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The Dynamics of an Impulsive Competitive System with Infinite Delay and Diffusion  [PDF]
Hairu Chen, Yuanfu Shao
Journal of Applied Mathematics and Physics (JAMP) , 2018, DOI: 10.4236/jamp.2018.66116
Abstract: In this paper, we consider an impulsive competitive system with infinite delay and diffusion. Firstly, on basis of inequality estimation techniques and comparison theorem of impulsive differential equations, we obtain some sufficient conditions for the permanence and extinction of the system. Then, we establish sufficient conditions for the globally attractive of the system by constructing appropriate Lyapunov function. Besides, under different impulsive conditions, we discuss the effect of time delay and diffusion on dynamic behavior of the competitive system.
The Effect of Impulsive Diffusion on Dynamics of a Stage-Structured Predator-Prey System  [PDF]
Jianjun Jiao
Discrete Dynamics in Nature and Society , 2010, DOI: 10.1155/2010/716932
Abstract: We investigate a predator-prey model with impulsive diffusion on predator and stage structure on prey. The globally attractive condition of prey-extinction periodic solution of the system is obtained by the stroboscopic map of the discrete dynamical system. The permanent condition of the system is also obtained by the theory of impulsive delay differential equation. The results indicate that the discrete time delay has influence on the dynamical behaviors of the system. Finally, some numerical simulations are carried out to support the analytic results. 1. Introduction The dispersal is a ubiquitous phenomenon in the natural world. It is well recognized that the spatial distribution of populations and population dynamics are much affected by spatial heterogeneity and population mobility [1]. The fragmented landscapes are common because the populations of most species occupy mosaic habitats and because of rapid destruction of natural habitats. Briggs and Hoopes [2] identify three mechanisms whereby limited dispersal of hosts and parasitoids combined with other features, such as spatial and temporal heterogeneity, can promote persistence and stability of populations. It is important for us to understand the ecological and evolutionary dynamics of populations mirrored by the large number of mathematical models devoting to it in the scientific literatures [3–6]. In recent years, the analysis of these models focuses on the coexistence of population and local (or global) stability of equilibria [7–13]. Spatial factors play a fundamental role on the persistence and stability of the population, although the complete results have not yet been obtained even in the simplest one-species case. Most previous papers focused on the population dynamical system modeled by the ordinary differential equations; if the population dynamics with the effects of spatial heterogeneity are modeled by a diffusion process, it will be very interesting. While in practice, it is often the case that diffusion occurs in regular pulse. For example, when winter comes, birds will migrate between patches in search for a better environment, whereas they do not diffuse in other seasons, and the excursion of foliage seeds occurs at fixed period of time every year. Thus, impulsive diffusion provides a more natural description. Lately theories of impulsive differential equations [14] have been introduced into population dynamics. Impulsive differential equations are found in almost a domain of applied science [15–24]. Newly, persistence and stability of population dynamical system involving time
Attractivity of nonlinear impulsive delay differential equations  [PDF]
Zhichun Yang,Daoyi Xu
International Journal of Stochastic Analysis , 2006, DOI: 10.1155/jamsa/2006/83152
Abstract: The attractivity of nonlinear differential equations with time delays and impulsive effects is discussed. We obtain some criteria to determine the attracting set and attracting basin of the impulsive delay system by developing an impulsive delay differential inequality and introducing the concept of nonlinear measure. Examples and their simulations illustrate the effectiveness of the results and different asymptotical behaviors between the impulsive system and the corresponding continuous system.
Stability and Stabilization of Impulsive Stochastic Delay Difference Equations
Kaining Wu,Xiaohua Ding,Liming Wang
Discrete Dynamics in Nature and Society , 2010, DOI: 10.1155/2010/592036
Abstract: When an impulsive control is adopted for a stochastic delay difference system (SDDS), there are at least two situations that should be contemplated. If the SDDS is stable, then what kind of impulse can the original system tolerate to keep stable? If the SDDS is unstable, then what kind of impulsive strategy should be taken to make the system stable? Using the Lyapunov-Razumikhin technique, we establish criteria for the stability of impulsive stochastic delay difference equations and these criteria answer those questions. As for applications, we consider a kind of impulsive stochastic delay difference equation and present some corollaries to our main results.
Asymptotic Behavior of Impulsive Infinite Delay Difference Equations with Continuous Variables  [cached]
Ma Zhixia,Xu Liguang
Advances in Difference Equations , 2009,
Abstract: A class of impulsive infinite delay difference equations with continuous variables is considered. By establishing an infinite delay difference inequality with impulsive initial conditions and using the properties of " -cone," we obtain the attracting and invariant sets of the equations.
Asymptotic Behavior of Impulsive Infinite Delay Difference Equations with Continuous Variables
Zhixia Ma,Liguang Xu
Advances in Difference Equations , 2009, DOI: 10.1155/2009/495972
Abstract: A class of impulsive infinite delay difference equations with continuous variables is considered. By establishing an infinite delay difference inequality with impulsive initial conditions and using the properties of “ -cone,” we obtain the attracting and invariant sets of the equations.
Linearized Oscillations in Nonlinear Neutral Delay Impulsive Differential Equations
I.O. Isaac,Zsolt Lipcsey
Journal of Modern Mathematics and Statistics , 2012,
Abstract: Our aim in this study, is to develop a linearized oscillation theory for nonlinear neutral delay impulsive differential equations. Precisely, we prove that a certain nonlinear neutral delay impulsive differential equation has the same oscillatory character as its associated linear impulsive equation.
Exponential Stability of Impulsive Stochastic Delay Differential Systems
Xiaotai Wu,Litan Yan,Wenbing Zhang,Liang Chen
Discrete Dynamics in Nature and Society , 2012, DOI: 10.1155/2012/296136
Abstract: This paper investigates the stability of stochastic delay differential systems with two kinds of impulses, that is, destabilizing impulses and stabilizing impulses. Both the th moment and almost sure exponential stability criteria are established by using the average impulsive interval. When the impulses are regarded as disturbances, a lower bound of average impulsive interval is obtained; it means that the impulses should not happen too frequently. On the other hand, when the impulses are used to stabilize the system, an upper bound of average impulsive interval is derived; namely, enough impulses are needed to stabilize the system. The effectiveness of the proposed results is illustrated by two examples.
Stability and Stabilization of Impulsive Stochastic Delay Differential Equations
Kaining Wu,Xiaohua Ding
Mathematical Problems in Engineering , 2012, DOI: 10.1155/2012/176375
Abstract: We consider the stability and stabilization of impulsive stochastic delay differential equations (ISDDEs). Using the Lyapunov-Razumikhin method, we obtain the sufficient conditions to guarantee the pth moment exponential stability of ISDDEs. Then the almost sure exponential stability is considered and the sufficient conditions of the almost sure exponential stability are obtained. Moreover, the stabilization problem of ISDDEs is studied and the criterion on impulsive stabilization of ISDDEs is established. At last, examples are presented to illustrate the correctness of our results.
A New Singular Impulsive Delay Differential Inequality and Its Application
Zhixia Ma,Xiaohu Wang
Journal of Inequalities and Applications , 2009, DOI: 10.1155/2009/461757
Abstract: A new singular impulsive delay differential inequality is established. Using this inequality, the invariant and attracting sets for impulsive neutral neural networks with delays are obtained. Our results can extend and improve earlier publications.
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