---

Journal of Applied Nonlinear Dynamics 9(3) (2020) 401--413 | DOI:10.5890/JAND.2020.09.005

Md. Nazmul Hasan$^{1}$, Md. Sharif Uddin$^{1}$, Md. Haider Ali Biswas$^{2}$

$^{1}$ Department of Mathematics, Jahangirnagar University, Saver, Dhaka, Bangladesh

$^{2}$ Mathematics Discipline, khulna University, Khulna, Bangladesh

Download Full Text PDF

 

Abstract

This paper deals with predator-prey eco-epidemiological model where an infectious disease among fish population is non-linearly transmitted in a reserve area. Then the disease is transmitted indirectly to the predator population during their feeding process. At the time of harvesting, both the susceptible and the infected prey population are harvested. When the nonlinear disease transmission and the inhibition effect on the dynamical behavior of the prey are measured, the number of infected individuals increases. The model in this study is analyzed in term of boundedness, and local and global stability under certain conditions and Hopf bifurcation. For the determine of optimal conditions, we have compared the theoretical results with numerical results for different sets of parameters.

Acknowledgments

The first author gratefully acknowledges the financial support provided by the University Grant Commission, Bangladesh (UGC/1,157/ M.Phil and PhD/2016/5343).

References

1. [1]	Ali, N. and Chakravarty, S. (2015), Stability analysis of a food chain model consisting of two competitive preys and one predator, Nonlinear Dyn., 82(3), 1303-1316.

2. [2]	Paker, C., Holt, D.R., Hudson, P.J., Lafferty, K.D. and Dobson, A.P. (2003), Keeping herds healthy and alert: implication of predator contr,ol for infectious disease, Ecol. Lett., 6, 797-802.

3. [3]	Chattapadhyay, J. and Arino, O. (1999), A predator-prey model with disease in the prey, Nonlinear Anal., 36, 747-766.

4. [4]	Lafferty, K.D. and Moris, A.K. (1996), Altered behavior of parasitised killifish in-creases susceptibility to predation by bird final hosts, Ecology, 77, 1390-1397.

5. [5]	Anderson, R.M. and May, R.M. (1982), Population Biology of Infectious Disease, Springer, Berlin.

6. [6]	Anderson, R.M. and May, R.M. (1991) Infectous Disease of Humans, Dynamics and Control, Oxford University Press, Oxford.

7. [7]	Aziz-Alaoui, M.A. (2002), Study of a Leslie-Gower-type tritrophic population model, Chaos Solitons Fractals, 14(8), 1275-1293.

8. [8]	Aziz-Alaoui, M.A. and Okiye, M.D. (2003), Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling type II schemes, Appl. Math. Lett., 16, 1069-1075.

9. [9]	Birkhoff, G. and Rota, G.C. (1982), Ordinary Differential Equations, Ginn, Boston (1999).

10. [10]	Dubey, B., Sharma, S., Sinha, P., and Shukla, J. (2009), Modelling the depletiopn of forestry resources by population and population pressure augmented industrialization, Appl. Math. Model, 33(7), 3002-3014.

11. [11]	Biswas, M.H.A. and de Pinho, M.R. (2013), A Maximum Principle for Optimal Control Problems with State and Mixed Constraints, Submitted to ESAIM: Control, Optimization and Calculus of Variations.

12. [12]	Biswas, M.H.A. (2011), A Necessary conditions for optimal control problems with and without state con- straints: a comparative study, WSEAS Transactions on Systems and Control, 6(6), 217-228.

13. [13]	Biswas, M.H.A., and de Pinho, M.R. (2011), A NonsmoothMaximum Principle for Optimal Control Problems with State and Mixed Constraints-Convex Case, Discrete and Continuous Dynamical Systems, (Special), 174- 183.

14. [14]	Biswas, M.H.A. and de Pinho, M.R. (2012), A variant of nonsmooth maximum principle for state constrained problems, IEEE 51st IEEE Conference on Decision and Control (CDC), 7685-7690.

15. [15]	Biswas, M.H.A. and Haque, M.M. (2016), Nonlinear Dynamical systems in Modeling and Control of Infectious Disease, Book chapter of differential and difference equations with applications, (164), 149-158.

16. [16]	Chattopadhyay, J., Pal, S., and Abdllaoui, A.E. (2003) Classical predator-prey system with infection of prey population-a mathematical model, Math. Methods Appl. Sci., 26, 1211-1222.

17. [17]	Das, K.P., Roy, S., and Chattapadhyay, J. (2009), Effect of disease-selective predation on prey infected by contact and external sources, Biosystems, 95(3), 188-199.