scholarly journals Global Asymptotic Stability of a Rational System

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Lin-Xia Hu ◽  
Xiu-Mei Jia
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Stability ◽  
Difference Equation ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Major Conclusion ◽  
Initial Value ◽  
Globally Asymptotically Stable ◽  
Rational System ◽  
The Difference

The main goal of this paper is to investigate the global asymptotic behavior of the difference equationxn+1=β1xn/A1+yn,yn+1=β2xn+γ2yn/xn+yn,n=0,1,2,…withβ1,β2,γ2,A1∈(0,∞)and the initial value(x0,y0)∈[0,∞)×[0,∞)such thatx0+y0≠0. The major conclusion shows that, in the case whereγ2<β2, if the unique positive equilibrium(x-,y-)exists, then it is globally asymptotically stable.

scholarly journals Global Behavior of a Discrete Survival Model with Several Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Meirong Xu ◽  
Yuzhen Wang
Keyword(s):  
Difference Equation ◽  
Sufficient Conditions ◽  
Survival Model ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
All Solutions ◽  
The Difference ◽  
Several Delays

The difference equationyn+1−yn=−αyn+∑j=1mβje−γjyn−kjis studied and some sufficient conditions which guarantee that all solutions of the equation are oscillatory, or that the positive equilibrium of the equation is globally asymptotically stable, are obtained.

scholarly journals Global Asymptotic Stability for a Fourth-Order Rational Difference Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-7
Author(s):  
Meseret Tuba Gülpinar ◽  
Mustafa Bayram
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Fourth Order ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Asymptotically Stable ◽  
Nontrivial Solutions ◽  
Globally Asymptotically Stable ◽  
Rational Difference Equation ◽  
Positive Equilibrium Point

Our aim is to investigate the global behavior of the following fourth-order rational difference equation: , where and the initial values . To verify that the positive equilibrium point of the equation is globally asymptotically stable, we used the rule of the successive lengths of positive and negative semicycles of nontrivial solutions of the aforementioned equation.

scholarly journals Dynamics of a Family of Nonlinear Delay Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Qiuli He ◽  
Taixiang Sun ◽  
Hongjian Xi
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

We study the global asymptotic stability of the following difference equation:xn+1=f(xn-k1,xn-k2,…,xn-ks;xn-m1,xn-m2,…,xn-mt),n=0,1,…,where0≤k1<k2<⋯<ksand0≤m1<m2<⋯<mtwith{k1,k2,…,ks}⋂‍{m1,m2,…,mt}=∅,the initial values are positive, andf∈C(Es+t,(0,+∞))withE∈{(0,+∞),[0,+∞)}. We give sufficient conditions under which the unique positive equilibriumx-of that equation is globally asymptotically stable.

scholarly journals Global Asymptotic Stability for Linear Fractional Difference Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
A. Brett ◽  
E. J. Janowski ◽  
M. R. S. Kulenović
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Positive Equilibrium ◽  
Real Numbers ◽  
Fractional Difference ◽  
Fractional Difference Equation ◽  
The Difference

Consider the difference equation xn+1=(α+∑i=0kaixn-i)/(β+∑i=0kbixn-i),  n=0,1,…, where all parameters α,β,ai,bi,  i=0,1,…,k, and the initial conditions xi,  i∈{-k,…,0} are nonnegative real numbers. We investigate the asymptotic behavior of the solutions of the considered equation. We give easy-to-check conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation.

Non Linear Difference Equation of Orlicz Type

2017 ◽  
Vol 14 (1) ◽  
pp. 306-313
Author(s):  
Awad. A Bakery ◽  
Afaf. R. Abou Elmatty
Keyword(s):  
Difference Equation ◽  
Equilibrium Point ◽  
Positive Solution ◽  
Orlicz Function ◽  
Sufficient Conditions ◽  
Linear Difference Equation ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Numerical Examples ◽  
The Difference

We give here the sufficient conditions on the positive solutions of the difference equation xn+1 = α+M((xn−1)/xn), n = 0, 1, …, where M is an Orlicz function, α∈ (0, ∞) with arbitrary positive initials x−1, x0 to be bounded, α-convergent and the equilibrium point to be globally asymptotically stable. Finally we present the condition for which every positive solution converges to a prime two periodic solution. Our results coincide with that known for the cases M(x) = x in Ref. [3] and M(x) = xk, where k ∈ (0, ∞) in Ref. [7]. We have given the solution of open problem proposed in Ref. [7] about the existence of the positive solution which eventually alternates above and below equilibrium and converges to the equilibrium point. Some numerical examples with figures will be given to show our results.

scholarly journals Global Asymptotic Stability of a Nonautonomous Difference Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Mehmet Gümüş ◽  
Özkan Öcalan
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Initial Values ◽  
Period 2 ◽  
Prime Period

We study the following nonautonomous difference equation:xn+1=(xnxn-1+pn)/(xn+xn-1),n=0,1,…, wherepn>0is a period-2 sequence and the initial valuesx-1,x0∈(0,∞). We show that the unique prime period-2 solution of the equation above is globally asymptotically stable.

scholarly journals Some Discussions on the Difference Equationxn+1=α+(xn-1m/xnk)

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Awad A. Bakery
Keyword(s):  
Periodic Solution ◽  
Difference Equation ◽  
Equilibrium Point ◽  
Positive Solution ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Numerical Examples ◽  
The Difference

We give in this work the sufficient conditions on the positive solutions of the difference equationxn+1=α+(xn-1m/xnk),  n=0,1,…, whereα,k, andm∈(0,∞)under positive initial conditionsx-1,  x0to be bounded,α-convergent, the equilibrium point to be globally asymptotically stable and that every positive solution converges to a prime two-periodic solution. Our results coincide with that known for the casesm=k=1of Amleh et al. (1999) andm=1of Hamza and Morsy (2009). We offer improving conditions in the case ofm=1of Gümüs and Öcalan (2012) and explain our results by some numerical examples with figures.

scholarly journals The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
M. R. S. Kulenović ◽  
Connor O’Loughlin ◽  
E. Pilav
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Invariant Manifolds ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Period Doubling ◽  
Positive Equilibrium ◽  
Equilibrium Solutions ◽  
The Difference

We present the bifurcation results for the difference equation x n + 1 = x n 2 / a x n 2 + x n − 1 2 + f where a and f are positive numbers and the initial conditions x − 1 and x 0 are nonnegative numbers. This difference equation is one of the perturbations of the sigmoid Beverton–Holt difference equation, which is a major mathematical model in population dynamics. We will show that this difference equation exhibits transcritical and Neimark–Sacker bifurcations but not flip (period-doubling) bifurcation since this difference equation cannot have period-two solutions. Furthermore, we give the asymptotic approximation of the invariant manifolds, stable, unstable, and center manifolds of the equilibrium solutions. We give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium.

scholarly journals Global Behavior ofxn+1=(α+βxn-k)/(γ+xn)

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Chenquan Gan ◽  
Xiaofan Yang ◽  
Wanping Liu
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Global Stability ◽  
Initial Conditions ◽  
Global Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Difference

This paper aims to investigate the global stability of negative solutions of the difference equationxn+1=(α+βxn-k)/(γ+xn),n=0,1,2,…, where the initial conditionsx-k,…,x0∈-∞,0,kis a positive integer, and the parametersβ,  γ<0,  α>0. By utilizing the invariant interval and periodic character of solutions, it is found that the unique negative equilibrium is globally asymptotically stable under some parameter conditions. Additionally, two examples are given to illustrate the main results in the end.

scholarly journals Globally asymptotic stability of a stochastic mutualism model with saturated response

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3893-3900
Author(s):  
Jingliang Lv ◽  
Heng Liu ◽  
Yifeng Zhang
Keyword(s):  
Asymptotic Stability ◽  
Stochastic Model ◽  
Positive Solution ◽  
Asymptotically Stable ◽  
Unique Positive Solution ◽  
Initial Value ◽  
Globally Asymptotically Stable ◽  
Globally Asymptotic Stability ◽  
Mutualism Model

A two-species stochastic mutualism model with saturated response is proposed and investigated in this paper. We demonstrate that there exists a unique positive solution to the model for any positive initial value. Under some conditions, we show that the stochastic model is globally asymptotically stable. Finally, we work out some figures to illustrate our results.

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