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Copyright of Journal of Guangxi Normal University - Natural Science Edition is the property of Gai Kan Bian Wei Hui and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we consider a stochastic food chain chemostat model with variable yields. First, we prove the stochastic model has a unique global positive solution. Second, by employing suitable Lyapunov functions, Itô ′ s formula and some other important inequalities, the existence of a unique ergodic stationary distribution of a stochastic food chain chemostat model is researched, which can help us better understand the statistical characteristics of stochastic food chain chemostat models. Second, we investigate the extinction of the microorganism and the bacteria. Moreover, the case of extinction for bacteria but persistence for microbial species is considered. Finally, some numerical simulations are carried out to illustrate our theoretical results and the influence of the variable yields on the microorganism and the bacteria. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic SIQR epidemic model with non-monotone incidence is investigated. First of all, we consider the disease-free equilibrium of the deterministic model is globally asymptotically stable by using the Lyapunov method. Secondly, the existence and uniqueness of positive solution to the stochastic model is obtained. Then, the sufficient condition for extinction of the stochastic model is established. Furthermore, a unique stationary distribution to stochastic model will exist by constructing proper Lyapunov function. Finally, numerical examples are carried out to illustrate the theoretical results, with the help of numerical simulations, we can see that the higher intensities of the white noise or the bigger of the quarantine rate can accelerate the extinction of the disease. This theoretically explains the significance of quarantine strength (or isolation measures) when an epidemic erupts. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stability of highly nonlinear hybrid neutral pantograph stochastic differential equations (NPSDEs) with general decay rate. By applying the method of the Lyapunov function, the pth moment and almost sure stability with general decay rate of solution for NPSDEs are derived. Finally, an example is presented to show the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamical behavior of a two-group SVIR epidemic model with random effect. Firstly, the two-group SVIR epidemic model with random perturbation of natural death rate is established. The existence and uniqueness of positive solution are proved by using stopping time theory and the Lyapunov analysis method. Secondly, a property of the system solution is obtained by using the law of strong numbers and the continuous local martingale. Finally, a new combination of Lyapunov functions is applied. The solution of the model we obtained is oscillating around a steady state if the basic reproduction number is less than one, which is the disease-free equilibrium of the corresponding deterministic model. A numerical simulation is presented to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics property of a stochastic HIV model with Beddington–DeAngelis functional response. It has a unique uninfected steady state. We prove that the model has a unique global positive solution. Furthermore, if the basic reproductive number is not larger than 1, the asymptotic behavior of the solution is stochastically stable. Otherwise, it fluctuates randomly around the infected steady state of the corresponding deterministic HIV model. Finally, some numerical simulations are carried out to verify our results. [ABSTRACT FROM AUTHOR]

The aim of this article is to consider the existence, uniqueness and uniformly exponential p-stability of the solution for ∇ -delay stochastic dynamic equations on time scales via Lyapunov functions. This work can be considered as a unification and generalization of stochastic difference and stochastic differential time-varying delay equations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the exponential stability of highly nonlinear hybrid neutral pantograph stochastic differential equations (NPSDEs). The aim of this paper is to establish exponential stability criteria for a class of hybrid NPSDEs without the linear growth condition. The methods of Lyapunov functions and M‐matrix are used to study exponential stability and boundedness of the hybrid NPSDEs. [ABSTRACT FROM AUTHOR]

The study of dynamics of epidemics, especially, double epidemics, has got paramount importance in the present days, since various health risks, an individual is exposed to due to the increasing global warming are unavoidable and/or due to limited medical resources available to handle suddenly developed disease. In view of this, we consider a study of an epidemic model with a double hypothesis, namely, SIR and SIRS mechanisms. First, we prove the local asymptotic stability of equilibrium points followed by the global stability of the disease for equilibrium established through the Lyapunov function. Next, we show the existence, extinction and persistence results of the stochastic system. Finally, some numerical examples are presented to support the theoretical results established. • Epidemic model with double hypothesis combined with SIR and SIRS transmission mechanisms. • Global existence of a positive solution. • Global stability of the disease-free equilibrium. • Extinction of the diseases. [ABSTRACT FROM AUTHOR]

This article studies the asymptotic behavior of a stochastic Chemostat model with Lotka–Volterra food chain in which the dilution rate was influenced by white noise. The long-time behavior of the model is studied. Using Lyapunov function and Itô's formula, we show that there is a unique positive solution to the system. Moreover, the sufficient conditions for some population dynamical properties including the boundedness in mean and the stochastically asymptotic stability of the washout equilibrium were obtained. Furthermore, we show how the solutions spiral around the predator-free equilibrium and the positive equilibrium of deterministic system. Besides, the existence of the stationary distribution is proved for the considered model. Numerical simulations are introduced finally to support the obtained results. [ABSTRACT FROM AUTHOR]

This paper studies a stochastic two-species Monod competition chemostat model which is subject to environment noises. Such noises are described by independent standard Brownian motions. It proves that the initial value problem of the model has a unique positive global solution. However, unlike the corresponding deterministic model, the stochastic model no longer has positive equilibrium points. The asymptotic behaviors and the steady state distributions are established by using Itô’s formula, Lyaponov method and Gronwall inequality. In addition, numerical simulations are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This article discusses a stochastic SIQS epidemic model with saturated incidence. We assume that random perturbations always fluctuate at the endemic equilibrium. The existence of a global positive solution is obtained by constructing a suitable Lyapunov function. Under some suitable conditions, we derive the stochastic boundedness and stochastic permanence of the solutions of a stochastic SIQS model. Some numerical simulations are carried out to check our results. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a stochastic SIR epidemic model with saturated incidence is investigated. Firstly, we prove that the system has a unique global positive solution with any positive initial value. Then we verify that random effect may lead the disease to extinction under a simple condition. Thirdly, we establish a sufficient condition for persistence in the mean of the disease. Moreover, we show that there is a stationary distribution to the stochastic system under certain parametric restrictions. Finally, some numerical simulations are carried out to confirm the analytical results. [ABSTRACT FROM AUTHOR]

Abstract: We study the stability of receding horizon control for continuous-time non-linear stochastic differential equations. We illustrate the results with a simulation example in which we employ receding horizon control to design an investment strategy to repay a debt. [Copyright &y& Elsevier]

In this paper, we present a differential infectivity SIR epidemic model with modified saturation incidences and stochastic perturbations. We show that the stochastic epidemic model has a unique global positive solution, and we utilize stochastic Lyapunov functions to show the asymptotic behavior of the solution. [ABSTRACT FROM AUTHOR]

For a stochastic functional differential equation (SFDE) to have a unique global solution it is in general required that the coefficients of the SFDE obey the local Lipschitz condition and the linear growth condition. However, there are many SFDEs in practice which do not obey the linear growth condition. The main aim of this paper is to establish existence-and-uniqueness theorems for SFDEs where the linear growth condition is replaced by more general Khasminskii-type conditions in terms of a pair of Laypunov-type functions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we obtain sufficient criteria for the existence of periodic solutions to deterministic SIR and SEIR epidemic models with modified saturation incidence rates by means of using the continuation theorem based on coincidence degree theory, and we show that the solution is unique and globally stable. Second, we discuss their corresponding stochastic epidemic models with random perturbation have a unique global positive solution respectively, and we utilize stochastic Lyapunov functions to investigate the asymptotic behavior of the solution. [Copyright &y& Elsevier]

The main aim of this article is to deal with the almost-sure stability of stochastic differential delay equations. Our improved theorems give better results while conditions imposed on the Lyapunov function are much weaker, thus, it is easier to find a right Lyapunov function in application. [ABSTRACT FROM PUBLISHER]

Abstract: This paper analyzes the asymptotic behaviors of the stochastic population model with the Allee effect. We apply the Feller’s test to obtain the criteria of the asymptotic behaviors for this model and compare our results with those obtained by Lyapunov functions. At last, the numerical simulations conform our results. [Copyright &y& Elsevier]

This paper discusses the issue of stability and stabilization for the non-linear stochastic systems with time-varying delay subject to passivity performance. Based on the fuzzy modelling approach, the non-linear systems can be represented by a stochastic T—S fuzzy model whose consequent part is described by linear differential equations with a multiplicative noise term. For analysing the stability of the stochastic T—S fuzzy model, the less conservative stability criterion derived by Itô's formula and the Lyapunov—Krasovskii function is proposed as a linear matrix inequality (LMI) problem. Furthermore, the passivity theory is employed to discuss the external disturbance effect on the considered system. Through applying the above techniques, the sufficiency conditions are derived to design fuzzy controllers such that the stochastic T—S fuzzy model with time delay is asymptotically stable and passive in the mean square. Finally, two numerical examples are proposed to show the application and effectiveness of the proposed fuzzy controller design method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a DI SIR epidemic model with two categories stochastic perturbations. The long time behavior of the two stochastic systems is studied. Mainly, we show how the solution goes around the infection-free equilibrium and the endemic equilibrium of deterministic system under different conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the two-group SIR model introduced by Guo, Li and Shuai [H.B. Guo, M.Y. Li, Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q. 14 (2006) 259–284], allowing random fluctuation around the endemic equilibrium. We prove the endemic equilibrium of the model with random perturbation is stochastic asymptotically stable in the large. In addition, the stability condition is obtained by the construction of Lyapunov function. Finally, numerical simulations are presented to illustrate our mathematical findings. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the stability analysis issue for stochastic delayed bidirectional associative memory (BAM) neural network with Markovian jumping parameters. Assume that the jumping parameters are generated from continue-time discrete-state homogeneous Markov process and the delays are time-invariant. By employing the Lyapunov stability theory, some inequality techniques and the stochastic analysis, sufficient conditions are derived to achieve the global exponential stability in the mean square of the stochastic BAM neural network. One example is also provided in the end of this paper to illustrate the effectiveness of our results. [Copyright &y& Elsevier]

In this paper, the robust exponential stability problem is investigated for a class of Markovian jumping genetic networks which involve both uncertain parameters and stochastic disturbances. Under the assumption that the jumping parameters are generated from a continuous-time discrete-state homogeneous Markov process, the stability problem is first studied for a deterministic genetic model. By constructing suitable Lyapunov functionals and conducting some stochastic analysis, the stability criteria are derived in the form of linear matrix inequalities (LMIs), which can be easily checked in practice. Then, based on the derived results, sufficient LMI conditions are obtained explicitly for an indeterministic genetic system where the parameter uncertainties are norm-bounded. An illustrative example is presented to demonstrate the effectiveness and usefulness of the proposed stability criteria. [ABSTRACT FROM AUTHOR]

In this paper, we will establish new results on the attraction for solutions to stochastic functional differential equations with respect to semimartingale. Most of the existing results stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in the study of attraction. Moreover, from our results on the attraction follow several new criteria on almost surely asymptotic stability and boundedness of the solutions. [ABSTRACT FROM AUTHOR]