathematics, 13(15), 2474. https://doi.org/10.3390/math13152474