Theory Of Impulsive Differential Equations Research Articles

Dynamical analysis of the pest control model with multiple delays and impulsive effects

In recent years, the suppression of disease population in pest control has gained popularity and remains a pressing issue in agricultural production. Additionally, the mathematical modeling method is often exploited to study the dynamics of different infectious diseases. A new pest control model with multiple delays and impulsive effects is developed and investigated. Firstly, the global attractivity of infection-free periodic solution of the system is proved by applying the theory of impulsive differential equation and comparative theorem. Secondly, the persistence of the infection-free periodic solution of the proposed model is also proved. Finally, the theoretical analysis results are validated by numerical simulations and some suggestions on how to better control the pests are presented.

A rumor propagation model in multilingual environment with time and state dependent impulsive control

In this paper we are interested in the problem of modeling, analysis and control of rumor spreading on online social networks. As the diversity of languages and the difference of user nodes in the network, a heterogeneous rumor spreading model with two language spreaders is proposed, and the impulsive isolation control strategy is considered. Firstly, the model with periodic impulsive isolation control and time-delay recovery rate is discussed. Based on the stability theory of impulsive differential equations, some conditions regarding the impulsive period and control gains are given for the rumor asymptotically disappearance and persistent existence. Secondly, a new type of event-triggered impulsive isolation control method is developed to address the tradeoff between restraining rumor and minimizing the cost of control, and some conditions are given to make the rumor disappear asymptotically. Furthermore, an improved event-triggered mechanism is established based on the fixed-time stability theory, which provides the conditions for the rumor to disappear within a fixed-time. Finally, some numerical simulations and an actual example are given to verify the rationality of the model and the effectiveness of the control methods.

Study on the Multi-Point Boundary Value Problem for Second-Order Nonlinear Impulsive Integro-Differential Equation

In the field of biological control, there are a large number of systems that gradually evolve over a certain period of time. However, due to some natural or human intervention behavior, the system state will be subjected to some relatively short time interference, so that the system state changes in an instant. This sudden change of state makes the system not simply described by continuous or discrete dynamical systems, but by means of impulse dynamical systems. In this paper, the multi-point boundary value problem of a class of second-order nonlinear impulsive integral differential equations is studied on the basis of impulsive differential equation theory. The main results of this kind of equation are obtained by using the fixed point theorem of strict set contraction operator. Under certain assumptions, the existence of the solution of the equation is proved by constructing the operator on the special cone. Finally, combined with the practical application, the theory was applied to the stability prediction of biological ecosystem, and the correctness of the conclusion was verified.

Dynamical Behavior of a Predator‐Prey System Incorporating a Prey Refuge with Impulse Effect

A predator‐prey model with Holling II functional response incorporating a prey refuge with impulse effect is considered in this paper. With the help of the Floquet theory of impulsive differential equations, local stability and global attractivity of the boundary periodic solution of the system are derived, and then sufficient conditions for global asymptotic stability of the boundary periodic solution are obtained. Next, the permanence of the system is proved by constructing a Lyapunov function. Further, by applying the bifurcation theory of impulsive differential equations, conditions under which the system has a positive periodic solution are obtained. Finally, numerical simulations are presented to illustrate the analytical results.

Catastrophe control of aphid populations model.

Considering the effect of the natural enemy on aphid populations, the corresponding model with delay is built. The model is analyzed using the qualitative theory of differential equations and catastrophe theory etc. For the outbreak phenomenon of aphid populations, the corresponding management model is proposed and the catastrophe controller is designed to keep the system in a virtuous cycle by means of the qualitative theory of impulsive differential equations. In the mean time, some simulations are carried to prove the results. The paper not only provides a new method for catastrophe control but also expands the application fields of catastrophe control.

Outer Synchronization of Complex-Variable Networks with Complex Coupling via Impulsive Pinning Control

In this paper, outer synchronization of complex-variable networks with complex coupling is considered. Sufficient conditions for achieving outer synchronization using static impulsive pinning controllers are first derived according to the Lyapunov function method and stability theory of impulsive differential equations. From these conditions, the necessary impulsive gains and intervals for given networks can be easily calculated. Further, an adaptive strategy is introduced to design universal controllers and avoid repeated calculations for different networks. Notably, the estimation algorithms of the impulsive gains and intervals are provided. Finally, three numerical examples are performed to verify the effectiveness of the main results.

Leader-Following Pinning Synchronization of Multiagent Systems With Impulsive Interlayer Coupling

This article is mainly concerned with leader-following pinning synchronization of multiagent systems in a multiplex network setting, where the interlayer couplings are impulsive. The goal is to enforce all the followers to asymptotically synchronize with the leader via impulsive pinning control. Firstly, based on the Lyapunov stability theory of impulsive differential equations, some general synchronization criteria are derived. The synchronization index is then defined. We explore how the synchronization index is effected by major network parameters, including the impulse strength, the impulse interval, the pinning gain and the number of layers. In particular, with equivalent impulse intervals, when both the impulse strength and the pinning gain are constant, it is found that the least number of pinned nodes needed for achieving complete synchronization (CS) decreases with decreasing impulse intervals, whereas it firstly decreases and then increases with increasing impulse strength and pinning gain. Furthermore, the synchronization region enlarges as the impulse interval decreases, while narrows as the number of layers increases. Numerical simulations verify the correctness of these results.

Finite-time cluster synchronization of coupled dynamical systems with impulsive effects

In our paper, the finite-time cluster synchronization problem is investigated for the coupled dynamical systems in networks. Based on impulsive differential equation theory and differential inequality method, two novel Lyapunov-based finite-time stability results are proposed and be used to obtain the finite-time cluster synchronization criteria for the coupled dynamical systems with synchronization and desynchronization impulsive effects, respectively. The settling time with respect to the average impulsive interval is estimated according to the sufficient synchronization conditions. It is illustrated that the introduced settling time is not only dependent on the initial conditions, but also dependent on the impulsive effects. Compared with the results without stabilizing impulses, the attractive domain of the finite-time stability can be enlarged by adding impulsive control input. Conversely, the smaller attractive domain can be obtained when the original system is subject to the destabilizing impulses. By using our criteria, the continuous feedback control can always be designed to finite-time stabilize the unstable impulsive system. Several existed results are extended and improved in the literature. Finally, typical numerical examples involving the large-scale complex network are outlined to exemplify the availability of the impulsive control and continuous feedback control, respectively.

Stability by fixed point theory of impulsive differential equations with delay

Abstract In this paper we ensure that for some class of impulsive differential equations with delay the zero solution is asymptotically stable by means of fixed point theory.

Adaptive Variable Impulsive Synchronization of Multi-agent Systems

This paper is concerned with the adaptive synchronization of multi-agent systems via variable impulsive control method. Based on the theory of impulsive differential equations, adaptive control technique, Lyapunov stability theory and comparison system method, some novel adaptive synchronization conditions with impulsive time window are given to realize the synchronization of multi-agent nonlinear systems with uncertain parameters. Different from the existing investigations of impulsive synchronization of multi-agent systems, the proposed impulsive control protocol with impulsive time window is more effective in practical systems. Finally, two simulation results demonstrate the validity of the theoretical results.

Global dynamics of a model for treating microorganisms in sewage by periodically adding microbial flocculants.

In this paper, a mathematical model for microbial treatment in livestock and poultry sewage is proposed and analyzed. We consider periodic addition of microbial flocculants to treat microorganisms such as Escherichia coli in sewage. Different from the traditional models, a class of composite dynamics models composed of impulsive differential equations is established. Our aim is to study the relationship between substrate, microorganisms and flocculants in sewage systems as well as the treatment strategies of microorganisms. Precisely, we first show the process of mathematical modeling by using impulsive differential equations. Then by using the theory of impulsive differential equations, the dynamics of the model is investigated. Our results show that the system has a microorganismsextinction periodic solution which is globally asymptotically stable when a certain threshold value is less than one, and the system is permanent when a certain threshold value is greater than one. Furthermore, the control strategy for microorganisms treatment is discussed. Finally, some numerical simulations are carried out to illustrate the theoretical results.

Dynamics of an Impulsive Stochastic Nonautonomous Chemostat Model with Two Different Growth Rates in a Polluted Environment

This paper proposes a novel impulsive stochastic nonautonomous chemostat model with the saturated and bilinear growth rates in a polluted environment. Using the theory of impulsive differential equations and Lyapunov functions method, we first investigate the dynamics of the stochastic system and establish the sufficient conditions for the extinction and the permanence of the microorganisms. Then we demonstrate that the stochastic periodic system has at least one nontrivial positive periodic solution. The results show that both impulsive toxicant input and stochastic noise have great effects on the survival and extinction of the microorganisms. Furthermore, a series of numerical simulations are presented to illustrate the performance of the theoretical results.

Leader-following consensus of multi-agent systems via event-based impulsive control

Leader-following consensus of multi-agent systems with general linear models is investigated via event-based impulsive control approach. Event-based impulsive controller and state-dependent triggering function are designed. Impulsive instants are determined by certain triggering events, that is, impulsive effect performs only at the instants when events occur, and so as controller update. Sufficient conditions on leader-following consensus are proposed by using stability theory of impulsive differential equations, matrix theory, and inequality technique. Zeno-behavior is also excluded for the concerned closed-loop system. Besides, a technique that used the state information of agents only at event instants is proposed to avoid continuous communication among agents. Finally, an example is presented to illustrate the effectiveness of the obtained theoretical results.

A Lycaon pictus impulsive state feedback control model with Allee effect and continuous time delay

Allee effect (i.e. sparse effect) is active when the population density is small. Our purpose is to study such an effect of this phenomenon on population dynamics. We investigate an impulsive state feedback control single-population model with Allee effect and continuous delay. We first qualitatively analyze the singularity of this model. Then we obtain sufficient conditions for the existence of an order-one periodic orbit by the geometric theory of impulsive differential equations for the survival of endangered populations and obtain the uniqueness of an order-one periodic orbit by the monotonicity of the subsequent function. Furthermore, we prove the orbital asymptotic stability of an order-one periodic orbit using the geometric properties of successor functions to confirm the robustness of this control. Finally, we verify the correctness of our theoretical results by using some numerical simulations. Our results show that the release of artificial captive African wild dog (Lycaon pictus) can effectively protect the African wild dog population with Allee effect.

Analysis and control of a non-smooth population model with impulsive effects

In this paper, a non-smooth population model with impulsive effects is proposed by combining discontinuity and non-smoothness. According to the qualitative theory of differential equations, the global analysis of the model is discussed. Using the theory of impulsive differential equations, the existence conditions of order one periodic solution are obtained. And the impulsive controllers are designed to make the pest populations stay at the refuge level. Some simulations are carried out to prove the results.

Effect of rhizosphere dispersal and impulsive input on the growth of wetland plant

Modeling the wetland plant–nutrient interaction is significant for elucidating the complex mechanisms of the nutrient absorption. In this paper, the effect of pollutant dispersal and impulsive input on the growth of wetland plant is presented. By using the theory of impulsive differential equations, we obtain that the wetland plant-eradication periodic solution is globally asymptotically stable if some conditions are satisfied. In addition, the permanent conditions are given by means of the comparison theorem of impulsive differential equations. Finally, numerical simulations are given to justify our results.

Dynamical analysis of a logistic model with impulsive Holling type-II harvesting

A logistic model with impulsive Holling type-II harvesting is proposed and investigated in this paper. Here, the species is harvested at fixed moments. By using the techniques derived from the theory of impulsive differential equations, sufficient conditions for both permanence and extinction of the system are established, respectively. Sufficient conditions which ensure the existence, uniqueness, and global attractivity of a positive periodic solution of the system are obtained. Our study shows that impulsive controls play an important role in maintaining the sustainable development of the ecological system. Compared with the linear impulsive capture or continuous nonlinear-type capture, our study shows that the nonlinear impulsive capture could lead to more complicated dynamic behaviors. Numeric simulations are carried out to show the feasibility of the main results. The results obtained here maybe useful to the practical biological economics management.

THE GEOMETRICAL ANALYSIS OF A PREDATOR-PREY MODEL WITH MULTI-STATE DEPENDENT IMPULSES

Starting from the practical problems of integrated pest management, we establish a predator-prey model for pest control with multi-state dependent impulsive, which adopts two different control methods for two different thresholds. By applying geometry theory of impulsive differential equations and the successor function, we obtain the existence of order one periodic solution. Then the stability of the order one periodic solution is studied by analogue of the Poincare criterion. Finally, some numerical simulations are exerted to show the feasibility of the results.

Adaptive‐impulsive consensus of multiagent systems

SummaryThis paper studies the leader‐following consensus problem of multiagent nonlinear systems with uncertain parameters and control gain error. On the basis of the theory of impulsive differential equations, the adaptive control technique, and the Lyapunov stability theory, some novel adaptive‐impulsive consensus conditions are given to realize the consensus of a class of multiagent nonlinear systems. Compared with the existing investigations on the impulsive consensus of multiagent systems, the proposed impulsive control protocol with uncertain parameters and control gain error is more rigorous and effective in practical systems. Four numerical simulations are verified to confirm the effectiveness of the proposed methods.

Global Mittag-Leffler Synchronization of Fractional-Order Neural Networks Via Impulsive Control

This paper aims at analyzing the impulsive synchronization of fractional-order neural works. Firstly, in view of control theory, by constructing a suitable impulsive response system with the designed controller, the synchronization error system between the drive system and the corresponding response system is given. Afterwards, based on the theory of impulsive differential equation, the theory of fractional differential equation, Lyapunov direct method, and inequality techniques, some effective sufficient criteria are established to guarantee the global Mittag-Leffler stability for the synchronization error system. Finally, several simulation examples are designed to demonstrate the effectiveness and feasibility of the obtained results.