Global dynamics of a competitive system of rational difference equations (Q2795248)

The authors consider the competitive system of rational difference equations NEWLINE\[NEWLINE x_{n+1}=\frac{\alpha +\beta x_{n-1}}{\gamma+\delta y_n^2},\quad y_{n+1}=\frac{\alpha_1+\beta_1y_{n-1}}{\gamma_1+\delta_1x_n^2},\quad n=0,1,\dots, \tag{\(*\)} NEWLINE\]NEWLINE where \(\alpha,\beta,\gamma,\delta, \alpha_1,\beta_1,\gamma_1,\delta_1\) and initial conditions \(x_0,x_{-1},y_0,y_{-1}\) are positive real numbers. They study boundedness character, existence and uniqueness of positive equilibrium point, local asymptotic stability and global stability of the unique positive equilibrium point and the rate of convergence solutions of the system. Some numerical examples which illustrate the results of the paper are given as well.NEWLINENEWLINEA typical statement given in the paper is:NEWLINENEWLINETheorem. (i) Every positive solution \(\{(x_n,y_n)\}\) of \((*)\) is bounded and persists, i.e., there exist positive constants \(L_1,L_2,U_1,U_2\) such that \(L_1\leq x_n\leq U_1\), \(L_2\leq y_n\leq U_2\), \(n=0,1,\dots.\) (ii) Let \(\{(x_n,y_n)\}\) be a positive solution of \((*)\). If \(\beta<\gamma\) and \(\gamma_1>\delta_1U_1^2\), then \(y_n\to 0\) as \(n\to \infty\). If \(\beta_1<\gamma_1\) and \(\beta>\gamma+\delta U_2^2\), then \(x_n\to \infty\) as \(n\to \infty\).