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.

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.

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 On a System of k-Difference Equations of Order Three

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ibrahim Yalçınkaya ◽  
Hijaz Ahmad ◽  
Durhasan Turgut Tollu ◽  
Yong-Min Li
Keyword(s):  
Positive Solutions ◽  
Difference Equations ◽  
Initial Conditions ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Numerical Examples ◽  
Theoretical Results

In this paper, we deal with the global behavior of the positive solutions of the system of k -difference equations u n + 1 1 = α 1 u n − 1 1 / β 1 + α 1 u n − 2 2 r 1 ,   u n + 1 2 = α 2 u n − 1 2 / β 2 + α 2 u n − 2 3 r 2 , … ,   u n + 1 k = α k u n − 1 k / β k + α k u n − 2 1 r k , n ∈ ℕ 0 , where the initial conditions u − l i l = 0,1,2 are nonnegative real numbers and the parameters α i , β i , γ i , and r i are positive real numbers for i = 1,2 , … , k , by extending some results in the literature. By the end of the paper, we give three numerical examples to support our theoretical results related to the system with some restrictions on the parameters.

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 On Stability Analysis of Higher-Order Rational Difference Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Abdul Khaliq ◽  
H. S. Alayachi ◽  
M. S. M. Noorani ◽  
A. Q. Khan
Keyword(s):  
Stability Analysis ◽  
Equilibrium Points ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Numerical Examples ◽  
Local Asymptotic Stability ◽  
Recursive Sequence ◽  
Rational Difference Equation ◽  
Theoretical Results

In this paper, we study the equilibrium points, local asymptotic stability of equilibrium points, global behavior of equilibrium points, boundedness and periodicity of the rational recursive sequence wn+1=wn−pα+βwn/γwn+δwn−r, where γwn≠−δwn−r for r∈0,∞, α, β, γ, δ∈0,∞, and r>p≥0. With initial values w−p,w−p+1,…,w−r,w−r+1,…,w−1, and w0 are positive real numbers. Some numerical examples are given to verify our theoretical results.

On some difference equations of exponential form

2018 ◽  
pp. 581-584
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Difference Equations ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
Exponential Form ◽  
Numerical Examples

In this paper, the local asymptotic behavior of positive solutions of some exponential difference equations x_(n+1)=(x_n+x_(n-k))/(1+x_(n-k) e^(x_(n-k) ) ) , k ∈ N, n=0,1,2,… are investigated where the initial conditions are arbitrary positive real numbers. Furthermore, some numerical examples are presented to verify our 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 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.

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 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>

Sign in / Sign up

Export Citation Format

Share Document