### 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.

$$ 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.

#### Keywords

Difference equation; Solutions; Convergence; Periodicity; Competitive

#### Full Text:

PDFDOI: http://dx.doi.org/10.3968%2Fj.sms.1923845220120501.1133

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