scholarly journals Behavior of a Competitive System of Second-Order Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Q. Din ◽  
T. F. Ibrahim ◽  
K. A. Khan
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Positive Equilibrium ◽  
Positive Real ◽  
Competitive System ◽  
Order Difference ◽  
Rational Difference Equations ◽  
Positive Equilibrium Point ◽  
Boundedness And Persistence ◽  
Theoretical Results

We study the boundedness and persistence, existence, and uniqueness of positive equilibrium, local and global behavior of positive equilibrium point, and rate of convergence of positive solutions of the following system of rational difference equations:xn+1=(α1+β1xn-1)/(a1+b1yn),yn+1=(α2+β2yn-1)/(a2+b2xn), where the parametersαi,βi,ai, andbifori∈{1,2}and initial conditionsx0,x-1,y0, andy-1are positive real numbers. Some numerical examples are given to verify our theoretical results.

scholarly journals Behavior of an Exponential System of Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
A. Q. Khan ◽  
M. N. Qureshi
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Positive Equilibrium ◽  
Qualitative Behavior ◽  
Positive Real ◽  
Exponential System ◽  
Rational Difference Equations ◽  
Boundedness Character ◽  
Positive Equilibrium Point ◽  
Theoretical Results

We study the qualitative behavior of the following exponential system of rational difference equations:xn+1 = αe-yn+βe-yn-1/γ+αxn+βxn-1,  yn+1 = α1e-xn+β1e-xn-1/γ1+α1yn+β1yn-1,  n = 0,1,…,whereα,β,γ,α1,β1, andγ1and initial conditionsx0,  x-1,  yo, and  y-1are positive real numbers. More precisely, we investigate the boundedness character and persistence, existence and uniqueness of positive equilibrium, local and global behavior, and rate of convergence of positive solutions that converges to unique positive equilibrium point of the system. Some numerical examples are given to verify our theoretical results.

scholarly journals Global dynamics of some system of second-order difference equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tran Hong Thai ◽  
Nguyen Anh Dai ◽  
Pham Tuan Anh
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Global Dynamics ◽  
Solution Existence ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Boundedness And Persistence ◽  
Invariant Rectangle ◽  
Theoretical Results

<p style='text-indent:20px;'>In this paper, we study the boundedness and persistence of positive solution, existence of invariant rectangle, local and global behavior, and rate of convergence of positive solutions of the following systems of exponential difference equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} x_{n+1} = \dfrac{\alpha_1+\beta_1e^{-x_{n-1}}}{\gamma_1+y_n},\ y_{n+1} = \dfrac{\alpha_2+\beta_2e^{-y_{n-1}}}{\gamma_2+x_n},\\ x_{n+1} = \dfrac{\alpha_1+\beta_1e^{-y_{n-1}}}{\gamma_1+x_n},\ y_{n+1} = \dfrac{\alpha_2+\beta_2e^{-x_{n-1}}}{\gamma_2+y_n}, \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where the parameters <inline-formula><tex-math id="M1">\begin{document}$ \alpha_i,\ \beta_i,\ \gamma_i $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ i \in \{1,2\} $\end{document}</tex-math></inline-formula> and the initial conditions <inline-formula><tex-math id="M3">\begin{document}$ x_{-1}, x_0, y_{-1}, y_0 $\end{document}</tex-math></inline-formula> are positive real numbers. Some numerical example are given to illustrate our theoretical results.</p>

scholarly journals Stability Analysis of a System of Exponential Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Q. Din ◽  
K. A. Khan ◽  
A. Nosheen
Keyword(s):  
Difference Equations ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Numerical Examples ◽  
Boundedness Character ◽  
Theoretical Results

We study the boundedness character and persistence, existence and uniqueness of positive equilibrium, local and global behavior, and rate of convergence of positive solutions of the following system of exponential difference equations:xn+1=(α1+β1e-xn+γ1e-xn-1)/(a1+b1yn+c1yn-1),yn+1=(α2+β2e-yn+γ2e-yn-1)/(a2+b2xn+c2xn-1), where the parametersαi, βi, γi, ai, bi, andcifori∈{1,2}and initial conditionsx0, x-1, y0, andy-1are positive real numbers. Furthermore, by constructing a discrete Lyapunov function, we obtain the global asymptotic stability of the positive equilibrium. Some numerical examples are given to verify our theoretical results.

Stability analysis of a discrete biological model

2016 ◽  
Vol 09 (02) ◽  
pp. 1650021 ◽  
Author(s):  
A. Q. Khan ◽  
M. N. Qureshi
Keyword(s):  
Equilibrium Point ◽  
Initial Conditions ◽  
Biological Model ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Numerical Examples ◽  
Positive Equilibrium Point ◽  
Theoretical Results

In this paper, we investigate the equilibrium point, local and global behavior of the unique positive equilibrium point, and rate of convergence of positive solutions of following discrete biological model: [Formula: see text] where parameters [Formula: see text] and the initial conditions [Formula: see text] are positive real numbers. Some numerical examples are given to verify theoretical results.

scholarly journals Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

2019 ◽  
pp. 1-12
Author(s):  
Abdualrazaq Sanbo ◽  
Elsayed M. Elsayed ◽  
Faris Alzahrani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Analytical Study ◽  
Initial Conditions ◽  
Specific Form ◽  
Positive Real ◽  
Rational Difference Equations ◽  
Special Cases ◽  
The Difference

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS OF DIFFERENCE EQUATION SYSTEM OF EXPONENTIAL FORM

Fractals ◽  
2020 ◽  
Vol 28 (06) ◽  
pp. 2050118
Author(s):  
ABDUL KHALIQ ◽  
MUHAMMAD ZUBAIR ◽  
A. Q. KHAN
Keyword(s):  
Initial Conditions ◽  
Equation System ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Exponential Form ◽  
Numerical Examples ◽  
Rational Difference Equations ◽  
Boundedness Character ◽  
Theoretical Results

In this paper, we study the boundedness character and persistence, local and global behavior, and rate of convergence of positive solutions of following system of rational difference equations [Formula: see text] wherein the parameters [Formula: see text] for [Formula: see text] and the initial conditions [Formula: see text] are positive real numbers. Some numerical examples are given to verify our theoretical results.

scholarly journals Analytical Study of a System of Dierence Equation

2019 ◽  
pp. 1-18
Author(s):  
Abdualrazaq Sanbo ◽  
Elsayed M. Elsayed
Keyword(s):  
Analytical Study ◽  
Initial Conditions ◽  
Qualitative Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Predator Prey ◽  
Rational Difference Equations ◽  
Predator Prey Model ◽  
Special Cases ◽  
Theoretical Results

We study the qualitative behavior of a predator-prey model, where the carrying capacity of the predators environment is proportional to the number of prey. The considered system is given by the following rational difference equations: where the initial conditions x-2; x-1; x0; y-2; y-1; y0 are arbitrary positive real numbers. Also, we give specic form of the solutions of some special cases of this equation. Some numerical examples are given to verify our theoretical results.

Global behavior of two third order rational difference equations with quadratic terms

2019 ◽  
Vol 69 (1) ◽  
pp. 147-158 ◽  
Author(s):  
R. Abo-Zeid
Keyword(s):  
Explicit Formula ◽  
Difference Equations ◽  
Initial Conditions ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Third Order ◽  
Rational Difference Equations ◽  
The Difference

Abstract In this paper, we determine the forbidden sets, introduce an explicit formula for the solutions and discuss the global behaviors of solutions of the difference equations $$\begin{array}{} \displaystyle x_{n+1}=\frac{ax_{n}x_{n-1}}{bx_{n-1}+ cx_{n-2}},\quad n=0,1,\ldots \end{array} $$ where a,b,c are positive real numbers and the initial conditions x−2,x−1,x0 are real numbers.

scholarly journals Explicit Solutions and Bifurcations for a System of Rational Difference Equations

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 69
Author(s):  
Bashir Al-Hdaibat ◽  
Saleem Al-Ashhab ◽  
Ramadan Sabra
Keyword(s):  
Hypergeometric Function ◽  
Difference Equations ◽  
Explicit Solution ◽  
Initial Conditions ◽  
Global Dynamics ◽  
Explicit Solutions ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equations

In this paper, we consider the explicit solution of the following system of nonlinear rational difference equations: x n + 1 = x n - 1 / x n - 1 + r , y n + 1 = x n - 1 y n / x n - 1 y n + r , with initial conditions x - 1 , x 0 and y 0 , which are arbitrary positive real numbers. By doing this, we encounter the hypergeometric function. We also investigate global dynamics of this system. The global dynamics of this system consists of two kind of bifurcations.

scholarly journals On the Solutions of a System of Third-Order Rational Difference Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
A. M. Alotaibi ◽  
M. S. M. Noorani ◽  
M. A. El-Moneam
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Theoretical Discussion ◽  
Real Numbers ◽  
Nonlinear Difference Equations ◽  
Positive Real ◽  
Third Order ◽  
Numerical Examples ◽  
Rational Difference Equations

The structure of the solutions for the system nonlinear difference equations xn+1=ynyn-2/(xn-1+yn-2), yn+1=xnxn-2/(±yn-1±xn-2), n=0,1,…, is clarified in which the initial conditions x-2, x-1, x0, y-2, y-1, y0 are considered as arbitrary positive real numbers. To exemplify the theoretical discussion, some numerical examples are presented.

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