Permanence and extinction of periodic predator–prey systems in a patchy environment with delay. (English) Zbl 1018.92033

Teng, Zhidong; Chen, Lansun

Nonlinear Anal., Real World Appl. 4, No. 2, 335-364 (2003).

Summary: This paper studies two species predator-prey Lotka-Volterra type dispersal systems with periodic coefficients and infinite delays, in which the prey species can disperse among \(n\)-patches, but the predator species is confined to one patch and cannot disperse. Sufficient and necessary conditions of integrable form for permanence, extinction and the existence of positive periodic solutions are established, respectively. Some well-known results on the nondelayed periodic predator-prey Lotka-Volterra type dispersal systems are improved and extended to the delayed case.

MSC:

92D40	Ecology
34C60	Qualitative investigation and simulation of ordinary differential equation models
34K25	Asymptotic theory of functional-differential equations

Keywords:

Lotka-Volterra system; dispersion; delay; predator-prey; permanence; extinction; periodic solution

References:

[1]	Beretta, E.; Solimano, F.; Takeuchi, Y., Global stability and periodic orbits for two-patch predator-prey diffusion-delay models, Math. Biosci., 85, 153-183 (1987) · Zbl 0634.92017
[2]	Cui, J.; Chen, L., The effect of diffusion on the time varying logistic population growth, Comput. Math. Appl., 36, 3, 1-9 (1998) · Zbl 0934.92025
[3]	Freedman, H. I.; Takeuchi, Y., Predator survival versus extinction as a function of dispersal in a predator-prey model with patchy environment, Appl. Anal., 31, 247-266 (1989) · Zbl 0641.92016
[4]	Freedman, H. I.; Takeuchi, Y., Global stability and predator dynamics in a model of prey dispersal in a patchy environment, Nonlinear Anal., 13, 993-1002 (1989) · Zbl 0685.92018
[5]	Hale, J. K., Theory of Functional Differential Equations (1977), Springer: Springer Heidelberg, New York
[6]	Hale, J. K.; Kato, J., Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21, 11-41 (1978) · Zbl 0383.34055
[7]	Kuang, Y.; Takeuchi, Y., Predator-prey dynamics in models of prey dispersal in two-patch environments, Math. Biosci., 120, 77-98 (1994) · Zbl 0793.92014
[8]	Lakshmikantham, V.; Matrosov, V. M.; Sivasundaram, S., Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 0721.34054
[9]	Levin, S. A., Dispersion and population interaction, Amer. Nature, 108, 207-228 (1974)
[10]	Sawano, K., Exponential asymptotic stability for functional differential equations with infinite retardations, Tohoku Math. J., 31, 363-382 (1979) · Zbl 0449.34053
[11]	Song, X.; Chen, L., Persistence and periodic orbits for two species predator-prey system with diffusion, Can. Appl. Math. Quart., 6, 233-244 (1998) · Zbl 0941.92032
[12]	Song, X.; Chen, L., Persistence and global stability for nonautonomous predator-prey system with diffusion and time delay, Comput. Math. Appl., 35, 6, 33-40 (1998) · Zbl 0903.92029
[13]	Song, X.; Chen, L., Harmless delays and global attractivity for nonautonomous predator-prey system with dispersion, Comput. Math. Appl., 39, 5-6, 33-42 (2000) · Zbl 0952.34063
[14]	Song, X.; Chen, L., Conditions for global attractivity of \(n\)-patches predator-prey dispersion-delay models, J. Math. Anal. Appl., 253, 1-15 (2001) · Zbl 0982.34061
[15]	Teng, Z., Uniform persistence of the periodic predator-prey Lotka-Volterra systems, Appl. Anal., 72, 339-352 (1999) · Zbl 1031.34045
[16]	Teng, Z., The almost periodic Kolmogorov competitive systems, Nonlinear Anal., 42, 1221-1230 (2000) · Zbl 1135.34319
[17]	Teng, Z.; Chen, L., Necessary and sufficient conditions for the existence of positive periodic solutions of periodic predator-prey systems, Acta Math. Sci., 18, 402-406 (1998), (in Chinese) · Zbl 0914.92019
[18]	Teng, Z.; Chen, L., The positive periodic solutions in periodic Kolmogorov type systems with delays, Acta Math. Appl. Sinica, 22, 446-456 (1999), (in Chinese) · Zbl 0976.34063
[19]	Wang, K.; Huang, Q., \(C_h\)-space and the boundedness and periodic solutions of functional differential equations with infinite delay, Sci. China, Ser. A, 3, 242-252 (1987), (in Chinese)
[20]	Wang, W.; Ma, Z., Asymptotic behavior of a predator-prey system with diffusion and delays, J. Math. Anal. Appl., 206, 191-204 (1997) · Zbl 0872.92019
[21]	Xu, R.; Chen, L., Persistence and stability for a two species ratio-dependent predator-prey system with time delay in a two-patch environment, Comput. Math. Appl., 40, 577-588 (2000) · Zbl 0949.92028
[22]	Yoshizawa, T., Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions (1975), Springer: Springer New York · Zbl 0304.34051