A nonstandard finite difference scheme for solving three-species food chain with fractional-order Lotka-Volterra model

Iranian Journal of Numerical Analysis and Optimization

مقاله 3، دوره 6، شماره 1 - شماره پیاپی 9، اردیبهشت 2016، صفحه 53-79   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.v6i1.41619

نویسندگان

S. Zibaei؛ M. Namjoo^*

Department of Mathematics, School of Mathematical Sciences, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.

چکیده

‎‎In this paper‎, ‎we introduce fractional-order for a model of tritrophic food chain Lotka-Volterra‎. ‎Moreover‎, ‎we discuss the stability analysis of fractional system‎. ‎The nonstandard finite difference (NSFD) scheme is implemented‎ ‎to study the dynamic behaviors in the fractional-order Lotka-Volterra system‎. ‎Numerical results show that the‎ ‎NSFD approach is easy to implement and accurate when applied to fractional -order Lotka-Volterra system‎.

کلیدواژه‌ها

Fractional differential equations؛ Lotka-Volterra model؛ prey-predator system؛ Nonstandard finite difference scheme؛ ‎ ‎Stability

مراجع

1. Ahmed, E., El-Sayed, A.M.A. and El-Saka, H.A.A. Equilibrium points, stability and numerical solutions of fractionalorder predator-prey and rabies models, J. Math. Anal. Appl. 325 (1) (2007) 542-553.

2. Ahmed, E., El-Sayed, A.M.A. and El-Saka, H.A.A. On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rossler, Chua and Chen systems, Phys. Lett. A

358 (2006) 1-4.

3. Ahmad, S. and Lazer, A.C. Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system, Nonlinear Anal. 40 (1)(2000) 37-49.

4. Arenas, A.J., Gonzalez-Parra, G. and Chen-Charpentier, B.M. A nonstandard numerical scheme of predictor-corrector type for epidemic models, Comput. Math. Appl. 59 (12) (2010) 3740-3749.

5. Chauvet, E., Paullet, J.E., Previte, J.P. and Walls, Z. A Lotka-Volterra three species food chain, Math. Mag. (75) (2002) 243-255.

6. El-Saka, H.A., Ahmed, E., Shehata, M.I. and El-Sayed, A.M.A. On stability, persistence, and Hopf bifurcation in fractional order dynamical systems, Nonlinear Dynam. 56 (1-2) (2009) 121-126.

7. El-Sayed, A.M.A., El-Mesiry, A.E.M. and El-Saka, H.A.A. On the fractional-order logistic equation, Appl. Math. Lett. 20 (7) (2007) 817-823.

8. Jordan, P.M. A nonstandard finite difference scheme for nonlinear heat transfer in a thin finite rod, J. Difference Equ. Appl. 9 (11) (2003) 1015-1021.

9. Kaslik, E. and Sivasundaram, S. Nonlinear dynamics and chaos in fractional-order neural networks, Neural Networks 32 (2012) 245-256.

10. Liao, C. and Ding X. Nonstandard finite difference variational integrators for multisymplectic PDEs, J. Appl. Math. 2012 (2012) Article ID 705179, 22 pages.

11. Matignon, D. Stability result on fractional differential equations with applications to control processing, Computational engineering in systems applications (1996) 963-968.

12. Mehmat, A.A., Secer, A. and Bayram, M. Stability, synchronization control and numerical solution of fractional Shimizu-Morioka dynamical system, Appl. Math. Inf. Sci. 8 (14) (2014) 1699-1705.

13. Mickens, R.E. Advances in the Applications of Nonstandard Finite Difference Schemes, Wiley-Interscience, Singapore, 2005.

14. Mickens, R.E. Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Numer. Methods Partial Differential Equations 23 (3) (2007)

672-691.

15. Murray, J.D. Mathematical Biology I, II, Third edition, Springer, 2003.

16. Podlubny, I. Fractional Differential Equations, Academic Press, New York, 1999.

17. Roeger, L.W. Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite difference schemes, Discrete Contin. Dyn. Syst. Ser. B 9 (2) (2008) 415-429.

18. Roeger, L.W. Local stability of Eulers and Kahans methods, J. Difference Equ. Appl. 10 (6) (2004) 601-614.

19. Van Den Driessche, P. and Zeeman, M.L. Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math. 58(1) (1998) 227-234.

20. Yan, X.P. and Li, W.T. Stability and Hopf bifurcation for a delayed cooperative system with diffusion effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 18 (2) (2008) 441-453.

21. Zibaei, S. and Namjoo, M. A NSFD Scheme for Lotka-Volterra Food Web Model, Iran. J. Sci. Technol. Trans. A Sci. 38 (4) (2014) 399-414.