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Some global results on quasipolynomial discrete systems. (English) Zbl 1130.39300
Summary: The quasipolynomial (QP) generalization of Lotka–Volterra discrete-time systems is considered. Use of the QP formalism is made for the investigation of various global dynamical properties of QP discrete-time systems including permanence, attractivity, dissipativity and chaos. The results obtained generalize previously known criteria for discrete Lotka–Volterra models.

MSC:
39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations
39A12 Discrete version of topics in analysis
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[1] Basson, M.; Fogarty, M.J., Harvesting in discrete-time predator – prey systems, Math. biosci., 141, 41-74, (1997) · Zbl 0880.92034
[2] Diamond, P., Chaotic behavior of systems of difference equations, Inter. J. sys. sci., 7, 953-956, (1976) · Zbl 0336.93004
[3] Dohtani, A., Occurrence of chaos in higher-dimensional discrete-time systems, SIAM J. appl. math., 52, 1707-1721, (1992) · Zbl 0774.93049
[4] Fisher, M.E.; Goh, B.S., Stability in a class of discrete-time models of interacting populations, J. math. biol., 4, 265-274, (1977) · Zbl 0381.92019
[5] Gamarra, J.G.P.; Solé, R.V., Complex discrete dynamics from simple continuous population models, Bull. math. biol., 64, 611-620, (2002) · Zbl 1334.92336
[6] Góra, P.; Boyarsky, A., Absolutely continuous invariant measures for the family of mappings \(x \rightarrow \mathit{rxe}^{- \mathit{bx}}\) with application to the belousov – zhabotinski reaction, Dynamic stability systems, 5, 65-81, (1990) · Zbl 0699.34049
[7] Hernández-Bermejo, B.; Brenig, L., Quasipolynomial generalization of lotka – volterra mappings, J. phys. A, 35, 5453-5469, (2002) · Zbl 1039.37010
[8] Hernández-Bermejo, B.; Brenig, L., Characterization and solvability of quasipolynomial symplectic mappings, J. phys. A, 37, 2191-2200, (2004) · Zbl 1040.37036
[9] Hernández-Bermejo, B.; Brenig, L., Conservative quasipolynomial maps, Phys. lett. A, 324, 425-436, (2004) · Zbl 1123.37302
[10] Hofbauer, J.; Hutson, V.; Jansen, W., Coexistence for systems governed by difference equations of lotka – volterra type, J. math. biol., 25, 553-570, (1987) · Zbl 0638.92019
[11] Lu, Z.; Wang, W., Permanence and global attractivity for lotka – volterra difference systems, J. math. biol., 39, 269-282, (1999) · Zbl 0945.92022
[12] Marotto, F.R., Snap-back repellers imply chaos in \(R^n\), J. math. anal. appl., 63, 199-223, (1978) · Zbl 0381.58004
[13] May, R.M., Simple mathematical models with very complicated dynamics, Nature, 261, 459-467, (1976) · Zbl 1369.37088
[14] May, R.M.; Oster, G.F., Bifurcations and dynamic complexity in simple ecological models, Am. nat., 110, 573-599, (1976)
[15] Moran, P.A.P., Some remarks on animal population dynamics, Biometrics, 6, 250-258, (1950)
[16] Rajasekar, S., Controlling of chaotic motion by chaos and noise signals in a logistic map and a bonhoeffer – van der Pol oscillator, Phys. rev. E, 51, 775-778, (1995)
[17] Udwadia, F.E.; Raju, N., Some global properties of a pair of coupled mapsquasi-symmetry, periodicity and syncronicity, Physica D, 111, 16-26, (1998) · Zbl 0932.37014
[18] Wendi, W.; Zhengyi, L., Global stability of discrete models of lotka – volterra type, Nonlinear anal. TMA, 35, 1019-1030, (1999) · Zbl 0919.92030
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