Uniform persistence and periodic solutions for a discrete predator-prey system with delays. (English) Zbl 1107.39017

Yang, Xitao

J. Math. Anal. Appl. 316, No. 1, 161-177 (2006).

This paper deals with a class of discrete predator-prey systems with delay. A sufficient condition for the uniform persistence of the system is first shown, and then, if the coefficients in the system are periodic, the existence of a periodic solution based on the uniform persistence result is obtained by generalizing the Yoshizawa’s theorem on the existence of periodic solution for ordinary differential equation to the difference equations with delays.

Reviewer: Wan-Tong Li (Lanzhou)

MSC:

39A11	Stability of difference equations (MSC2000)
39A12	Discrete version of topics in analysis
34K13	Periodic solutions to functional-differential equations

Keywords:

periodic solution; ultimate boundedness; delay; Yoshizawa’s theorem

References:

[1]	Agarwal, R.P., Difference equations and inequalities: theory, method and applications, Monogr. textbooks pure appl. math., vol. 228, (2000), Dekker New York
[2]	Agarwal, R.P.; Wong, P.J.Y., Advance topics in difference equations, (1997), Kluwer Acad. Publ. Dordrecht · Zbl 0914.39005
[3]	Chen, Y.; Zhou, Z., Stable periodic solution of a discrete periodic lotka – volterra competition system, J. math. anal. appl., 277, 358-366, (2003) · Zbl 1019.39004
[4]	Freedman, H.I., Deterministic mathematics models in population ecology, (1980), Dekker New York · Zbl 0448.92023
[5]	Hale, J.K.; Waltman, P., Persistence in infinite-dimensional system, SIAM J. math. anal., 20, 388-395, (1989) · Zbl 0692.34053
[6]	Horn, W.A., Some fixed-point theorem for compact mapping and flows on a Banach space, Trans. amer. math. soc., 149, 391-404, (1970) · Zbl 0201.46203
[7]	Hotbauer, J.; Hutson, V.; Jansen, W., Coexistence for system governed by difference equations of lotka – volterra type, J. math. biol., 25, 553-570, (1987) · Zbl 0638.92019
[8]	Hutson, V.; Moran, W., Persistence of species obeying difference equation, J. math. biol., 151, 203-213, (1982) · Zbl 0495.92015
[9]	Jia, J.W., Persistence and periodic solution for the nonautonomous predator – prey system with type III functional response, J. biomath., 16, 763-783, (2001)
[10]	Murry, J.D., Mathematical biology, (1989), Springer New York
[11]	Rui, X.; Chaplain, M.A.J.; Davidson, F.A., Periodic solutions of a discrete nonautonomous lotka – volterra predator – prey model with time delays, Discrete contin. dynam. syst. ser. B, 4, 823-831, (2004) · Zbl 1116.92072
[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, K., Persistence for nonautonomous predator – prey systems with infinite delay, Acta math. sinica, 40, 321-332, (1997) · Zbl 0907.92028
[14]	Wang, L.L.; Li, W.T., Existence and global stability of positive periodic solutions of a predator – prey system with delays, Appl. math. comput., 146, 167-185, (2003) · Zbl 1029.92025
[15]	Wang, L.L.; Li, W.T.; Zhao, P.H., Existence and global stability of positive periodic solutions of a discrete predator – prey system with delays, Adv. differential equations, 5, 321-336, (2004) · Zbl 1081.39007
[16]	Wang, W.; Ma, Z., Harmless delays for uniform persistence, J. math. anal. appl., 158, 256-268, (1991) · Zbl 0731.34085
[17]	Zhao, Y.; Hutson, V., Permanence in Kolmogorov periodic predator – prey models with diffusion, Nonlinear anal., 23, 651-668, (1994) · Zbl 0823.92031
[18]	Zhou, Z.; Zou, X., Stable periodic solutions in a discrete periodic logistic equation, Appl. math. lett., 16, 165-171, (2003) · Zbl 1049.39017