scholarly journals Global attractivity for a nonautonomous Nicholson’s equation with mixed monotonicities

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 589-607
Author(s):  
Teresa Faria ◽  
Henrique C Prates
Keyword(s):  
Local Stability ◽  
Schwarzian Derivative ◽  
Global Attractivity ◽  
Recent Literature ◽  
Sufficient Conditions ◽  
Time Dependent ◽  
Positive Equilibrium ◽  
Time Varying ◽  
Monotone Functions ◽  
Open Problems

Abstract We consider a Nicholson’s equation with multiple pairs of time-varying delays and nonlinear terms given by mixed monotone functions. Sufficient conditions for the permanence, local stability and global attractivity of its positive equilibrium K are established. The main novelty here is the construction of a suitable auxiliary difference equation x n+1 = h(x n ) with h having negative Schwarzian derivative, and its application to derive the attractivity of K for a model with one or more pairs of time-dependent delays. Our criteria depend on the size of some delays, improve results in recent literature and provide answers to open problems.

scholarly journals On Global Attractivity of a Class of Nonautonomous Difference Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Wanping Liu ◽  
Xiaofan Yang ◽  
Jianqiu Cao
Keyword(s):  
Positive Solution ◽  
Difference Equations ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Open Problems ◽  
The Family ◽  
Recursive Equations

We mainly investigate the global behavior to the family of higher-order nonautonomous recursive equations given byyn=(p+ryn-s)/(q+ϕn(yn-1,yn-2,…,yn-m)+yn-s),n∈ℕ0, withp≥0,r,q>0,s,m∈ℕand positive initial values, and present some sufficient conditions for the parameters and mapsϕn:(ℝ+)m→ℝ+,n∈ℕ0, under which every positive solution to the equation converges to zero or a unique positive equilibrium. Our main result in the paper extends some related results from the work of Gibbons et al. (2000), Iričanin (2007), and Stević (vol. 33, no. 12, pages 1767–1774, 2002; vol. 6, no. 3, pages 405–414, 2002; vol. 9, no. 4, pages 583–593, 2005). Besides, several examples and open problems are presented in the end.

scholarly journals Global Stability of a Discrete Mutualism Model

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Kun Yang ◽  
Xiangdong Xie ◽  
Fengde Chen
Keyword(s):  
Iterative Method ◽  
Comparison Principle ◽  
Local Stability ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Global Asymptotical Stability ◽  
Interior Equilibrium ◽  
Linear Approximation Method ◽  
Mutualism Model

A discrete mutualism model is studied in this paper. By using the linear approximation method, the local stability of the interior equilibrium of the system is investigated. By using the iterative method and the comparison principle of difference equations, sufficient conditions which ensure the global asymptotical stability of the interior equilibrium of the system are obtained. The conditions which ensure the local stability of the positive equilibrium is enough to ensure the global attractivity are proved.

Dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey

2017 ◽  
Vol 10 (08) ◽  
pp. 1750119 ◽  
Author(s):  
Wensheng Yang
Keyword(s):  
Lyapunov Function ◽  
Iterative Method ◽  
Functional Response ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

The dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey is considered in this work. By applying the comparison principle, linearized method, Lyapunov function and iterative method, we are able to achieve sufficient conditions of the permanence, the local stability and global stability of the boundary equilibria and the positive equilibrium, respectively. Our result complements and supplements some known ones.

QUALITATIVE ANALYSIS FOR THE MERKIN ENZYME REACTION SYSTEM

2002 ◽  
Vol 10 (02) ◽  
pp. 167-182
Author(s):  
YUQUAN WANG ◽  
ZUORUI SHEN
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Local Stability ◽  
Bifurcation Theory ◽  
Sufficient Conditions ◽  
Reaction System ◽  
Enzyme Reaction ◽  
Qualitative Theory ◽  
Positive Equilibrium ◽  
Hopf Bifurcations

Applying qualitative theory and Hopf bifurcation theory, we detailedly discuss the Merkin enzyme reaction system, and the sufficient conditions derived for the global stability of the unique positive equilibrium, the local stability of three equilibria and the existence of limit cycles. Meanwhile, we show that the Hopf bifurcations may occur. Using MATLAB software, we present three examples to simulate these conclusions in this paper.

GLOBAL STABILITY OF A CHEMOSTAT MODEL WITH DELAYED RESPONSE IN GROWTH AND A LETHAL EXTERNAL INHIBITOR

2008 ◽  
Vol 01 (04) ◽  
pp. 503-520 ◽  
Author(s):  
ZHIQI LU ◽  
JINGJING WU
Keyword(s):  
Global Stability ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Delayed Response ◽  
Qualitative Properties ◽  
Chemostat Model ◽  
Stability And Instability ◽  
Discrete Delays ◽  
Globally Stable

A competition model between two species with a lethal inhibitor in a chemostat is analyzed. Discrete delays are used to describe the nutrient conversion process. The proved qualitative properties of the solution are positivity, boundedness. By analyzing the local stability of equilibria, it is found that the conditions for stability and instability of the boundary equilibria are similar to those in [9]. In addition, the global asymptotic behavior of the system is discussed and the sufficient conditions for the global stability of the boundary equilibria are obtained. Moreover, by numerical simulation, it is interesting to find that the positive equilibrium may be globally stable.

scholarly journals Delay-dependent attractivity on a tick population dynamics model incorporating two distinctive time-varying delays

2021 ◽  
Vol 477 (2251) ◽  
pp. 20210302
Author(s):  
Chuangxia Huang ◽  
Jian Zhang ◽  
Jinde Cao
Keyword(s):  
Population Dynamics ◽  
Global Attractivity ◽  
Positive Equilibrium ◽  
Tick Population ◽  
Time Varying ◽  
Dynamics Model ◽  
Population Dynamics Model ◽  
All Solutions ◽  
Positive Equilibrium Point ◽  
Delay Dependent

In this paper, we aim to investigate the influence of delay on the global attractivity of a tick population dynamics model incorporating two distinctive time-varying delays. By exploiting some differential inequality techniques and with the aid of the fluctuation lemma, we first prove the persistence and positiveness for all solutions of the addressed equation. Consequently, a delay-dependent criterion is derived to assure the global attractivity of the positive equilibrium point. And lastly, some numerical simulations are presented to verify that the obtained results improve and complement some existing ones.

scholarly journals Boundedness and Global Attractivity of a Higher-Order Nonlinear Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-17
Author(s):  
Xiu-Mei Jia ◽  
Wan-Tong Li
Keyword(s):  
Difference Equation ◽  
Global Attractor ◽  
Positive Solutions ◽  
Local Stability ◽  
Global Attractivity ◽  
Initial Conditions ◽  
Higher Order ◽  
Nonlinear Difference Equation ◽  
Positive Equilibrium ◽  
Prime Period

We investigate the local stability, prime period-two solutions, boundedness, invariant intervals, and global attractivity of all positive solutions of the following difference equation: , , where the parameters and the initial conditions . We show that the unique positive equilibrium of this equation is a global attractor under certain conditions.

scholarly journals Global Attractivity for Lasota–Wazewska-Type System with Patch Structure and Multiple Time-Varying Delays

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
Zhiwen Long ◽  
Yanxiang Tan
Keyword(s):  
Global Attractivity ◽  
Type System ◽  
Small Time ◽  
Positive Equilibrium ◽  
Multiple Time ◽  
Time Varying ◽  
Patch Structure ◽  
Small Time Delay ◽  
Inequality Techniques ◽  
Theoretical Results

This paper aims to study the asymptotic behavior of Lasota–Wazewska-type system with patch structure and multiple time-varying delays. Based on the fluctuation lemma and some differential inequality techniques, we prove that the positive equilibrium is a global attractor of the addressed system with small time delay. Finally, we provide an example to illustrate the feasibility of the theoretical results.

scholarly journals Hopf Bifurcation Analysis for a Computer Virus Model with Two Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Computer Virus ◽  
Positive Equilibrium ◽  
Local Hopf Bifurcation ◽  
Two Delays ◽  
Virus Model ◽  
Normal Form Method ◽  
Manifold Theory

This paper is concerned with a computer virus model with two delays. Its dynamics are studied in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the positive equilibrium and existence of the local Hopf bifurcation are obtained by regarding the possible combinations of the two delays as a bifurcation parameter. Furthermore, explicit formulae for determining direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are obtained by using the normal form method and center manifold theory. Finally, some numerical simulations are presented to support the theoretical results.

scholarly journals Hopf Bifurcation of a Predator-Prey System with Delays and Stage Structure for the Prey

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
The Stability

This paper is concerned with a Holling type III predator-prey system with stage structure for the prey population and two time delays. The main result is given in terms of local stability and bifurcation. By choosing the time delay as a bifurcation parameter, sufficient conditions for the local stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation with respect to both delays are obtained. In particular, explicit formulas that can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form method and center manifold theorem. Finally, numerical simulations supporting the theoretical analysis are also included.

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