Behaviour of two-dimensional competitive system of nonlinear difference equations of higher order (Q2286565)

Summary: We generalise a recent result of \textit{M. Mansour} et al. [Discrete Dyn. Nat. Soc. 2012, Article ID 406821, 17 p. (2012; Zbl 1248.39014)] and study other related systems that deal with the dynamics of a competitive population model described by a system of nonlinear difference equations. Particularly, we consider a discrete-competitive system of the form \[x_{n+1} = f(x_{n-(2k-1)},y_{n-(k-1)}),y_{n+1} = g(x_{n-(2k-1)},y_{n-(k-1)}),\quad n \in \mathbb{N}_0,\] where \(k\in\mathbb{N}\) and \(f:\mathbb{R}\setminus \mathcal{F}_f \rightarrow\mathbb{R}\) and \(g: \mathbb R \setminus \mathcal{F}_g \rightarrow\mathbb{R}\), where \(\mathcal{F}_f\) and \(\mathcal{F}_g\) denote the forbidden sets of \(f\) and \(g\), respectively. This work, in turn, generalises several other results on system of nonlinear difference equations. See, for example, the work of \textit{M. Alghamdi} et al. [``On the solutions of some systems of second order rational difference equations'', Life Sci. J. 10, No. 3, 344--351 (2013; \url{doi:10.7537/marslsj100313.53})], \textit{E. M. Elsayed} [Math. Comput. Modelling 55, No. 3--4, 378--384 (2012; Zbl 1255.39003)], \textit{T. F. Ibrahim} et al. [``Behavior of two-dimensional competitive system of nonlinear rational recursive sequences'' (submitted)], \textit{A. S. Kurbanli} [Discrete Dyn. Nat. Soc. 2011, Article ID 932362, 12 p. (2011; Zbl 1217.39015); Adv. Difference Equ. 2011, Paper No. 40, 8 p. (2011; Zbl 1271.39009)] and \textit{N. Touafek} and \textit{E. M. Elsayed} [Math. Comput. Modelling 55, No. 7--8, 1987--1997 (2012; Zbl 1255.39011); Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 55(103), No. 2, 217--224 (2012; Zbl 1274.39032)]. Furthermore, the one-dimensional case of the given system provides a generalisation of a series of paper of Elsayed on nonlinear difference equations.