Basins of Attraction of an Anti-competitive Discrete Rational System
DOI:
https://doi.org/10.5644/SJM.08.2.07Keywords:
Difference equations, anti-competitive, map, stability, stable setAbstract
We investigate the global asymptotic behavior of solutions of the following
anti-competitive system of difference equations
\begin{equation*}
x_{n+1}=\frac{\gamma _{1}y_{n}}{A_{1}+x_{n}},\quad
y_{n+1}=\frac{\beta _{2}x_{n}+\gamma _{2}y_{n}}{y_{n}},\quad
n=0,1,\dots,
\end{equation*}
where the parameters $\gamma _1,\gamma _2,\beta _2,A_{1}$ are positive numbers and the initial conditions $x_{0}\geq 0,y_{0}>0$. We find the basins of attraction of all attractors of the system, which are the equilibrium point and period-two solutions.
2000 Mathematics Subject Classification. 37E30, 37G99, 39A10, 39A11
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References
A. Brett, M. Gari´c-Demirovi´c, M. R. S. Kulenovi´c and M. Nurkanovi´c, Global behavior of two competitive rational systems of difference equations in the plane, Comm. Appl. Nonlinear Anal., 16 (2009), 1–18.
E. Camouzis, M. R. S. Kulenovi´c, G. Ladas and O. Merino, Rational systems in the plane-open problems and conjectures, J. Difference Equ. Appl., 15 (2009), 303–323.
D. Clark and M. R. S. Kulenovi´c, On a coupled system of rational difference equations, Comput. Math. Appl., 43 (2002), 849–867.
D. Clark, M. R. S. Kulenovi´c, and J. F. Selgrade, Global asymptotic behavior of a two dimensional difference equation modelling competition, Nonlinear Anal., TMA, 52 (2003), 1765–1776.
C. A. Clark, M. R. S. Kulenovi´c, and J.F. Selgrade, On a system of rational difference equations, J. Difference Equ. Appl., 11 (2005), 565–580.
M. Garic-Demirovi´c, M. R. S. Kulenovi´c and M. Nurkanovi´c, Global behavior of four competitive rational systems of difference equations in the plane, Discrete Dyn. Nat. Soc., (2009), Art. ID 153058, 34 pages.
M. Gari´c-Demirovi´c and M. Nurkanovi´c, Dynamics of an anti-competitive two dimensional rational system of difference equations, Sarajevo J. Math., 7 (19) (2011), 39–56.
M. Hirsch and H. Smith, Monotone Dynamical Systems, Handbook of Differential Equations, Ordinary Differential Equations, (second volume), 239–357, Elsevier B. V., Amsterdam, 2005.
S. Kalabuˇsi´c and M. R. S. Kulenovi´c, Dynamics of certain anti-competitive systems of rational difference equations in the plane, J. Difference Equ. Appl., 17 (2011), 1599–1615.
S. Kalabuˇsi´c, M. R. S. Kulenovi´c and E. Pilav, Global dynamics of anti-competitive systems in the plane, (submitted).
M. R. S. Kulenovi´c and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman and Hall/CRC, Boca Raton, London, 2002.
M. R. S. Kulenovi´c and O. Merino, Competitive-exclusion versus competitivecoexistence for systems in the plane, Discrete Cont. Dyn. Syst. Ser. B, 6 (2006), 1141–1156.
M. R. S. Kulenovi´c and O. Merino, A global attractivity result for maps with invariant boxes, Discrete Cont. Dyn. Syst. Ser. B, 6 (2006), 97–110.
M. R. S. Kulenovi´c and O. Merino, Global bifurcation for discrete competitive systems in the plane, Discrete Cont. Dyn. Syst. Ser. B, 12 (2009), 133–149.
M. R. S. Kulenovi´c and O. Merino, Invariant manifolds for competitive discrete systems in the plane, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (8) (2010), 2471–2486.
M. R. S. Kulenovi´c and M. Nurkanovi´c, Asymptotic behavior of two dimensional linear fractional system of difference equations, Rad. Mat., 11 (2002), 59–78.
M. R. S. Kulenovi´c and M. Nurkanovi´c, Asymptotic behavior of a competitive system of linear fractional difference equations, J. Inequal. Appl., 2 (2005), 127–144.
M. R. S. Kulenovi´c and M. Nurkanovi´c, Asymptotic behavior of a system of linear fractional difference equations, Adv. Difference Equ., (2006), Art. ID 19756, 13 pp.
M. R. S. Kulenovi´c and M. Nurkanovi´c, Basins of attraction of an anti-competitive system of difference equations, Comm. Appl. Nonlinear Anal., 19 (2) (2012), 41–53.
C. Robinson, Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, 1995.
H. L. Smith, Planar competitive and cooperative difference equations, J. Difference Equ. Appl., 3 (1998), 335–357.
H. L. Smith, The discrete dynamics of monotonically decomposable maps, J. Math. Biology, 53 (2006), 747–758.
