An Eco-Epidemiological Model Involving Prey Refuge and Prey Harvesting with Beddington–DeAngelis, Crowley–Martin and Holling Type II Functional Responses †
Abstract
:1. Introduction
2. Mathematical Formulation of the Model
3. Positive Invariance and Boundedness
4. Boundary Equilibrium Points
- is the point of trivial Equilibrium. Here, (0, 0, 0) exists.
- , diseased prey and no predator Equilibria, exists for < r.
- is the equilibria with no predator, where = and= . exists for and .
- is the equilibria with no disease, where = and= . exists for and .
- is the equilibria of interior which is positive, by system (2) . It exists for , , , . Where,=,=,=.
5. Local Stability
- 1.
- The equilibria of trivial point is locally stable if ; otherwise, it is unstable.
- 2.
- The equilibria without infection and predator is locally asymptotically stable if , , .
- 3.
- The equilibria with no predator is locally asymptotically stable if , , and .
- The eigenvalues of are , , . Hence, it is locally asymptotically stable when . If not, it is unstable.
- The eigen values of are , ,. Hence, it is locally asymptotically stable if , , . If not, it is unstable.
- The Jacobian matrix is
6. Global Stability
7. Hopf Bifurcation Analysis
- 1.
- 2.
- where ρ is the zero of the characteristic equation, which equates to the equilibrium point’s positive value.
8. Numerical Simulation
9. Conclusions and Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Parameters | Environmental Illustration |
---|---|
Susceptible prey, Infected prey, Predator | |
, , | Infection rate, Growth rate of prey, refuge of prey |
K, , | Carrying capacity, Predator’s handling time, harvesting effort |
Half-saturation constant among infected prey and predators | |
Rate of predation on susceptible prey, Conversion of prey to predators | |
Magnitude of interference among predators by Crowley and Beddington | |
Capture rate by predator on susceptible prey | |
Death rate of infected prey and predators | |
, | Catchability coefficients of susceptible and infected prey |
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Thangavel, M.; Thangaraj, N.G.; Manickasundaram, S.P.; Arunachalam, Y. An Eco-Epidemiological Model Involving Prey Refuge and Prey Harvesting with Beddington–DeAngelis, Crowley–Martin and Holling Type II Functional Responses. Eng. Proc. 2023, 56, 325. https://doi.org/10.3390/ASEC2023-15812
Thangavel M, Thangaraj NG, Manickasundaram SP, Arunachalam Y. An Eco-Epidemiological Model Involving Prey Refuge and Prey Harvesting with Beddington–DeAngelis, Crowley–Martin and Holling Type II Functional Responses. Engineering Proceedings. 2023; 56(1):325. https://doi.org/10.3390/ASEC2023-15812
Chicago/Turabian StyleThangavel, Megala, Nandha Gopal Thangaraj, Siva Pradeep Manickasundaram, and Yasotha Arunachalam. 2023. "An Eco-Epidemiological Model Involving Prey Refuge and Prey Harvesting with Beddington–DeAngelis, Crowley–Martin and Holling Type II Functional Responses" Engineering Proceedings 56, no. 1: 325. https://doi.org/10.3390/ASEC2023-15812