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Global dynamics of some system of second-order difference equations. (English) Zbl 1479.39013

The authors investigate the qualitative behavior of the systems of exponential difference equations \[ x_{n+1}=\frac{\alpha_1+\beta_1e^{-x_{n-1}}}{\gamma_1+y_n}, \quad y_{n+1}=\frac{\alpha_2+\beta_2e^{-y_{n-1}}}{\gamma_2+x_n}, \] \[ x_{n+1}=\frac{\alpha_1+\beta_1e^{-y_{n-1}}}{\gamma_1+x_n}, \quad y_{n+1}=\frac{\alpha_2+\beta_2e^{-x_{n-1}}}{\gamma_2+y_n}, \] where the parameters \(\alpha_i, \beta_i, \) and \( \gamma_i \) for \( i\in \{1, 2\} \) and the initial conditions \( x_{-1}, x_0, y_{-1} \) and \( y_0 \) are positive real numbers. They prove the boundedness and the persistence of positive solutions of this system. Moreover, it is shown that the unique positive equilibrium point of the system is globally asymptotically stable under certain conditions on the parameters. Furthermore, the rate of convergence of the positive solutions which converge to their unique positive equilibrium point is computed. Finally, some illustrative numerical examples are provided.

MSC:

39A22 Growth, boundedness, comparison of solutions to difference equations
39A30 Stability theory for difference equations
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[1] R. P. Agarwal, Difference Equations and Inequalities, \(2^{nd}\) edition, Dekker, New York, 2000. · Zbl 0952.39001
[2] Q. Din, Global stability of a population models, Chaos, Solitons & Fractals, 59, 119-128 (2014) · Zbl 1348.92123 · doi:10.1016/j.chaos.2013.12.008
[3] Q. Din; E. M. Elsayed, Stability analysis of a discrete ecological model, Computational Ecology and Software, 4, 89-103 (2014)
[4] H. El-Metwally; E. A. Grove; G. Ladas; R. Levins; M. Radin, On the difference equation \(x_{n+1} = \alpha + \beta x_{n-1}e^{-x_n}\), Nonlinear Anal., 47, 4623-4634 (2001) · Zbl 1042.39506 · doi:10.1016/S0362-546X(01)00575-2
[5] N. Fotiades; G. Papaschinopoulos, Existence, uniqueness and attractivity of prime period two solution for a difference equation of exponential form, Appl. Math. Comput., 218, 11648-11653 (2012) · Zbl 1280.39011 · doi:10.1016/j.amc.2012.05.047
[6] E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC, Boca Raton, Fla, 2005. · Zbl 1078.39009
[7] E. A. Grove; G. Ladas; N. R. Prokup; R. Levins, On the global behavior of solutions of a biological model, Comm. Appl. Nonlinear Anal., 7, 33-46 (2000) · Zbl 1110.39300
[8] T. Hong Thai, Asymptotic behavior of the solution of a system of difference equations, Int. J. Difference Equ., 13, 157-171 (2018)
[9] T. Hong Thai; V. Van Khuong, Stability analysis of a system of second-order difference equations, Math. Methods Appl. Sci., 39, 3691-3700 (2016) · Zbl 1357.39010 · doi:10.1002/mma.3816
[10] V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. · Zbl 0787.39001
[11] M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman and Hall/CRC, Boca Raton, Fla, 2002. · Zbl 0981.39011
[12] I. Ozturk; F. Bozkurt; S. Ozen, On the difference equation \(y_{n+1} = \dfrac{\alpha+\beta e^{-y_n}}{\gamma + y_{n-1}} \), Appl. Math. Comput., 181, 1387-1393 (2006) · Zbl 1108.39012 · doi:10.1016/j.amc.2006.03.007
[13] G. Papaschinopoluos; G. Ellina; K. B. Papadopoulos, Asymptotic behavior of the positive solutions of an exponential type system of difference equations, Appl. Math. Comput., 245, 181-190 (2014) · Zbl 1335.39021 · doi:10.1016/j.amc.2014.07.074
[14] G. Papaschinopoluos; N. Fotiades; C. J. Schinas, On a system of difference equations including negative exponential terms, J. Difference Equ. Appl., 20, 717-732 (2014) · Zbl 1291.39040 · doi:10.1080/10236198.2013.814647
[15] G. Papaschinopoluos; M. A. Radin; C. J. Schinas, On the system of two difference equations of exponential form: \(x_{n+1} = a+b x_{n-1}e^{-y_n}, y_{n+1} = c+d y_{n-1}e^{-x_n}\), Math. Comput. Modelling, 54, 2969-2977 (2011) · Zbl 1235.39006 · doi:10.1016/j.mcm.2011.07.019
[16] G. Papaschinopoluos; M. Radin; C. J. Schinas, Study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form, Appl. Math. Comput., 218, 5310-5318 (2012) · Zbl 1245.39011 · doi:10.1016/j.amc.2011.11.014
[17] G. Papaschinopoluos; C. J. Schinas, On the dynamics of two exponential type systems of difference equations, Comput. Math. Appl., 64, 2326-2334 (2012) · Zbl 1274.39026 · doi:10.1016/j.camwa.2012.04.002
[18] M. Pituk, More on Poincaré’s and Peron’s theorems for difference equations, J. Difference Equ. Appl., 8, 201-216 (2002) · Zbl 1002.39014 · doi:10.1080/10236190211954
[19] H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. · Zbl 1020.39007
[20] W. Wang; H. Feng, On the dynamics of positive solutions for the difference equation in a new population model, J. Nonlinear Sci. Appl., 9, 1748-1754 (2016) · Zbl 1334.39038 · doi:10.22436/jnsa.009.04.30
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