×

zbMATH — the first resource for mathematics

Permanence and global attractivity of a discrete semi-ratio dependent predator-prey system with Holling II type functional response. (English) Zbl 1213.49046
Summary: We propose a discrete semi-ratio dependent predator-prey system with Holling II type functional response. For general nonautonomous case, sufficient conditions which ensure the permanence and the global stability of the system are obtained; for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system are obtained.

MSC:
49N75 Pursuit and evasion games
49M25 Discrete approximations in optimal control
34C25 Periodic solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Chen, F.D.: Permance and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems. Appl. Math. Comput. 59, 804–814 (1991)
[2] 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
[3] Fan, M., Wang, K.: Periodic solution of a discrete time nonautonomous ratio-dependent predator-prey system. Math. Comput. Model. 35, 951–961 (2002) · Zbl 1050.39022
[4] Berryman, A.A.: The origins and evolution of predator-prey theory. Ecology 75, 1530–1535 (1992)
[5] Huo, H.F., Li, W.T.: Stable periodic solution of the discrete periodic Leslie-Gower predator-prey model. Math. Comput. Model. 34, 261–269 (2004) · Zbl 1067.39008
[6] Yang, X.T.: Uniform persistence and periodic solutions for a discrete predator-prey system with delays. J. Math. Anal. Appl. 316, 161–177 (2006) · Zbl 1107.39017
[7] Fang, Y.H., Li, W.T.: Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response. J. Math. Anal. Appl. 299, 357–374 (2004) · Zbl 1063.39013
[8] Fan, M., Wang, Q.: Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator-prey systems. Discrete Contin. Dyn. Syst. Ser. B 4, 563–574 (2004) · Zbl 1100.92064
[9] Wang, Q., Fan, M., Wang, K.: Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey systems with functional responses. J. Math. Anal. Appl. 278, 443–471 (2003) · Zbl 1029.34042
[10] Arrowsmith, D.K., Place, C.M.: Dynamical Systems. Chapman and Hall, London (1992)
[11] Beltrami, E.: Mathematics for Dynamical Modelling. Academic Press, San Diego (1987) · Zbl 0625.58012
[12] Huo, H.F., Li, W.T.: Permanence and global stability for nonautonomous discrete model of plankton allelopathy. Appl. Math. Lett. 17, 1007–1013 (2004) · Zbl 1067.39009
[13] Chen, Y.M., Zhou, Z.: Stable periodic solution of a discrete periodic Lotka-Volterra competition system. J. Math. Anal. Appl. 277, 358–366 (2003) · Zbl 1019.39004
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.