Existence and global stability of positive periodic solutions of a predator-prey system with delays

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 Liapunov functional, some sufficient conditions are obtained.

[1]  R. Macarthur,et al.  Graphical Representation and Stability Conditions of Predator-Prey Interactions , 1963, The American Naturalist.

[2]  S. Ruan.  Absolute stability, conditional stability and bifurcation in Kolmogrov-type predator-prey systems with discrete delays , 2001 .

[3]  A Hastings,et al.  Delays in recruitment at different trophic levels: Effects on stability , 1984, Journal of mathematical biology.

[4]  Jia Jian-wen,et al.  Persistence and Periodic Solution for the Nonautonomous Predator-Prey System with Type III Functional Response , 2001 .

[5]  Yang Kuang,et al.  Periodic Solutions of Periodic Delay Lotka–Volterra Equations and Systems☆ , 2001 .

[6]  K. Gopalsamy.  Harmless delays in model systems , 1983 .

[7]  Robert M. May,et al.  Time‐Delay Versus Stability in Population Models with Two and Three Trophic Levels , 1973 .

[8]  R. Gaines,et al.  Coincidence Degree and Nonlinear Differential Equations , 1977 .

[9]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

[10]  H. I. Freedman.  Deterministic mathematical models in population ecology , 1982 .

[11]  Sze-Bi Hsu,et al.  Global Stability for a Class of Predator-Prey Systems , 1995, SIAM J. Appl. Math..

[12]  Alan A. Berryman,et al.  The Orgins and Evolution of Predator‐Prey Theory , 1992 .

[13]  K. Gopalsamy,et al.  Delayed responses and stability in two-species systems , 1984, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[14]  Yang Kuang,et al.  Convergence Results in a Well-Known Delayed Predator-Prey System , 1996 .

[15]  N. Macdonald.  Time lags in biological models , 1978 .

[16]  Peixuan Weng,et al.  Global attractivity in a periodic competition system with feedback controls , 1996 .

[17]  Li Yongkun,et al.  PERIODIC SOLUTIONS OF A PERIODIC DELAY PREDATOR-PREY SYSTEM , 1999 .

[18]  C. S. Holling,et al.  The functional response of predators to prey density and its role in mimicry and population regulation. , 1965 .

[19]  Ke Wang,et al.  Periodicity in a Delayed Ratio-Dependent Predator–Prey System☆☆☆ , 2001 .

[20]  M. Rosenzweig.  Paradox of Enrichment: Destabilization of Exploitation Ecosystems in Ecological Time , 1971, Science.