Dynamics of a Discrete Three Species Food Chain System

The main purpose of this paper is to investigate a discrete time three–species food chain system with ratio dependence. By employing coincidence degree theory and analysis techniques, sufficient conditions for existence of periodic solutions are established.

• Kejun Zhuang
• Zhaohui Wen

• Food chain
• ratio–dependent
• coincidence degree
• periodic solutions.

[1] Haifeng Huo, Wantong Li, J J Nieto. Periodic solutions of delayed
predator-prey model with the Beddington-DeAngelis functional response.
Chaos, Solitons and Fractals, 33(2007), 505-512.
[2] Changyong Xu, Meijuan Wang. Permanence for a delayed discrete three-
level food-chain model with Beddington-DeAngelis functional response.
Applied Mathematics and Computation, 187(2007), 1109-1119.
[3] A. Maiti, A.K. Pal, G.P. Samanta. Effect of time-delay on a food chain
model. Applied Mathematics and Computation, 200(2008), 189-203.
[4] Chengjun Sun, Michel Loreau. Dynamics of a three-species food chain
model with adaptive traits. Chaos, Solitons and Fractals, 41(2009), 2812-
2819.
[5] Zhijun Zeng. Dynamics of a non-autonomous ratio-dependent food chain
model. Applied Mathematics and Computation, 215(2009), 1274-1287.
[6] Raid Kamel Naji, Ranjit Kumar Upadhyay, Vikas Rai. Dynamical consequences
of predator interference in a tri-trophic model food chain.
Nonlinear Analysis: Real World Applications, 11(2010), 809-818.
[7] Fengyan Wang, Guoping Pang. Chaos and Hopf bifurcation of a hybrid
ratio-dependent three species food chain. Chaos, Solitons and Fractals,
(36)2008, 1366-1376.
[8] Pingzhou Liu, K. Gopalsamy. Global stability and chaos in a population
model with piecewise constant arguments. Applied Mathematics and
Computation, 101(1999), 63-88.
[9] Bingbing Zhang, Meng Fan. A remark on the application of coincidence
degree to periodicity of dynamic equtions on time scales. J. Northeast
Normal University(Natural Science Edition), 39(2007), 1-3.(in Chinese)
[10] R E Gaines, J L Mawhin. Coincidence Degree and Nonlinear Differential
Equations. Lecture Notes in Mathematics, Berlin: Springer-Verlag, 1977.