Conditions For Global Attractivity Research Articles

A Sufficient Condition for Global Attractivity of Nonlinear Delay Differential Equations

A Sufficient Condition for Global Attractivity of Nonlinear Delay Differential Equations

Optimal harvesting strategy for stochastic hybrid delay Lotka-Volterra systems with Lévy noise in a polluted environment.

This paper concerns the dynamics of two stochastic hybrid delay Lotka-Volterra systems with harvesting and Lévy noise in a polluted environment (i.e., predator-prey system and competitive system). For every system, sufficient and necessary conditions for persistence in mean and extinction of each species are established. Then, sufficient conditions for global attractivity of the systems are obtained. Finally, sufficient and necessary conditions for the existence of optimal harvesting strategy are provided. The accurate expressions for the optimal harvesting effort (OHE) and the maximum of expectation of sustainable yield (MESY) are given. Our results show that the dynamic behaviors and optimal harvesting strategy are closely correlated with both time delays and three types of environmental noises (namely white Gaussian noises, telephone noises and Lévy noises).

Stochastic stability and impulsive vaccination of multicompartment nonlinear epidemic model with incidence rate

In this work, we consider a multicompartment nonlinear epidemic model with temporary immunity and a saturated incidence rate. N(t) at time t, this population is divide into seven sub-classes. N(t) = S(t) + E(t) + I(t) + I1(t) + I2(t) + I3(t) + Q(t). where S(t),E(t); I(t); I(t); I1(t),I2(t); I3(t) and Q(t) denote the sizes of the population susceptible to disease, exposed, infectious members and quarantine members with the possibility of infection through temporary immunity, respectively.The stability of a disease-free status equilibrium and the existence of endemic equilibrium determined by the ratio called the basic reproductive number. The multicompartment non linear epidemic model with saturated rate has been studied the stochastic stability of the free disease equilibrium under certain conditions, and obtain the conditions of global attractivity of the infection.

Global attractivity for uncertain differential systems

<abstract><p>This paper studies global attractivity for uncertain differential systems, which are effective tools to solve the problems with uncertainty. And They have been applied in many areas. This article presents several global attractivity concepts. Based on the knowledge of uncertainty theory, some sufficient conditions of global attractivity for linear uncertain differential systems are given. In particular, the attractivity on the solutions and $ \alpha $-path of uncertain differential systems is studied. Last, as an application of attractivity, an interest rate model with uncertainty is shown.</p></abstract>

Stochastic persistence and global attractivity of a two-predator one-prey system with S-type distributed time delays

<abstract><p>In this paper, well-posedness and asymptotic behaviors of a stochastic two-predator one-prey system with S-type distributed time delays are studied by using stochastic analytical techniques. First, the existence and uniqueness of global positive solution with positive initial condition is proved. Second, sufficient conditions for persistence in mean and extinction of each species are obtained. Then, sufficient conditions for global attractivity are established. Finally, some numerical simulations are provided to support the analytical results.</p></abstract>

Global Dynamics of a Stochastic Two-Prey One-Predator Model with <i>S</i>-Type Distributed Delays and Lévy Noises

In this paper, a stochastic two-prey one-predator model with S-type distributed time delays and Lévy noises is considered. Using the comparison theorem and Ito’s formula, sufficient conditions of persistence in the mean and extinct for each population are established. Then, conditions of global attractivity and stability in distribution by Barbalat’s conclusion are also obtained. Furthermore, Euler numerical simulation method is given to demonstrate our conclusions.

Dynamics analysis of a stochastic non-autonomous one-predator\u2013two-prey system with Beddington\u2013DeAngelis functional response and impulsive perturbations

In this paper, we explore a stochastic non-autonomous one-predator–two-prey system with Beddington–DeAngelis functional response and impulsive perturbations. First, by using Itô’s formula, exponential martingale inequality, Chebyshev’s inequality and other mathematical skills, we establish some sufficient conditions for extinction, non-persistence in the mean, weak persistence, persistence in the mean and stochastic permanence of the solution of the stochastic system. Then the limit of the average in time of the sample path of the solution is estimated by two constants. Afterwards, the lower-growth rate and the upper-growth rate of the positive solution are estimated. In addition, sufficient conditions for global attractivity of the system are established. Finally, we carry out some simulations to verify our main results and explain the biological implications: the large stochastic interference is disadvantageous for the persistence of the population and the strong impulsive harvesting can lead to extinct of the population.

Sufficient and necessary conditions for global attractivity and stability of a class of discrete Hopfield-type neural networks with time delays.

In this paper, a class of discrete Hopfield-type neural networks with time delays is consid-ered. Sufficient and necessary conditions for global attractivity and stability of the equilibrium of the discrete Hopfield-type neural networks are given by using a class of n-dimensional nonlinear algebraic equations and M-matrix theory.

Dynamics and Optimal Harvesting Control for a Stochastic One‐Predator‐Two‐Prey Time Delay System with Jumps

We consider a stochastic one‐predator‐two‐prey harvesting model with time delays and Lévy jumps in this paper. Using the comparison theorem of stochastic differential equations and asymptotic approaches, sufficient conditions for persistence in mean and extinction of three species are derived. By analyzing the asymptotic invariant distribution, we study the variation of the persistent level of a population. Then we obtain the conditions of global attractivity and stability in distribution. Furthermore, making use of Hessian matrix method and optimal harvesting theory of differential equations, the explicit forms of optimal harvesting effort and maximum expectation of sustainable yield are obtained. Some numerical simulations are given to illustrate the theoretical results.

Stability and Synchronization Analysis of Discrete-Time Delayed Neural Networks with Discontinuous Activations

In this paper, both stability and synchronization of discrete-time delayed neural networks with discontinuous activations are investigated. By means of functional differential inclusions and Kakutani’s fixed point theorem, conditions are derived to ensure the existence and uniqueness of the solution. Besides, we obtain some novel sufficient conditions for global attractivity and asymptotic stability of the discontinuous discrete-time neural networks via the Halanay-type inequality and the comparison principle. Furthermore, under weaker conditions than Lipschitz conditions, we also select two different controllers to guarantee synchronization of the discrete-time neural networks with discontinuous activations. Finally, some examples with numerical simulation are given to demonstrate the effectiveness of the obtained results.

Global convergence of a class of networks on time scales

Abstract In this paper, we propose a class of simplified background neural networks model with two subnetworks on time scales. Some basic dynamic properties including positive invariance, boundedness, global attractivity and complete convergence of networks are analyzed. The main contributions in this paper are listed as follows: (1) the global attractive set of the model is verified and conditions for global attractivity are derived. (2) Complete convergence for the new networks are proved by constructing a novel energy function on time scales. Finally, three simulation examples are presented to illustrate the feasibility and effectiveness of the obtained results.

Dynamics of a stochastic predator-prey model with two competitive preys and one predator in a polluted environment

In this paper, we propose and study a stochastic predator-prey model with two competitive preys and one predator in a polluted environment. We first carry out the survival analysis and establish sufficient criteria for the extinction, non-persistence, weak persistence in the mean and strong persistence in the mean. The threshold between weak persistence in the mean and extinction is obtained for each population. Then, we derive sufficient conditions for global attractivity of the studied model. Numerical simulations are carried out to verify the theoretical results.

The dynamics of a time delayed epidemic model on a population with birth pulse

An epidemic model with time delay and impulse is proposed to describe if time delay plays an important role in the spread of an infectious disease. Mathematical analyses with regard to the local, global stability of equilibrium are performed. We give out the conditions for global attractivity of the disease-free equilibrium and permanence of the system, respectively.

Sufficient conditions for global attractivity of a class of neutral Hopfield-type neural networks

In this paper, a class of simple n-dimensional neutral Hopfield-type neural networks is considered. Based on a class of nonlinear algebraic equations with n variables and some computational techniques in M-matrix, some sufficient conditions for global attractivity of the equilibrium of the n-dimensional neutral Hopfield-type neural networks are established. Numerical simulations are given to summarize the applications.

The Effect of Impulsive Vaccination on Delayed SEIRS Epidemic Model Incorporating Saturation Recovery

A delay SEIRS model with pulse vaccination incorporating a general media coverage function and saturation recovery is investigated. Using the discrete dynamical system determined by the stroboscopic map, we obtain the existence of the disease-free periodic solution and its exact expression. Further, using the comparison theorem, we establish the sufficient conditions of global attractivity of the disease-free periodic solution. Moreover, we show that the disease is uniformly persistent if vaccination rate is less thanθ*. Finally, we discuss the effect of media coverage on controlling disease.

A STAGE-STRUCTURED PREDATOR-PREY SI MODEL WITH DISEASE IN THE PREY AND IMPULSIVE EFFECTS

This paper aims to develop a high-dimensional SI model with stage structure for both the prey (pest) and the predator, and then to investigate the dynamics of it. The model can be used for the study of Integrated Pest Management (IPM) which is a combination of constant pulse releasing of animal enemies and diseased pests at two different fixed moments. Firstly, we use analytical techniques for impulsive delay differential equations to obtain the conditions for global attractivity of the ‘pest-free’ periodic solution and permanence of the population model. It shows that the conditions strongly depend on time delay, impulsive release of animal enemies and infective pests. Secondly, we present a pest management strategy in which the pest population is kept under the economic threshold level (ETL) when the pest population is permanent. Finally, numerical analysis is presented to illustrate our main conclusion.

Dynamics of a Stage Structured Pest Control Model in a Polluted Environment with Pulse Pollution Input

By using pollution model and impulsive delay differential equation, we formulate a pest control model with stage structure for natural enemy in a polluted environment by introducing a constant periodic pollutant input and killing pest at different fixed moments and investigate the dynamics of such a system. We assume only that the natural enemies are affected by pollution, and we choose the method to kill the pest without harming natural enemies. Sufficient conditions for global attractivity of the natural enemy-extinction periodic solution and permanence of the system are obtained. Numerical simulations are presented to confirm our theoretical results.

On a stochastic logistic equation with impulsive perturbations

A stochastic logistic model with impulsive perturbations is proposed and investigated. First, we give a new definition of a solution of an impulsive stochastic differential equation (ISDE), which is more convenient for use than the existing one. Using this definition, we show that our model has a global and positive solution and obtain its explicit expression. Then we establish the sufficient conditions for extinction, non-persistence in the mean, weak persistence, persistence in the mean and stochastic permanence of the solution. The critical value between weak persistence and extinction is obtained. In addition, the limit of the average in time of the sample path of the solution is estimated by two constants. Afterwards, the lower-growth rate and the upper-growth rate of the solution are estimated. Finally, sufficient conditions for global attractivity are established.

Convergence analysis of a class of simplified background neural networks with two subnetworks

Motivated by Salinas's (2003) original discovery in [1] and inspired by Oja's (1982) seminal work in [2], in this paper, we propose a class of simplified background neural networks model with two subnetworks. Some basic dynamic properties including boundedness, global attractivity, stability, and complete convergence are analyzed rigorously. The main contributions in this paper are as follows: (1) The boundedness of the new model is verified and conditions for global attractivity are derived. (2) Conditions on asymptotically stable of equilibrium points are obtained. (3) Complete convergence for the new network is proved by constructing a novel energy function. Finally, numerical examples demonstrate our theoretical results.

Dynamics of a discretized SIR epidemic model with pulse vaccination and time delay

We derive a discretized SIR epidemic model with pulse vaccination and time delay from the original continuous model. The sufficient conditions for global attractivity of an infection-free periodic solution and permanence of our model are obtained. Improving discretization, our results are corresponding to those in the original continuous model.