Volume 15, Issue 1 (4-2020)
IJMSI 2020, 15(1): 135-159 |
Back to browse issues page
Abstract:
A mathematical model describing the dynamics of a delayed stage structure prey - predator system with prey refuge is considered. The existence, uniqueness and bounded- ness of the solution are discussed. All the feasibl e equilibrium points are determined. The stability analysis of them are investigated. By employ ing the time delay as the bifurcation parameter, we observed the existence of Hopf bifurcation at the positive equilibrium. The stability and direction of the Hopf bifurcation are determined by utilizing the normal form method and the center manifold reduction. Numerical simulations are given to support the analytic results.
Type of Study: Research paper |
Subject:
Special
References
1. H.I. Freedman, P. Waltman,, Persistence in models of three interacting predator prey populations, Math. Biosci., 68, (1984), 213-231. [DOI:10.1016/0025-5564(84)90032-4]
2. Y. Xiao, L. Chen, Modelling and analysis of predator- prey with disease in the prey,Mathematical Biosciences, 171(1), (2001), 59-82. [DOI:10.1016/S0025-5564(01)00049-9]
3. B. Dubey, R.K. Upadhyay, Persistence and Extinction of One-Prey and Two-Predators System, Nonlinear Analysis: Modelling and Control, 9(4),307-329. (2004). [DOI:10.15388/NA.2004.9.4.15147]
4. K.P. Das, A Mathematical Study of a predator-prey Dynamics with Disease in Predator, International Scholarly Research Network ISRN Applied Mathematics, (2011), 669-684. [DOI:10.5402/2011/807486]
5. S. Sharma, G. P. Samanta, A ratio-dependent predator-prey model with Allee eect and disease in prey, J. Appl. Math. Comput., 47, (2015), 345-364. [DOI:10.1007/s12190-014-0779-0]
6. R. K. Naji and R. M. Hussien, The Dynamics of Epidemic Model with Two Types of Infectious Diseases and Vertical Transmission, Journal of applied mathematics, 2016, (2016). [DOI:10.1155/2016/4907964]
7. A. J. Lotka, Elements of physical biology, Williams & Wilkins, Princeton, N. J., 1925.
8. V. Volterra, Variazioni e uttuazioni del numero dindividui in specie conviventi, Mem.Accad. Lincei Roma, 2,(1926), 31-113.
9. W. Wang, L. Chen, A predator-prey system with stage-structure for predator, Computers Math., Applic., 33(8), 83-91, (1997). [DOI:10.1016/S0898-1221(97)00056-4]
10. J. A. Cui, Y. Takeuchi, A predator-prey system with a stage structure for the prey, Mathematical and Computer Modelling, 11, (2006), 26-1132.
11. X. Song, L. Chen, Optimal harvesting and stability for a two-species competitive system with stage structure, Mathematical Biosciences, 170, (2001), 173-186. [DOI:10.1016/S0025-5564(00)00068-7]
12. T.K. Kar, U.K. Pahari, Modelling and analysis of a prey-predator system with stage structure and harvesting, Nonlinear Analysis: Real World Applications, 8, (2007), 601609. [DOI:10.1016/j.nonrwa.2006.01.004]
13. R. Shi and L. Chen, The study of a ratio-dependent predator-prey model with stage structure in the prey, Nonlinear Dyn, 58, (2009), 443-451. [DOI:10.1007/s11071-009-9491-2]
14. S. Xu., Dynamics of a general prey-predator model with prey-stage structure and diusive effects, Computers & Mathematics with Applications, 68(3),(2014), 405-423. [DOI:10.1016/j.camwa.2014.06.016]
15. P. Hao, J. Wei and D. Fan, Analysis of dynamics in an eco-epidemiological model with stage structure, Advances in Difference Equations 2016(1),(2016). [DOI:10.1186/s13662-016-0956-6]
16. Raid K. Naji and Salam J. Majeed, The Dynamical Analysis of a Prey-Predator Model with a Refuge-Stage Structure Prey Population, International Journal of Differential Equations, 2016, (2016). [DOI:10.1155/2016/2010464]
17. W.G. Aiello and H. I. Freedman,A time-delay model of single-species growth with stage structure, Mathematical Biosciences, 101, (1990), 139-153. [DOI:10.1016/0025-5564(90)90019-U]
18. Y. Kuang, Delay differential equations with applications in population dynamics, Boston: Academic Press , 1993.
19. K. Ye, X. Song, Predator-prey system with stage structure and delay, Appl. Math. J. Chinese Univ. Ser. B, 18(2), (2003), 143-150. [DOI:10.1007/s11766-003-0018-1]
20. M. Bandyopadhyaya and S. Banerjee,A stage-structured prey-predator model with discrete time delay, Applied Mathematics and Computation, 182,1385-1398.(2006). [DOI:10.1016/j.amc.2006.05.025]
21. Y. Chen and S. Changming, Stability and Hopf bifurcation analysis in a prey-predator system with stage-structure for prey and time delay, Chaos, Solitons and Fractals, 38,1104-1114.(2008). [DOI:10.1016/j.chaos.2007.01.035]
22. L. Wang, R. Xu, G. Feng, A stage-structured prey-predator system with time delay, J. Appl. Math. Comput., 33, (2010), 267-281. [DOI:10.1007/s12190-009-0286-x]
23. H. L. Smith, An introduction to delay di erential equations with applications to the life sciences, Texts in Applied Mathematics. New York: Springer, 2011. [DOI:10.1007/978-1-4419-7646-8_1]
24. X. K. Sun, H. F, Huo and X. B. Zhang, A Predator-Prey Model with Functional Response and Stage Structure for Prey, Abstract and Applied Analysis, 2012, (2012). [DOI:10.1155/2012/628103]
25. Q. Gao and Z. Jin, A Time Delay Predator-Prey System with Three-Stage-Structure, The Scienti c World Journal,2014, (2014). [DOI:10.1155/2014/512838]
26. J.F.M. Al-Omari, A stage-structured predator-prey model with distributed maturation delay and harvesting, journal of biological dynamics, 9(1),278-287.(2015). [DOI:10.1080/17513758.2015.1088080]
27. B. Hassard, N. Kazarino and Y. H. Wan, Theory and applications of Hopf bifurcation. London mathematical society lecture note series, Cambridge University Press, 41.(1981).
| Rights and permissions | |
 |
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. |