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ratio-dependent; predator-prey system; periodic solution; a priori estimate
In this paper, sharp a priori estimate of the periodic solutions is obtained for the discrete analogue of the continuous time ratio-dependent predator-prey system, which is governed by nonautonomous difference equations, modelling the dynamics of the $n-1$ competing preys and one predator having nonoverlapping generations. Based on more precise a priori estimate and the continuation theorem of the coincidence degree, an easily verifiable sufficient criterion of the existence of positive periodic solutions is established. The result obtained in this paper greatly improves the existing results.
[1] Arditi, R., Ginzburg, L. R.: Coupling in predator-prey dynamics: Ratio-dependence. J. Theor. Biol. 139 (1989), 311-326. DOI 10.1016/S0022-5193(89)80211-5
[2] Ding, X., Lu, C., Liu, M. Z.: Periodic solutions for a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay. Nonliear Anal., Real World Appl. 9 (2008), 762-775. MR 2392373 | Zbl 1152.34046
[3] Ding, X., Lu, C.: Existence of positive periodic solution for ratio-dependent $N$-species difference system. Appl. Math. Modelling 33 (2009), 2748-2756. DOI 10.1016/j.apm.2008.08.008 | MR 2502144 | Zbl 1205.39001
[4] Fan, M., Wang, K.: Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system. Math. Comput. Modelling 35 (2002), 951-961. DOI 10.1016/S0895-7177(02)00062-6 | MR 1910673 | Zbl 1050.39022
[5] Fan, M., Wang, Q., Zhou, X.: Dynamics of a non-autonomous ratio-dependent predator-prey system. Proc. R. Soc. Edinb. A 133 (2003), 97-118. MR 1960049
[6] Fan, M., Wang, Q.: Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator-prey system. Discrete Contin. Dyn. Syst. B 4 (2004), 563-574. DOI 10.3934/dcdsb.2004.4.563 | MR 2073960
[7] Freedman, H. I., Mathsen, R. M.: Persistence in predator-prey systems with ratio-dependent predator influence. Bull. Math. Biol. 55 (1993), 817-827. DOI 10.1007/BF02460674 | Zbl 0771.92017
[8] Gaines, R. E., Mawhin, J. L.: Coincidence Degree, and Nonlinear Differential Equations. Lect. Notes Math., Vol. 568 Springer Berlin (1977). DOI 10.1007/BFb0089537 | MR 0637067 | Zbl 0339.47031
[9] Hsu, S.-B., Hwang, T.-W.: Global stability for a class of predator-prey systems. SIAM J. Appl. Math. 55 (1995), 763-783. DOI 10.1137/S0036139993253201 | MR 1331585
[10] Hsu, S.-B., Hwang, T.-W., Kuang, Y.: Global analysis of Michaelis-Menten type ratio-dependent predator-prey system. J. Math. Biol. 42 (2001), 489-506. DOI 10.1007/s002850100079 | MR 1845589
[11] Jost, C., Arino, O., Arditi, R.: About deterministic extinction in ratio-dependent \hbox{predator}-prey models. Bull. Math. Biol. 61 (1999), 19-32. DOI 10.1006/bulm.1998.0072
[12] Kuang, Y.: Rich dynamics of Gause-type ratio-dependent predator-prey systems. Fields Inst. Commun. 21 (1999), 325-337. MR 1662624
[13] Kuang, Y., Beretta, E.: Global qualitative analysis of a ratio-dependent predator-prey systems. J. Math. Biol. 36 (1998), 389-406. DOI 10.1007/s002850050105 | MR 1624192
[14] Xiao, D., Ruan, S.: Global dynamics of a ratio-dependent predator-prey system. J. Math. Biol. 43 (2001), 268-290. DOI 10.1007/s002850100097 | MR 1868217 | Zbl 1007.34031
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