On the dynamics of the nonlinear rational difference equation $ { x_{n+1}} = \frac{{\alpha {x_{n-m}}} \ \ + \ \ \delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}} \ \ { x_{n-l}} \ \ \left({{x_{n-k}} \ \ + \ \ {x_{n-l}}} \ \ \right) }} $

Abstract

In this paper, we discuss some qualitative properties of the positive solutions to the following rational nonlinear difference equation $ { x_{n+1}} = \frac{{\alpha {x_{n-m}}} \ \ + \ \ \delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}} \ \ { x_{n-l}} \ \ \left({{x_{n-k}} \ \ + \ \ {x_{n-l}}} \ \ \right) }} $, $ n = 0, 1, 2, ... $ where the parameters $ \alpha, \beta, \gamma, \delta \in (0, \infty) $, while $ m, k, l $ are positive integers, such that $ m < k < l. $ The initial conditions $ {x_{-m}}, ..., {x_{-k}}, ..., {x_{-l}}, ..., {x_{-1}}, ..., {x_{0}} $ are arbitrary positive real numbers. We will give some numerical examples to illustrate our results.

Keywords

• difference equations
• rational difference equations
• qualitative properties of solutions of difference equations
• equilibrium
• oscillates
• prime period two solution
• globally asymptotically stable
• points