Existence of positive periodic solution for ratio-dependent N-species difference system. (English) Zbl 1205.39001

Summary: We propose a food web model consisting of \(n-1\) competing preys and one predator. Firstly, following the same idea and method in [M. Fan and K. Wang, Math. Comput. Modelling 35, No. 9–10, 951–961 (2002; Zbl 1050.39022)], we derive the discrete time analogue of the model. Then, the easily verifiable sufficient criteria are established for the existence of positive periodic solutions of this analogue, the approach is based on the coincidence degree and the related continuation theorem as well as some prior estimates. Meanwhile, some numerical simulations are carried out to support the theoretical analysis of the research.


39A10 Additive difference equations
92D25 Population dynamics (general)


Zbl 1050.39022
Full Text: DOI


[1] Arditi, R.; Ginzburg, L. R., Coupling in predator-prey dynamics: ratio-dependence, J. Theor. Biol., 139, 311-326 (1989)
[2] Freedman, H. I.; Mathsen, R. M., Persistence in predator-prey systems with ratio-dependent predator influence, Bull. Math. Biol., 55, 817-827 (1993) · Zbl 0771.92017
[3] Hsu, S. B.; Hwang, T. W.; Kuang, Y., Global analysis of Michaelis-Menten type ratio-dependent predator-prey system, J. Math. Biol., 42, 489-506 (2001) · Zbl 0984.92035
[4] Jost, C.; Arino, C.; Arditi, R., About deterministic extinction in ratio-dependent predator-prey models, Bull. Math. Biol., 61, 19-32 (1999) · Zbl 1323.92173
[5] Kuang, Y., Rich dynamics of Gause-type ratio-dependent predator-prey systems, Fields Inst. Commun., 21, 325-337 (1999) · Zbl 0920.92032
[6] Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator-prey systems, J. Math. Biol., 36, 389-406 (1998) · Zbl 0895.92032
[7] Xiao, D. M.; Ruan, S. G., Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43, 3, 268-290 (2001) · Zbl 1007.34031
[8] Chen, Fengde; Li, Zhong; Huang, Yunjin, Note on the permanence of a competitive system with infinite delay and feedback controls, Nonlinear Anal., 58, 8, 680-687 (2007) · Zbl 1152.34366
[9] Fan, M.; Wang, K.; Wang, Q.; Zhou, X. F., Dynamics of a nonautonomous ratio-dependent predator-prey system, Proc. R. Soc. Edinburgh A, 133, 97-118 (2003) · Zbl 1032.34044
[10] Freedman, H. I., Deterministic Mathematical Models in Population Ecology (1980), Marcel Dekker: Marcel Dekker New York · Zbl 0448.92023
[11] Goh, B. S., Management, Analysis of Biological Populations (1980), Elsevier Scientific: Elsevier Scientific The Netherlands · Zbl 0453.92015
[12] Murry, J. D., Mathematical Biology (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0682.92001
[13] Fan, M.; Wang, K., Periodic solutions of discrete time nonautonomous ratio-dependent predator-prey system, Math. Comput. Model., 35, 951-961 (2002) · Zbl 1050.39022
[14] Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0326.34021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.