Periodic solutions of a discrete time Lotka–Volterra type food-chain model with delays. (English) Zbl 1081.92043

Summary: A periodic discrete time three trophic level Lotka-Volterra type food-chain model is investigated. By using R. E. Gaines and J. L. Mawhin’s continuation theorem of coincidence degree theory [Coincidence degree, and nonlinear differential equations. (1977; Zbl 0339.47031)], sufficient conditions are derived for the existence of positive periodic solutions of the model.


92D40 Ecology
39A10 Additive difference equations
39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis


Zbl 0339.47031
Full Text: DOI


[1] Agarwal, R. P., Difference Equations and Inequalities: Theory, Methods and Applications, Monographs and Textbooks in Pure and Applied Mathematics, No. 228 (2000), Marcel Dekker: Marcel Dekker New York
[2] Crone, E. E., Delayed density dependence and the stability of interacting populations and subpopulations, Theoret. Population Biol., 51, 67-76 (1997) · Zbl 0882.92025
[3] Cushing, J. M., Periodic time-dependent predator-prey system, SIAM J. Appl. Math., 32, 82-95 (1977) · Zbl 0348.34031
[4] Fan, M.; Wang, K., Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system, Math. Comput. Model., 35, 951-961 (2002) · Zbl 1050.39022
[5] Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0326.34021
[6] Goh, B. S., Global stability in two species interactions, J. Math. Biol., 3, 313-318 (1976) · Zbl 0362.92013
[7] Hastings, A., Global stability in two species systems, J. Math. Biol., 5, 399-403 (1978) · Zbl 0382.92008
[8] He, X., Stability and delays in a predator-prey system, J. Math. Anal. Appl., 198, 355-370 (1996) · Zbl 0873.34062
[9] Lu, Z.; Wang, W., Permanence and global attractivity for Lotka-Volterra difference systems, J. Math. Biol., 22, 269-282 (1999) · Zbl 0945.92022
[10] Murray, J. D., Mathematical Biology (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0682.92001
[11] Saito, Y.; Hara, T.; Ma, W. B., Harmless delays for permanence and impersistence of a Lotka-Volterra discrete predator-prey system, Nonlinear Anal., 50, 703-715 (2002) · Zbl 1005.39013
[12] Saito, Y.; Ma, W.; Hara, T., A necessary and sufficient condition for permanence of a Lotka-Volterra discrete system with delays, J. Math. Anal. Appl., 256, 162-174 (2001) · Zbl 0976.92031
[13] Wang, W.; Lu, Z., Global stability of discrete models of Lotka-Volterra type, Nonlinear Anal., 35, 1019-1030 (1999) · Zbl 0919.92030
[14] Wang, W.; Mulone, G.; Salemi, F.; Salone, V., Global stability of discrete population models with delays and fluctuating environment, J. Math. Anal. Appl., 264, 147-167 (2001) · Zbl 1006.92025
[15] Zhang, R.; Wang, Z.; Chen, Y.; Wu, J., Periodic solutions of a single psecies discrete population model with periodic harvest/stock, Comput. Math. Appl., 39, 77-90 (2000) · Zbl 0970.92019
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.