Persistence in nonautonomous predator-prey systems with infinite delays

This paper studies the general nonautonomous predator-prey Lotka-Volterra systems with infinite delays. The sufficient and necessary conditions of integrable form on the permanence and persistence of species are established. A very interesting and important property of two-species predator-prey systems is discovered, that is, the permanence of species and the existence of a persistent solution are each other equivalent. Particularly, for the periodic system with delays, applying these results, the sufficient and necessary conditions on the permanence and the existence of positive periodic solutions are obtained. Some well-known results on the nondelayed periodic predator-prey Lotka-Volterra systems are strongly improved and extended to the delayed case.

[1]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[2]  Wan-Tong Li,et al.  Periodic solutions of delayed ratio-dependent predator–prey models with monotonic or nonmonotonic functional responses , 2004 .

[3]  Wan-Tong Li,et al.  Periodic solutions of delayed Leslie-Gower predator-prey models , 2004, Appl. Math. Comput..

[4]  Xinyu Song,et al.  Conditions for Global Attractivity of n-Patches Predator–Prey Dispersion-Delay Models , 2001 .

[5]  Wan-Tong Li,et al.  Existence and global stability of positive periodic solutions of a predator-prey system with delays , 2003, Appl. Math. Comput..

[6]  Ke Wang,et al.  Positive Periodic Solutions of Predator-Prey Systems with Infinite Delay , 1980 .

[7]  Zhidong Teng,et al.  Permanence criteria in non-autonomous predator–prey Kolmogorov systems and its applications , 2004 .

[8]  Fordyce A. Davidson,et al.  Periodic solutions of a predator-prey model with stage structure for predator , 2004, Appl. Math. Comput..

[9]  Zhidong Teng,et al.  Nonautonomous Lotka–Volterra Systems with Delays , 2002 .

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

[11]  Xinyu Song,et al.  Persistence and global stability for nonautonomous predator-prey system with diffusion and time delay , 1998 .

[12]  Fordyce A. Davidson,et al.  Periodic solution of a Lotka-Volterra predator-prey model with dispersion and time delays , 2004, Appl. Math. Comput..

[13]  Yang Kuang.  Global stability in delayed nonautonomous Lotka-Volterra type systems without saturated equilibria , 1996 .

[14]  Wan-Tong Li,et al.  Periodic solution of a delayed predator-prey system without dominating instantaneous negative feedback , 2004, Appl. Math. Comput..

[15]  Feng Wang,et al.  Positive periodic solution of two-species ratio-dependent predator-prey system with time delay in two-patch environment , 2004, Appl. Math. Comput..

[16]  Yang Kuang,et al.  Periodic Solutions in Periodic Delayed Gause-Type Predator-Prey Systems , 2001 .

[17]  Lansun Chen,et al.  Harmless delays and global attractivity for nonautonomous predator-prey system with dispersion☆ , 2000 .

[18]  Teng Zhi-dong.  Uniform persistence of the periodic predator-prey lotka-volterra systems , 1999 .

[19]  Zhidong Teng,et al.  Permanence for non-autonomous food chain systems with delay , 2003 .

[20]  Zhidong Teng,et al.  Permanence and extinction of periodic predator-prey systems in a patchy environment with delay , 2003 .

[21]  Yang Kuang,et al.  Permanence in Kolmogorov-Type Systems of Nonautonomous Functional Differential Equations , 1996 .

[22]  Fordyce A. Davidson,et al.  Periodic solutions for a delayed predator-prey model of prey dispersal in two-patch environments , 2004 .

[23]  Ke Wang,et al.  Global Existence of Positive Periodic Solutions of Periodic Predator–Prey System with Infinite Delays , 2001 .