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Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system. (English) Zbl 1050.39022
Summary: With the help of differential equations with piecewise constant arguments, we first propose a discrete analogue of continuous time ratio-dependent predator-prey system, which is governed by nonautonomous difference equations, modeling the dynamics of the prey and the predator having nonoverlapping generations. Then, easily verifiable sufficient criteria are established for the existence of positive periodic solutions. The approach is based on the coincidence degree and the related continuation theorem as well as some a priori estimates.

39A12Discrete version of topics in analysis
39A11Stability of difference equations (MSC2000)
92D25Population dynamics (general)
Full Text: DOI
[1] Chen, L. S.; Jing, Z. J.: Existence and uniqueness of limit cycles for differential equations of predator-prey interactions. Chinese sciences bulletin 24, No. 9, 521-523 (1984)
[2] Cheng, K. S.: Uniqueness of a limit cycle for a predator-prey system. SIAM J. Math. anal. 12, 541-548 (1981) · Zbl 0471.92021
[3] Freedman, H. I.: Deterministic mathematical models in population ecology. (1980) · Zbl 0448.92023
[4] Hsu, S. B.; Huang, T. W.: Global stability for a class of predator-prey systems. SIAM J. Appl. math. 55, 763-783 (1995) · Zbl 0832.34035
[5] Kuang, Y.; Freedman, H. I.: Uniqueness of limit cycles in gause type models of predator prey system. Math. biosci. 88, 67-84 (1988) · Zbl 0642.92016
[6] May, R. M.: Stability and complexity in model ecosystems. (1974)
[7] Smith, J. Maynard: Models in ecology. (1974) · Zbl 0312.92001
[8] Rosenzweig, M. L.: Why the prey curve has a hump. Amer. nat. 103, 81-87 (1969)
[9] Rosenzweig, M. L.: Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171, 385-387 (1969)
[10] Rosenzweig, M. L.; Macarthur, R. H.: Graphical representation and stability conditions of predator-prey interactions. Amer. naturalist 47, 209-223 (1963)
[11] Akcakaya, H. R.: Population cycles of mammals: evidence for a ratio-dependent predation hypothesis. Eco. monogr. 62, 119-142 (1992)
[12] Arditi, R.; Ginzburg, L. R.: Coupling in predator-prey dynamics: ratio-dependence. J. theoretical biology 139, 311-326 (1989)
[13] Arditi, R.; Ginzburg, L. R.; Akcakaya, H. R.: Variation in plankton densities among lakes: A case for ratio-dependent models. American naturalist 138, 1287-1296 (1991)
[14] Arditi, R.; Perrin, N.; Saiah, H.: Functional response and heterogeneities: an experimental test with cladocerans. Oikos 60, 69-75 (1991)
[15] Arditi, R.; Saiah, H.: Empirical evidence of the role of heterogeneity in ratio-dependent consumption. Ecology 73, 1544-1551 (1992)
[16] Ginzburg, L. R.; Akcakaya, H. R.: Consequences of ratio-dependent predation for steady state properties of ecosystems. Ecology 73, 1536-1543 (1992)
[17] Gutierrez, A. P.: The physiological basis of ratio-dependent predator-prey theory: A metabolic pool model of Nicholson’s blowflies as an example. Ecology 73, 1552-1563 (1992)
[18] Hanski, I.: The functional response of predator: worries about scale. Tree 6, 141-142 (1991)
[19] Berryman, A. A.: The origins and evolution of predator-prey theory. Ecology 73, 1530-1535 (1992)
[20] Lundberg, P.; Fryxell, J. M.: Expected population density versus productivity in ratio-dependent and prey dependent models. American naturalist 147, 153-161 (1995)
[21] Freedman, H. I.; Mathsen, R. M.: Persistence in predator-prey systems with ratio-dependent predator influence. Bull. math. Biol. 55, 817-827 (1993) · Zbl 0771.92017
[22] S.B. Hsu, T.W. Hwang and Y. Kuang, Global analysis of Michaelis-Menten type ratio-dependent predator-prey system, J. Math. Biol. (to appear). · Zbl 0984.92035
[23] Jost, C.; Arino, C.; Arditi, R.: About deterministic extinction in ratio-dependent predator-prey models. Bull. math. Biol. 61, 19-32 (1999) · Zbl 1323.92173
[24] Kuang, Y.: Rich dynamics of gause-type ratio-dependent predator-prey systems. Fields institute communications 21, 325-337 (1999) · Zbl 0920.92032
[25] Kuang, Y.; Beretta, E.: Global qualitative analysis of a ratio-dependent predator-prey systems. J. math. Biol. 36, 389-406 (1998) · Zbl 0895.92032
[26] D.M. Xiao and S.G. Ruan, Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol. (to appear). · Zbl 1007.34031
[27] M. Fan, K. Wang, Q. Wang and X.F. Zou, Dynamics of a nonautonomous ratio-dependent predator-prey system, Journal of Dynamics of Continuous, Discrete and Impulsive Systems (submitted). · Zbl 1032.34044
[28] Agarwal, R. P.: Difference equations and inequalities: theory, methods and applications, monographs and textbooks in pure and applied mathematics, no. 228. (2000)
[29] Goh, B. S.: Management and analysis of biological populations. (1980)
[30] Murry, J. D.: Mathematical biology. (1989)
[31] Wiener, J.: Differential equations with piecewise constant delays, trends in theory and practice of nonlinear differential equations. Lecture notes in pure and appl. Math. 90 (1984)
[32] Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations. (1977) · Zbl 0339.47031
[33] M. Fan and K. Wang, Periodicity in a delayed ratio-dependent predator-prey system, J. Math. Anal. Appl. 262 (1), 179--190. · Zbl 0994.34058
[34] Beretta, E.; Kuang, Y.: Global analysis in some delayed ratio-dependent predator-prey systems. Nonlinear analysis TMA 32, No. 3, 381-408 (1998) · Zbl 0946.34061
[35] Mohamad, S.; Gopalsamy, K.: Extremes stability and almost periodicity in a discrete logistic equation. Tohoku math. J. 52, 107-125 (2000) · Zbl 0954.39005