Dynamic analysis of the fractional predator-prey system based on the Mittag-Leffler function

Document Type : Original Paper


Department of Mathematics, Sahand University of Technology, Tabriz, Iran


‎In this paper, the dynamic behavior of a fractional-order predator-prey system based on the Mittag-Leffler function is investigated. First, we study the existence, uniqueness, non-negativity, and boundedness for the solution of this fractional-order system. Then, we show that this system has two different equilibrium points. Some sufficient conditions to ensure the global asymmetric stability of these points are also proposed by using the Lyapunov function. Finally, we present some numerical simulations to confirm the analytical results.


Main Subjects

[1] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, in: fractional differential equations, to methods of their solution and some of their applications, Academic Press, New York, 1999 .
[2] J. Sabatier, M. Aoun, A. Oustaloup, G. Gregoire, F. Ragot, P. Roy, Fractional system identification for lead acid battery state of charge estimation, Signal processing 86 (2006) 2645–2657.
[3] J. D. Gabano, T. Poinot, Fractional modelling and identification of thermal systems, Signal Processing 91 (2011) 531–541.
[4] D. Baleanu, Fractional calculus: models and numerical methods, World Scientific, 2012.
[5] A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Science 20 (2016) 763–769.
[6] D. Baleanu , A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Communications in Nonlinear Science and Numerical Simulation 59 (2018) 444–462.
[7] D. Baleanu, A. Jajarmi, M. Hajipour, On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel, Nonlinear Dynamics 94 (2018) 397–414.
[8] V. Volterra, Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Mem. Acad. Lincei Roma. 2 (1926) 31–113.
[9] J. D. Murray, Mathematical Biology, Spring-Verlag, New York, Berlin, 1993.
[10] J. P. Tripathi , S. Abbas, M. Thakur, Dynamical analysis of a prey–predator model with Beddington–DeAngelis type function response incorporating a prey refuge, Nonlinear Dynamics 80 (2015) 177–96.
[11] Y. Huang , F. Chen, L. Zhong, Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge, Applied Mathematics and Computation 182 (2006) 672–83.
[12] E. Ahmed, A. M. El-Sayed H. A. El-Saka. Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, Journal of Mathematical Analysis and Applications 325 (2007) 542–553.
[13] H. L. Li, L. Zhang, C. Hu, Y. L. Jiang Z. Teng, Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge, Journal of Applied Mathematics and Computing 54 (2016) 435–49.
[14] A. A. Elsadany, A. E. Matouk, Dynamical behaviors of fractional-order Lotka-Volterra predator-prey model and its discretization, Journal of Applied Mathematics and Computing 49 (2015) 269–83.
[15] F. A. Rihan, S. Lakshmanan A. H Hashish, Rakkiyappan R. Ahmed E., Fractional-order delayed predator-prey systems with Holling type-II functional response, Nonlinear Dynamics 80 (2015) 777–89.
[16] B. K. Lenka, S. Banerjee, Sufficient conditions for asymptotic stability and stabilization of autonomous fractional order systems, Communications in Nonlinear Science and Numerical Simulation 56 (2018) 365–79.