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Basin of attraction through invariant curves and dominant functions. (English) Zbl 1418.39013
Summary: We study a second-order difference equation of the form \(z_{n + 1} = z_n F(z_{n - 1}) + h\), where both \(F(z)\) and \(z F(z)\) are decreasing. We consider a set of invariant curves at \(h = 1\) and use it to characterize the behaviour of solutions when \(h > 1\) and when \(0 < h < 1\). The case \(h > 1\) is related to the Y2K problem. For \(0 < h < 1\), we study the stability of the equilibrium solutions and find an invariant region where solutions are attracted to the stable equilibrium. In particular, for certain range of the parameters, a subset of the basin of attraction of the stable equilibrium is achieved by bounding positive solutions using the iteration of dominant functions with attracting equilibria.
MSC:
39A30 Stability theory for difference equations
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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[1] Kocic, V. L.; Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, (1993), Dordrecht, The Netherlands: Kluwer Academic Publishers, Dordrecht, The Netherlands · Zbl 0787.39001
[2] Murray, J. D., Mathematical Biology, (1989), New York, NY, USA: Springer, New York, NY, USA · Zbl 0682.92001
[3] Mueller, L. D.; Joshi, A., Stability in Model Populations, (2000), Princeton, NJ, USA: Princeton University Press, Princeton, NJ, USA
[4] Pielou, E. C., Population and Community Ecology, (1974), New York, NY, USA: Gordon and Breach, New York, NY, USA · Zbl 0349.92024
[5] Levin, S. A.; May, R. M., A note on difference-delay equations, Theoretical Population Biology, 9, 2, 178-187, (1976) · Zbl 0338.92021
[6] Liz, E.; Tkachenko, V.; Trofimchuk, S., Global stability in discrete population models with delayed-density dependence, Mathematical Biosciences, 199, 1, 26-37, (2006) · Zbl 1086.92045
[7] Abu-Saris, R.; AlSharawi, Z.; Rhouma, M. B., The dynamics of some discrete models with delay under the effect of constant yield harvesting, Chaos, Solitons & Fractals, 54, 26-38, (2013) · Zbl 1341.92056
[8] AlSharawi, Z., A global attractor in some discrete contest competition models with delay under the effect of periodic stocking, Abstract and Applied Analysis, 2013, (2013) · Zbl 1297.39018
[9] AlSharawi, Z.; Rhouma, M. B., The Beverton-Holt model with periodic and conditional harvesting, Journal of Biological Dynamics, 3, 5, 463-478, (2009) · Zbl 1342.91025
[10] Arriola, L., First integrals for difference equations, Nonlinear Analysis: Theory, Methods & Applications, 30, 1191-1196, (1997) · Zbl 0893.39003
[11] Grove, E. A.; Ladas, G., Periodicities in Nonlinear Difference Equations, (2000), Boca Raton, Fla, USA: CRC Press, Boca Raton, Fla, USA · Zbl 1078.39009
[12] Grove, E. A.; Kocic, V. L.; Ladas, G., Classification of invariants for certain difference equations, Advances in Difference Equations: Proceedings of the Second International Conference on Difference Equations, Hungary, 1995, (1997), Gordon and Breach · Zbl 0890.39015
[13] Nesemann, T., Invariants and Liapunov functions for nonautonomous systems, Computers & Mathematics with Applications, 42, 3–5, 385-392, (2001) · Zbl 1001.39024
[14] Camouzis, E.; Ladas, G., Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, (2008), Chapman & Hall, CRC Press · Zbl 1133.39302
[15] Nussbaum, R. D., Global stability, two conjectures and Maple, Nonlinear Analysis: Theory, Methods & Applications, 66, 5, 1064-1090, (2007) · Zbl 1121.39004
[16] Cull, P., Enveloping implies global stability, Difference Equations and Discrete Dynamical Systems (Proceedings of the 9th Annual International Conference on Difference Equations and Applications, Los Angeles, Calif, USA, 2004), 71-85, (2005), World Science Publisher · Zbl 1094.39005
[17] El-Morshedy, H. A.; López, V. J., Global attractors for difference equations dominated by one-dimensional maps, Journal of Difference Equations and Applications, 14, 4, 391-410, (2008) · Zbl 1142.39009
[18] Lopez, V. J., The Y2K problem revisited, Journal of Difference Equations and Applications, 16, 1, 105-119, (2010) · Zbl 1206.39012
[19] Merino, O., Global attractivity of the equilibrium of a difference equation: an elementary proof assisted by computer algebra system, Journal of Difference Equations and Applications, 17, 1, 33-41, (2011) · Zbl 1216.39022
[20] Kulenovic, M. R.; Ladas, G., Dynamics of Second Order Rational Difference Equations, (2002), Boca Raton, Fla, USA: Chapman and Hall/CRC, Boca Raton, Fla, USA · Zbl 0981.39011
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