On a Competitive System of Rational Difference Equations

Year 2019, Volume: 2 Issue: 4, 224 - 228, 26.12.2019

Mehmet Gümüş

https://doi.org/10.32323/ujma.649122

Abstract

This paper aims to investigate the global stability and the rate of convergence of positive solutions that converge to the equilibrium point of the system of difference equations in the modeling competitive populations in the form $$ x_{n+1}^{(1)}=\frac{\alpha x_{n-2}^{(1)}}{\beta +\gamma \prod\limits_{i=0}^{2}x_{n-i}^{(2)}},\text{ }x_{n+1}^{(2)}=\frac{\alpha _{1}x_{n-2}^{(2)}}{\beta _{1}+\gamma _{1}\prod\limits_{i=0}^{2}x_{n-i}^{(1)} }\text{, }n=0,1,... $$ where the parameters $\alpha ,\beta ,\gamma ,\alpha _{1},\beta _{1},\gamma _{1}$ are positive numbers and the initial conditions $ x_{-i}^{(1)},x_{-i}^{(2)}$ are arbitrary non-negative numbers for $i\in \{0,1,2\}$.

Keywords

System of difference equation, global asymptotic stability, equilibrium, rate of convergence

References

• [1] D. Burgic, M. R. S. Kulenovic and M. Nurkanovic, Global Dynamics of a Rational System of Difference Equations in the plane, Comm. Appl. Nonlinear Anal., 15 (2008), 71-84.
• [2] D. Burgic and M. Nurkanovic, The Rational System of Nonlinear Difference Equations in the Modeling Competitive Populations, 15th International Research/Expert Conference, Trends in the Devolpment of Machinery and Associated Tehnology, TMT, (2011).
• [3] D. Burgic and A. Huskanovic, The Rational System of Equations in the Modeling Anti-Competitive Populations, 18th International Research/Expert Conference, Trends in the Devolpment of Machinery and Associated Tehnology, TMT, (2014).
• [4] A. Brett, M. Garic-Demirovic, M. R. S. Kulenovic and M. Nurkanovic, Global behavior of two competitive rational systems of difference equations in the plane, Commun. Appl. Nonlinear Anal., 16 (2009), 1-18.
• [5] D. Clark, M. R. S. Kulenovic, and J. F. Selgrade, Global asymptotic behavior of a two dimensional difference equation modelling competition, Nonlinear Analysis. Theory, Methods & Applications, 52 (7) (2003), 1765–1776.
• [6] M. Gocen and M. Guneysu, The Global Attractivity of some ratinal difference equations, J. Comp. Anal. Appl., 25(7) (2018), 1233-1243.
• [7] M. Gocen and A. Cebeci, On the periodic solutions of some systems of higher order difference equations, Rocky Mountain Journal of Mathematics, 48(3) (2018), 845-858.
• [8] T. F. Ibrahim, Two-dimensional fractional system of nonlinear difference equations in the modeling competitive populations, International Journal of Basic & Applied Sciences, 12 (5) (2012), 103-121.
• [9] S. Kalabusic and M. R. S. Kulenovi´c, Rate of convergence of solutions of rational difference equation of second order, Advances in Difference Equations, 2 (2004): 1-19.
• [10] V. Kocic and G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht, (1993).
• [11] M. R. S. Kulenovic and G. Ladas, Dynamics of second order rational difference equations, Chapman & Hall/CRC, Boca Raton, London (2001).
• [12] M. R. S. Kulenovic and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman & Hall/CRC, Boca Raton, London (2002):
• [13] M. Pituk, More on Poincar´e’s and Perron’s Theorems for Difference Equations, Journal of Difference Equations and Applications, 8 (3) (2002), 201-216.
• [14] J. F. Selgrade and M. Ziehe, Convergence to equilibrium in a genetic model with differential viability between the sexes, Journal of Mathematical Biology, 25 (5) (1987), 477–490.
• [15] Q. Zhang, L. Yang and J. Liu, Dynamics of a system of rational third-order difference equation, Advances in Difference Equations, 136 (2012), 1-6.