
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) }} $
Author(s) -
Abeer M. Alotaibi,
AUTHOR_ID,
M. A. El-Moneam,
AUTHOR_ID
Publication year - 2022
Publication title -
aims mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.329
H-Index - 15
ISSN - 2473-6988
DOI - 10.3934/math.2022411
Subject(s) - beta (programming language) , physics , combinatorics , mathematics , computer science , programming language
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.