Existence and global stability of positive periodic solutions of a predator-prey system with delays. (English) Zbl 1029.92025

Summary: This paper studies the existence, global stability and uniform persistence of positive periodic solutions of a periodic predator-prey system with Holling type III functional response. By using the continuation theorem of coincidence degree theory and Lyapunov functionals, some sufficient conditions are obtained.


92D40 Ecology
34C25 Periodic solutions to ordinary differential equations
92D25 Population dynamics (general)
34D05 Asymptotic properties of solutions to ordinary differential equations
Full Text: DOI


[1] Barbălat, I., Systemes d’equations differentielle d’oscillations nonlineaires, Rev. Roumaine Math. Pures Appl., 4, 267-270 (1959) · Zbl 0090.06601
[2] Beretta, E.; Kuang, Y., Convergence results in a well-known delayed predator-prey system, J. Math. Anal. Appl., 204, 840-853 (1996) · Zbl 0876.92021
[3] Berryman, A. A., The origins and evolution of predator-prey theory, Ecology, 75, 1530-1535 (1992)
[4] Cushing, J. M., Intergrodifferential Equations and Delay Models in Population Dynamics (1977), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0363.92014
[5] Fan, M.; Wang, K., Periodicity in a delayed ratio-dependent predator-prey system, J. Math. Anal. Appl., 262, 179-190 (2001) · Zbl 0994.34058
[6] Freedman, H. I., Deterministic Mathematical Models in Population Ecology (1980), Marcel Dekker: Marcel Dekker New York · Zbl 0448.92023
[7] Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0326.34021
[8] Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0752.34039
[9] Gopalsamy, K., Harmless delay in model system, Bull. Math. Biol., 45, 295-309 (1983) · Zbl 0514.34060
[10] Gopalsamy, K., Delayed responses and stability in two-species systems, J. Austral. Math. Soc., Ser. B, 25, 473-500 (1984) · Zbl 0552.92016
[11] Hastings, A., Delays in recruitment at different trophic levels: effects on stability, J. Math. Biol., 21, 35-44 (1984) · Zbl 0547.92014
[12] Holling, C. S., The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Ent. Sec. Can., 45, 1-60 (1965)
[13] H.F. Huo, W.T. Li, S.S. Cheng, Periodic solutions of two-species competition model with delays, Int. J. Appl. Math., in press; H.F. Huo, W.T. Li, S.S. Cheng, Periodic solutions of two-species competition model with delays, Int. J. Appl. Math., in press
[14] Huo, H. F.; Li, W. T.; Cheng, S. S., Periodic solutions of two-species diffusive models with continuous time delays, Demonstntio Mathematica, 35, 2, 433-466 (2002) · Zbl 1013.92035
[15] Hsu, S. B.; Huang, T. W., Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55, 3, 763-783 (1995) · Zbl 0832.34035
[16] Jia, J. W., Persistence and periodic solution for the nonautonomous predator-prey system with type III functional response, J. Biomath., 16, 1, 59-62 (2001) · Zbl 1057.34510
[17] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002
[18] Li, Y. K., Periodic solutions of a periodic delay predator-prey system, Proc. Amer. Math. Soc., 127, 1331-1335 (1999) · Zbl 0917.34057
[19] Li, Y. K.; Kuang, Y., Periodic solutions of periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl., 255, 260-280 (2001) · Zbl 1024.34062
[20] MacDonald, N., Time Lags in Biological Models (1978), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0403.92020
[21] May, R. M., Time delay versus stability in population models with two and three trophic levels, Ecology, 4, 315-325 (1973)
[22] Rosenzweig, M. L.; MacArthur, R., Graphical representation and stability conditions of predator-prey interactions, Amer. Nat., 97, 209-223 (1963)
[23] Rosenzweig, M. L., Paradox of enrichment: destabilization of exploitation ecosystems in ecological time, Science, 171, 385-387 (1969)
[24] Ruan, S., Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays, Quart. Appl. Math., 59, 159-173 (2001) · Zbl 1035.34084
[25] Weng, P. X., Global attractivity in a periodic competition system with feedback controls, Acta Appl. Math., 12, 11-21 (1996) · Zbl 0859.34061
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.