Globally Asymptotic Stability of a Delayed Integro-Differential Equation With Nonlocal Diffusion

2017 ◽  
Vol 60 (2) ◽  
pp. 436-448 ◽  
Author(s):  
Peixuan Weng ◽  
Li Liu
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Age Structure ◽  
Population Model ◽  
Zero Solution ◽  
Comparison Method ◽  
Nonlocal Diffusion ◽  
Positive Equilibrium ◽  
Globally Asymptotic Stability ◽  
Analytical Technique

AbstractWe study a population model with nonlocal diòusion, which is a delayed integro-diòerential equation with double nonlinearity and two integrable kernels. By comparison method and analytical technique, we obtain globally asymptotic stability of the zero solution and the positive equilibrium. The results obtained reveal that the globally asymptotic stability only depends on the property of nonlinearity. As an application, we discuss an example for a population model with age structure.

scholarly journals Asymptotic stability of systems with impulses by the direct method of Lyapunov

1988 ◽  
Vol 38 (1) ◽  
pp. 113-123 ◽  
Author(s):  
G.K. Kulev ◽  
D.D. Bainov
Keyword(s):  
Asymptotic Stability ◽  
Direct Method ◽  
Zero Solution ◽  
Globally Asymptotic Stability ◽  
Auxiliary Functions ◽  
Piecewise Continuous

In the present paper the asymptotic and globally asymptotic stability of the zero solution of systems with impulses are investigated. For this purpose piecewise continuous auxiliary functions are used which are analogous to Lyapunov's functions. The theorem of Marachkov on the asymptotic stability of systems without impulses is generalised. The results obtained are formulated in four theorems.

Dynamical Analysis of a Phytoplankton–Zooplankton System with Harvesting Term and Holling III Functional Response

2018 ◽  
Vol 28 (13) ◽  
pp. 1850162 ◽  
Author(s):  
Zhichao Jiang ◽  
Wenzhi Zhang ◽  
Jing Zhang ◽  
Tongqian Zhang
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Release Rate ◽  
Global Asymptotic Stability ◽  
Dynamical Analysis ◽  
Positive Equilibrium ◽  
Diffusion Term ◽  
Stability Of Equilibrium ◽  
Local Hopf Bifurcation ◽  
Ode System

A toxin-producing phytoplankton and zooplankton system is investigated. Considering that zooplankton can be harvested for food in some bodies of water, the harvesting term is introduced to zooplankton population. Firstly, from the ordinary differential equation (ODE) system, we obtain the global asymptotic stability of equilibrium and optimal capture problem. Secondly, based on the ODE system, the diffusion term is introduced and the global asymptotic stability of the steady state solution is obtained. As a result, the diffusion cannot affect the global asymptotic stability of equilibrium, and Turing instability cannot occur. Once again, a delayed differential equation (DDE) system is put forward. The global asymptotic stability of boundary equilibrium and the existence of local Hopf bifurcation at positive equilibrium are discussed. Furthermore, it is proved that there exists at least one positive periodic solution as delay varies in some region by using the global Hopf result of Wu for functional differential equations. Lastly, some numerical simulations are carried out for supporting the theoretical analyses and the positive impacts of harvesting effort, and the release rate of toxin is given. The unstable interval of the positive equilibrium becomes smaller and smaller with the increase of harvesting effort or the release rate of toxin.

scholarly journals ASYMPTOTIC STABILITY OF A NEUTRAL DIFFERENTIAL EQUATION

2002 ◽  
Vol 45 (2) ◽  
pp. 333-347 ◽  
Author(s):  
X. H. Tang ◽  
Xingfu Zou
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Neutral Delay ◽  
Neutral Delay Differential Equation ◽  
Primary 34K20 ◽  
All Solutions ◽  
Delay Differential ◽  
T Tau

AbstractThe uniform stability of the zero solution and the asymptotic behaviour of all solutions of the neutral delay differential equation$$ [x(t)-P(t)x(t-\tau)]'+Q(t)x(t-\sigma)=0,\quad t\ge t_0, $$are investigated, where $\tau,\sigma\in(0,\infty)$, $P\in C([t_0,\infty),\mathbb{R})$, and $Q\in C([t_0,\infty), [0,\infty))$. The obtained sufficient conditions improve the existing results in the literature.AMS 2000 Mathematics subject classification: Primary 34K20; 34K15; 34K40

On the uniform stability for a ‘food-limited’ population model with time delay

1995 ◽  
Vol 125 (5) ◽  
pp. 991-1002 ◽  
Author(s):  
Joseph W.-H. So ◽  
J. S. Yu
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Population Model ◽  
Sufficient Conditions ◽  
Uniform Stability ◽  
Positive Equilibrium

In this paper, we provide sufficient conditions which guarantee the uniform stability as well as asymptotic stability of the positive equilibrium for a food limited population model with time delay.

scholarly journals Stability of solutions and the problem of Aizerman  for sixth-order differential equations

2020 ◽  
pp. 49-58
Author(s):  
Boris S. Kalitine
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Sixth Order ◽  
Stability Of Solutions ◽  
Stability And Instability ◽  
Linear Differential ◽  
Aizerman Problem

This article is devoted to the investigation of stability of equilibrium of ordinary differential equations using the method of semi-definite Lyapunov’s functions. Types of scalar nonlinear sixth-order differential equations for which regular constant auxiliary functions are used are emphasized. Sufficient conditions of global asymptotic stability and instability of the zero solution have been obtained and it has been established that the Aizerman problem has a positive solution concerning the roots of the corresponding linear differential equation. Studies highlight the advantages of using semi-definite functions compared to definitely positive Lyapunov’s functions.

scholarly journals Existence and Globally Asymptotic Stability of Equilibrium Solution for Fractional-Order Hybrid BAM Neural Networks with Distributed Delays and Impulses

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Hai Zhang ◽  
Renyu Ye ◽  
Jinde Cao ◽  
Ahmed Alsaedi
Keyword(s):  
Neural Networks ◽  
Asymptotic Stability ◽  
Fractional Order ◽  
Lyapunov Functional ◽  
Equilibrium Solution ◽  
Distributed Delays ◽  
Bam Neural Networks ◽  
Network Systems ◽  
Globally Asymptotic Stability ◽  
Stability Of Equilibrium

This paper investigates the existence and globally asymptotic stability of equilibrium solution for Riemann-Liouville fractional-order hybrid BAM neural networks with distributed delays and impulses. The factors of such network systems including the distributed delays, impulsive effects, and two different fractional-order derivatives between the U-layer and V-layer are taken into account synchronously. Based on the contraction mapping principle, the sufficient conditions are derived to ensure the existence and uniqueness of the equilibrium solution for such network systems. By constructing a novel Lyapunov functional composed of fractional integral and definite integral terms, the globally asymptotic stability criteria of the equilibrium solution are obtained, which are dependent on the order of fractional derivative and network parameters. The advantage of our constructed method is that one may directly calculate integer-order derivative of the Lyapunov functional. A numerical example is also presented to show the validity and feasibility of the theoretical results.

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