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.     

Robust stability in distribution of Boolean networks under multi-bits stochastic function perturbations

In this work, robust stability in distribution of Boolean networks (BNs) is studied under multi-bits probabilistic and markovian function perturbations. Firstly, definition of multi-bits stochastic function perturbations is given and an identification matrix is introduced to present each case. Then, by viewing each case as a switching subsystem, BNs under multi-bits stochastic function perturbations can be equivalently converted into stochastic switching systems. After constructing respective transition probability matrices which can unify multi-bits probabilistic and markovian function perturbations in a consolidated framework, robust stability in distribution can be handled. On such basis, necessary and sufficient conditions for robust stability in distribution of BNs under stochastic function perturbations are given respectively. Finally, two numerical examples are presented to verify the validity of our theoretical results.     

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.     

Generalized average dwell time approach to stability and input-to-state stability of hybrid impulsive stochastic differential systems

This paper is concerned with the stability properties of a class of impulsive stochastic differential systems with Markovian switching. Employing the generalized average dwell time (gADT) approach, some criteria on the global asymptotic stability in probability and the stochastic input-to-state stability of the systems under consideration are established. Two numerical examples are given to illustrate the effectiveness of the theoretical results, as well as the effects of the impulses and the Markovian switching on the systems stability.     

On the global asymptotic stability and oscillation of solutions in a stochastic business cycle model

We discuss a second order nonlinear stochastic difference equation which is constructed of a business cycle model with organized labor considered. A global asymptotic mean square stability criterion is obtained by Lyapunov function method. We also prove a theorem on the almost sure oscillation of the solutions for the difference equation with state-independent stochastic perturbations.     

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.     

Asymptotic stability of a two-group stochastic SEIR model with infinite delays

A two-group stochastic SEIR epidemic model with infinite delays is proposed and investigated. Sufficient conditions for asymptotic stability are established. Some simulation figures are introduced to support the results.     

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.     

Dynamics of a stochastic three species prey-predator model with intraguild predation

Intraguild predation is ubiquitous in many ecological communities. This paper is concerned with a stochastic three species prey-predator model with intraguild predation. The model involves a prey, an intermediate predator which preys on only prey and an omnivorous top predator which preys on both prey and intermediate predator. First, we show the existence of a unique positive global solution of the model. Then we mainly establish the sufficient conditions for the extinction and persistence in the mean of each population. Moreover, we show that the model is stable in distribution. Finally, some numerical simulations are given to illustrate the main results.     

Stability analysis of a stochastic delayed SIR epidemic model with general incidence rate

In this paper, a stochastic delayed epidemic model with a generalized incidence rate is proposed and discussed. The positivity of solutions is established. A linearized form of the model is given and the stability conditions of the endemic equilibrium are obtained by using the technique of Lyapunov functionals.     

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 a multiple infected compartments model for waterborne diseases

In this paper, mathematical analysis is carried out for a multiple infected compartments model for waterborne diseases, such as cholera, giardia, and rotavirus. The model accounts for both person-to-person and water-to-person transmission routes. Global stability of the equilibria is studied. In terms of the basic reproduction number $R_0$, we prove that, if $R_0⩽1$, then the disease-free equilibrium is globally asymptotically stable and the infection always disappears; whereas if $R_0>1$, there exists a unique endemic equilibrium which is globally asymptotically stable for the corresponding fast–slow system. Numerical simulations verify our theoretical results and present that the decay rate of waterborne pathogens has a significant impact on the epidemic growth rate. Also, we observe numerically that the unique endemic equilibrium is globally asymptotically stable for the whole system. This statement indicates that the present method need to be improved by other techniques.     

Stationary distribution of a stochastic food chain chemostat model with general response functions

In this paper, we focus on a food chain chemostat model with general response functions, perturbed by white noise. Under appropriate assumptions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution by using stochastic Lyapunov analysis method. Our main effort is to construct the suitable Lyapunov function.     

Global asymptotical stability for a fishery model with seasonal harvesting

A new fishery model is proposed by using the strategy of seasonal harvesting. Sufficient and necessary conditions are established to ensure the existence of a unique equilibrium or a periodic solution by the approach of Poincar′e maps. It is shown that the equilibrium or the periodic solution is globally asymptotically stable. Numerical examples are provided to demonstrate the model dynamics and some biological implications are given as well.     

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.