Stability Analysis in a Fractional Order Delayed Predator-Prey Model

In this paper, we study the stability of a fractional order delayed predator-prey model. By using the Laplace transform, we introduce a characteristic equation for the above system. It is shown that if all roots of the characteristic equation have negative parts, then the equilibrium of the above fractional order predator-prey system is Lyapunov globally asymptotical stable. An example is given to show the effectiveness of the approach presented in this paper.





References:
<p>[1] R.M. May, Time delay versus stability in population models with two
and three trophic levels. Ecology 54 (2) (1973) 315-325.
[2] Y.L. Song and J. Wei, Local Hopf bifurcation and global periodic
solutions in a delayed predator-prey system. J. Math. Anal. Appl. 301
(1) (2005) 1-21.
[3] X.P. Yan and W.T. Li, Hopf bifurcation and global periodic solutions
in a delayed predator-prey system. Appl. Math. Comput. 177 (1) (2006)
427-445.
[4] T. Faria, Stability and bifurcation for a delayed predator-prey model and
the effect of diffusion. J. Math. Anal. Appl. 254 (2) (2001) 433-463.
[5] Y. Kuang, Delay Differential Equations With Applications in Population
Dynamics. Academic Press, INC, 1993.
[6] Yan, X. P. and C.H. Zhang, Hopf bifurcation in a delayed Lokta-Volterra
predator-prey system. Nonlinear Anal.: Real World Appl. 9 (1) (2008)
114-127.
[7] C.J. Xu, X.H. Tang, M.X. Liao and X.F. He, Bifurcation analysis in a
delayed Lokta-Volterra predator-prey model with two delays. Nonlinear
Dyn. 66 (1-2) (2011) 169-183.
[8] Eva Kaslik and Seenith Sivasundaram, Nonlinear dynamics and chaos in
fractional-order neural networks. Neural netw. 32 (2012) 245C256.
[9] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and
Fractional Differential Equations. Wiley Interscience, New York, 1993.
[10] I. Podlubny, Fractional Differential Equations. Academic Press, New
York, 1999.
[11] P.L. Butzer and U. Westphal, An Introduction to the Fractional Calculus.
World Scientific, Singapore, 2000.
[12] O.P. Agrawal, J.A. Tenreiro Machado and J. Sabatier, Introduction,
Nonlinear Dyn. 38 (1-2) (2204) 1-2.
[13] S. Kempfle, I. Schafer and H. Beyer, Fractional calculus: theory and
applications. Nonlinear Dyn. 29(1-4) (2002) 99-127.
[14] M.W. Spong, A theorem on neutral delay systems. Systems & Control
Letters, 6 (4) (1985)291-294.
[15] Y. He, M. Wu, J.H. She, G.P. Liu, Delay-dependent robust stability
criteria for uncertain neutral systems with mixed delays. Systems &
Control Letters 51 (1) (2004) 57-65.
[16] C.H. Lien, K.W. Yu, Y.-J. Chung, Y.F. Lin, L.Y. Chung and J.D. Chen,
Exponential stability analysis for uncertain switched neutral systems with
interval time-varying state delay. Nonlinear Anal.: Hybrid Syst. 3 (3)
(2009) 334-342.
[17] L.L. Xiong, S.M. Zhong, M. Ye and S.L. Wu, New stability and
stabilization for switched neutral control systems. Chaos, Solitons &
Fractals 42 (3) (2009) 1800-1811.
[18] X.G. Liu, M. Wu, Ralph Martin and M.L. Tang, Stability analysis for
neutral systems with mixed delays. J. Comput. Appl. Math. 202 (2) (2007)
478-497.
[19] L.L. Xiong, S.M. Zhong and J.K. Tian, Novel robust stability criteria
of uncertain neutral systems with discrete and distributed delays. Chaos,
Solitons & Fractals 40 (2) (2009) 771-777.
[20] W.J. Xiong and J.L. Liang. Novel stability criteria for neutral systems
with multiple time delays. Chaos, solitions & Fractals 32 (5) (2007) 1735-
1741.
[21] H. Li, H.B. Li, S.M. Zhong. Some new simple stability criteria of linear
neutral systems with a single delay. J. Comput. Appl. Math. 200 (1)
(2007) 441-447.
[22] L.L. Xiong, Y. Zhao and T. Jiang, Stability analsysi of linear fractional
order neutral system with multiple delays by algebraic approach. World
Academy of Science, Engineering and Technology 76 (2011) 983-986.
[23] W.H. Deng, C.P. Li and J.H. L¨u, Stbility analysis of linear fractional
differential system with multiple time delays. Nonlinear Dyn. 48 (4)
(2007) 409-416.
[24] E.J. Muth, Transform Methods with Applications to Engineering and
Operations Research. Prentice-Hall, New Jersey, 1977.