Global asymptotic stability of a stochastic Lotka-Volterra model with infinite delays

In this paper, sufficient criteria for global asymptotic stability of a general stochastic Lotka-Volterra system with infinite delays are established. Some simulation figures are introduced to support the analytical findings.     

A notion of stochastic input-to-state stability and its application to stability of cascaded stochastic nonlinear systems

In this paper, the property of practical input-to-state stability and its application to stability of cascaded nonlinear systems are investigated in the stochastic framework. Firstly, the notion of （practical） stochastic input-to-state stability with respect to a stochastic input is introduced, and then by the method of changing supply functions, （a） an （practical） SISS-Lyapunov function for the overall system is obtained from the corresponding Lyapunov functions for cascaded （practical） SISS subsystems.     

Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation

This paper discusses a randomized non-autonomous logistic equation , where B(t) is a 1-dimensional standard Brownian motion. In [D.Q. Jiang, N.Z. Shi, A note on non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl. 303 (2005) 164-172], the authors show that E[1/N(t)] has a unique positive T-periodic solution E[1/Np(t)] provided a(t), b(t) and α(t) are continuous T-periodic functions, a(t)>0, b(t)>0 and . We show that this equation is stochastically permanent and the solution Np(t) is globally attractive provided a(t), b(t) and α(t) are continuous T-periodic functions, a(t)>0, b(t)>0 and mint∈[0,T]a(t)>maxt∈[0,T]α²(t). By the way, the similar results of a generalized non-autonomous logistic equation with random perturbation are yielded.     

Global stability of a stage-structured epidemic model with a nonlinear incidence

In this paper, a stage-structured epidemic model with a nonlinear incidence with a factor S^p is investigated. By using limit theory of differential equations and Theorem of Busenberg and van den Driessche, global dynamics of the model is rigorously established. We prove that if the basic reproduction number R₀ is less than one, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if R₀ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Numerical simulations support our analytical results and illustrate the effect of p on the dynamic behavior of the model.     

On asymptotic stability and instability with respect to a fading stochastic perturbation

We develop necessary and sufficient conditions for the a.s. asymptotic stability of solutions of a scalar, non-linear stochastic equation with state-independent stochastic perturbations that fade in intensity. These conditions are formulated in terms of the intensity function: roughly speaking, we show that as long as the perturbations fade quicker than some identifiable critical rate, the stability of the underlying deterministic equation is unaffected. These results improve on those of Chan and Williams; for example, we remove the monotonicity requirement on the drift coefficient and relax it on the intensity of the stochastic perturbation. We also employ different analytic techniques.     

Mean square global asymptotic stability of stochastic recurrent neural networks with distributed delays

By constructing suitable Lyapunov functionals and combining with matrix inequality technique, a new simple sufficient condition is presented for the global asymptotic stability in the mean square of delayed neural networks.     

A sharp global stability result for a discrete population model

We get a sharp global stability result for a first order difference equation modelling the growth of bobwhite quail populations. The corresponding higher-dimensional model is also discussed, and our stability conditions improve other recent results for the same equation.     

Stationary distribution and extinction of a stochastic SIRI epidemic model with relapse

In this paper, we study the dynamics of a stochastic Susceptible-Infective-Removed-Infective (SIRI) epidemic model with relapse. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of an ergodic stationary distribution to the model. Moreover, sufficient conditions for extinction of the disease are also obtained.     

Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates

In this paper, we introduce a basic reproduction number for a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. Then, we establish that global dynamics are completely determined by the basic reproduction number R₀. It shows that, the basic reproduction number R₀ is a global threshold parameter in the sense that if it is less than or equal to one, the disease free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Finally, two numerical examples are also included to illustrate the effectiveness of the proposed result.     

Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems

In this paper, we analyze the robustness of global exponential stable stochastic delayed systems subject to the uncertainty in parameter matrices. Given a globally exponentially stable systems, the problem to be addressed here is how much uncertainty in parameter matrices the systems can withstand to be globally exponentially stable. The upper bounds of the parameter uncertainty intensity are characterized by using transcendental equation for the systems to sustain global exponential stability. Moreover, we prove theoretically that, the globally exponentially stable systems, if additive uncertainties in parameter matrices are smaller than the upper bounds arrived at here, then the perturbed systems are guaranteed to also be globally exponentially stable. Two numerical examples are provided here to illustrate the theoretical results.     

Stability analysis of a stochastic logistic model with nonlinear diffusion term

This paper presents an asymptotic analysis of a stochastic logistic population model with nonlinear diffusion term. The classical probability method is applied to obtain the criteria of asymptotic behavior for the considered model. The numerical simulations validate the efficiency of the theory analysis.     

Stochastic asymptotical stability for stochastic impulsive differential equations and it is application to chaos synchronization

In this paper, the stochastic asymptotical stability of stochastic impulsive differential equations is studied, and a comparison theory about the stochastic asymptotical stability of trivial solution is established. From the comparison theory, we can find out whether the stochastic impulsive differential system is stochastic asymptotically stable by studying the stability of a deterministic comparison system. As an application of this theory, we study the problem of chaos synchronization in Chua circuit using impulsive method. Finally, numerical simulation is employed to verify the feasibility of our method.     

On almost sure stability conditions of linear switching stochastic differential systems

In modeling practical systems, it can be efficient to apply Poisson process and Wiener process to represent the abrupt changes and the environmental noise, respectively. Therefore, we consider the systems affected by these random processes and investigate their joint effects on stability. In order to apply Lyapunov stability method, we formulate the action of the infinitesimal generator corresponding to such a system. Then, we derive the almost sure stability conditions by using some fundamental convergence theorem. To illustrate the theoretical results, we construct an example to show that it is possible to achieve stabilization by using random perturbations.     

Global stability of a rational symmetric difference equation

Motivated by ideas from papers: C. Cinar, I. Yalçinkaya, S. Stević, A note on global asymptotic stability of a family of rational equations, Rostock. Math. Kolloq. 59 (2004), 41–49, and S. Stević, Global stability and asymptotics of some classes of rational difference equations, J. Math. Anal. Appl. 316 (2006), 60–68, here we confirm a conjecture on a rational symmetric difference equation from the paper: K. Berenhaut, J. Foley, S. Stević, The global attractivity of the rational difference equation , Appl. Math. Lett. 20 (2007), 54–58.     

The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion

In this paper, we consider a class of impulsive stochastic differential equations driven by G-Brownian motion (IGSDEs in short). By means of the G-Lyapunov function method, some criteria on p-th moment stability and p-th moment asymptotical stability for the trivial solutions of IGSDEs are established. An example is presented to illustrate the efficiency of the obtained results.     

Oscillation and global asymptotic stability in a discrete epidemic model

In this paper, we study the oscillation, global asymptotic stability, and other properties of the positive solutions of the difference equation

     

Global stability in a diffusive Holling-Tanner predator-prey model

A diffusive Holling-Tanner predator-prey model with no-flux boundary condition is considered, and it is proved that the unique constant equilibrium is globally asymptotically stable under a new simpler parameter condition.     

Almost sure exponential stability of stochastic reaction diffusion systems

The Lyapunov direct method, as the most effective measure of studying stability theory for ordinary differential systems and stochastic ordinary differential systems, has not been generalized to research concerning stochastic partial differential systems owing to the emptiness of the corresponding Ito differential formula. The goal of this paper is just employing the Lyapunov direct method to investigate the stability of Ito stochastic reaction diffusion systems, including asymptotical stability in probability and almost sure exponential stability. The obtained results extend the conclusions of [X.X. Liao, X.R. Mao, Exponential stability and instability of stochastic neural networks, Stochastic Analysis and Applications 14 (2) (1996) 165-185; X.X. Liao, S.Z. Yang, S.J. Cheng, Y.L. Fu, Stability of general neural networks with reaction-diffusion, Science in China (F) 44 (5) (2001) 389-395].     

Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination

This paper deals with global dynamics of an SIRS epidemic model for infections with non permanent acquired immunity. The SIRS model studied here incorporates a preventive vaccination and generalized non-linear incidence rate as well as the disease-related death. Lyapunov functions are used to show that the disease-free equilibrium state is globally asymptotically stable when the basic reproduction number is less than or equal to one, and that there is an endemic equilibrium state which is globally asymptotically stable when it is greater than one.