Periodic Solutions for a Food Chain System with Monod–Haldane Functional Response on Time Scales
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 32870
Periodic Solutions for a Food Chain System with Monod–Haldane Functional Response on Time Scales

Authors: Kejun Zhuang, Hailong Zhu


In this paper, the three species food chain model on time scales is established. The Monod–Haldane functional response and time delay are considered. With the help of coincidence degree theory, existence of periodic solutions is investigated, which unifies the continuous and discrete analogies.

Keywords: Food chain system, periodic solution, time scales, coincidence degree.

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 8440


[1] Jean-Pierre Francoise, Jaume Llibre, Analytical study of a triple Hopf bifurcation in a tritrophic food chain model, Applied Mathematics and Computation, 217(2011), 7146–7154.
[2] B.Mukhopadhyay, R.Bhattacharyya, A stage–structured food chain model with stage dependent predation: Existence of codimension one and codimension two bifurcations, Nonlinear Analysis: RWA, 12(2011), 3056– 3072.
[3] 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.
[4] Rui Xu, Lansun Chen, Feilong Hao, Periodic solutions of a discrete time Lotka–Volterra type food–chain model with delays, Applied Mathematics and Computation, 171(2005), 91–103.
[5] Weiming Wang, Hailing Wang, Zhenqing Li, The dynamic complexity of a three–species Beddington–type food chain with impulsive control strategy, Chaos, Solitons and Fractals, 32(2007), 1772–1785.
[6] J.F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhabitory substrates, Biotechnol Bioengrg. 10(1986), 707–723.
[7] Stefan Hilger, Analysis on measure chains–a unified approach to continuous and discrete calculus, Results Math., 18(1990), 18–56.
[8] Martin Bohner, Meng Fan, Jiming Zhang, Periodicity of scalar dynamic equations and applications to population models, J. Math. Anal. Appl, 330(2007), 1–9.
[9] Yu Tong, Zhenjie Liu, Zhiying Gao, Yonghong Wang, Existence of periodic solutions for a predator–prey system with sparse effect and functional response on time scales, Commun. Nonlinear Sci. Numer. Simulat., 17(2012), 3360–3366.
[10] Kejun Zhuang, Periodic solutions for a three species predator–prey system on time scales, Int. J. Dynamical Systems and Differential Equations, 3(2011), 3-11.
[11] Martin Bohner, Meng Fan, Jiming Zhang, Existence of periodic solutions in predator–prey and competition dynamic systems, Nonl. Anal: RWA, 7(2006), 1193–1204.
[12] Martin Bohner, Allan Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Boston: Birkh¨auser, 2001.
[13] 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)
[14] R. E. Gaines, J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics, Berlin: Springer–Verlag, 1977.