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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.

MSC:
39A12 Discrete version of topics in analysis
39A11 Stability of difference equations (MSC2000)
92D25 Population dynamics (general)
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[1] Chen, L.S.; Jing, Z.J., Existence and uniqueness of limit cycles for differential equations of predator-prey interactions, Chinese sciences bulletin, 24, 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), Marcel Dekker New York · 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), Princeton University Press
[7] Smith, J.Maynard, Models in ecology, (1974), Cambridge University Press Cambridge · 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), Marcel Dekker New York
[29] Goh, B.S., Management and analysis of biological populations, (1980), Elsevier Scientific The Netherlands · Zbl 0453.92015
[30] Murry, J.D., Mathematical biology, (1989), Springer-Verlag New York
[31] Wiener, J., ()
[32] Gaines, R.E.; Mawhin, J.L., Coincidence degree and nonlinear differential equations, (1977), Springer-Verlag Berlin · Zbl 0326.34021
[33] M. Fan and K. Wang, Periodicity in a delayed ratio-dependent predator-prey system, J. Math. Anal. Appl.\bf262 (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, 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
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