Asymptotic Properties of Stochastic Functional Kolmogorov-Type System

2008 ◽  
Vol 106 (2) ◽  
pp. 251-263 ◽  
Author(s):  
Fuke Wu ◽  
Yangzi Hu
Keyword(s):  
Asymptotic Properties ◽  
Type System ◽  
Kolmogorov Type ◽  
Stochastic Functional

Unbounded delay stochastic functional Kolmogorov-type system

2010 ◽  
Vol 140 (6) ◽  
pp. 1309-1334 ◽  
Author(s):  
Fuke Wu
Keyword(s):  
Existence Theorems ◽  
Type System ◽  
Environmental Noise ◽  
Population System ◽  
Kolmogorov Type ◽  
Unbounded Delay ◽  
Functional Differential ◽  
The Moment ◽  
Image Position ◽  
Stochastic Functional

In general, population systems are often subject to environmental noise. To examine whether the presence of such noise affects these systems significantly, we perturb the functional Kolmogorov-type systeminto the stochastic functional differential equationWe show that different environmental noise structures have different effects on the population system with unbounded delay. Under two classes of different environmental noise perturbations, we establish existence theorems of the global positive solution to the unbounded delay stochastic functional Kolmogorov-type system. As the desired results for population dynamics, we also examine asymptotic boundedness, including the moment boundedness, stochastically ultimate boundedness and the moment average boundedness in time. To illustrate our idea more clearly, as a special case we also discuss a Lotka–Volterra system with unbounded delay.

scholarly journals Asymptotic properties and stabilization of a neutral type system with constant delay

2021 ◽  
Vol 17 (1) ◽  
pp. 81-96
Author(s):  
Boris G. Grebenshchikov ◽  
Keyword(s):  
Asymptotic Properties ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Type System ◽  
Original System ◽  
Order System ◽  
Constant Delay ◽  
Exponential Factor ◽  
Difference Systems ◽  
The Right

The problem of obtaining sufficient conditions for the asymptotic stability for a certain class of linear systems of a neutral type with constant delay is analyzed in the article. Some coefficients of these systems in the right side have an exponential factor. As a consequence, the study of the stability of such systems with the help of the Lyapunov—Krasovskii functionals is not possible; methods of receiving asymptotic appreciations lead to extremely rough results. By applying the apparatus of difference systems and the properties of simpler systems, which the author examined previous, sufficient conditions for the exponential stability of such systems are obtained. As an example, a second-order system is considered. The graphs of the solutions of the corresponding system, both without neutral members and with the original system where the right-hand side contains neutral terms, are provided. On the basis of theory difference systems, the author proposes an algorithm of stabilization for some systems of a similar type.

On stable integral manifolds for impulsive Kolmogorov systems of fractional order

2017 ◽  
Vol 31 (15) ◽  
pp. 1750168
Author(s):  
Gani Stamov ◽  
Ivanka Stamova
Keyword(s):  
Fractional Order ◽  
Lyapunov Functions ◽  
Type System ◽  
Fractional Integrals ◽  
Order System ◽  
Kolmogorov Type ◽  
Fractal Media ◽  
Integral Manifolds ◽  
Existence And Stability ◽  
Caputo Fractional Order Derivative

In this paper, an impulsive Kolmogorov-type system using the Caputo fractional-order derivative is developed. The fractional-order system displays many interesting dynamic behaviors and fractional integrals can be used to describe the fractal media. The existence and stability of integral manifolds for the impulsive fractional model are considered. The main results are proved by means of piecewise continuous Lyapunov functions and the new fractional comparison principle. The impulses are realized at variable impulsive moments of time and can be considered as a control. Finally, an example is given to illustrate our results.

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