Closed Form Solution of a Symmetric Competitive System of Rational Difference Equations

Abstract

In this paper, we will study a symmetric competitive three-dimensional system of difference equations in the form:
$$ x_{n+1} = \frac {x_n}{z_n y_n}~ \& ~y_{n+1} = \frac {y_n}{x_n z_n}~ \& ~z_{n+1} = \frac {z_n}{y_n x_n}
\eqno{(1)} $$ where the initial values $x_0$, $y_0$, and $z_0$ are nonzero real numbers. Moreover, we have studied periodicity of solutions for this system. Finally we will give some numerical examples as applications.