# American Institute of Mathematical Sciences

## Journals

DCDS-B
This paper discusses the stochastic Kolmogorov system with time-varying delay. Under two classes of sufficient conditions, this paper solves the non-explosion, the moment boundedness and the polynomial pathwise growth simultaneously. This is an important improvement for the existing results, since the moment boundedness and the polynomial pathwise growth do not imply each in general. Moreover, these two classes of conditions only depends on the parameters of the system and are easier to be used. Finally, a two-dimensional Komogorov model is examined to illustrate the efficiency of our result.
keywords: non-explosion time-varying delay polynomial pathwise growth. moment boundedness Stochastic Kolmogorov system
DCDS-B
The existence of a random attactor is established for a mean-square random dynamical system (MS-RDS) generated by a stochastic delay equation (SDDE) with random delay for which the drift term is dominated by a nondelay component satisfying a one-sided dissipative Lipschitz condition. It is shown by Razumikhin-type techniques that the solution of this SDDE is ultimately bounded in the mean-square sense and that solutions for different initial values converge exponentially together as time increases in the mean-square sense. Consequently, similar boundedness and convergence properties hold for the MS-RDS and imply the existence of a mean-square random attractor for the MS-RDS that consists of a single stochastic process.
keywords: Razumikhin-type theorem random delay ultimate boundedness. mean-square random dynamical system (MS-RDS) Mean-square random attractor
DCDS-B
In general, population systems are often subject to environmental noise. To examine whether the presence of such noise affects these systems significantly, this paper perturbs the Lotka--Volterra system

$\dot{x}(t)=\mbox{diag}(x_1(t), \cdots, x_n(t))(r+Ax(t)+B\int_{-\infty}^0x(t+\theta)d\mu(\theta))$

into the corresponding stochastic system

$dx(t)=\mbox{diag}(x_1(t), \cdots, x_n(t))[(r+Ax(t)+B\int_{-\infty}^0x(t+\theta)d\mu(\theta))dt+\beta dw(t)].$

This paper obtains one condition under which the above stochastic system has a global almost surely positive solution and gives the asymptotic pathwise estimation of this solution. This paper also shows that when the noise is sufficiently large, the solution of this stochastic system will converge to zero with probability one. This reveals that the sufficiently large noise may make the population extinct.

DCDS
The main aim of this paper is to establish the LaSalle-type theorem to locate limit sets for neutral stochastic functional differential equations with infinite delay, from which some criteria on attraction, boundedness and the almost sure stability with general decay rate and robustness are obtained. To make our theory more applicable, by the $M$-matrix theory, this paper also examines some conditions under which attraction and stability are guaranteed. These conditions also show that attraction and stability are robust with respect to stochastic perturbations. By specializing the general decay rate as the exponential decay rate and the polynomial decay rate, this paper examines two neutral stochastic integral-differential equations and shows that they are exponentially attractive and polynomially stable, respectively.
keywords: LaSalle-type theorem attractor neutral stochastic functional differential equations the Itô formula. infinite delay
MCRF
To treat networked systems involving uncertainty due to randomness with both continuous dynamics and discrete events, this paper focuses on diffusions modulated by a continuous-time Markov chain. In our paper [19], we considered ordinary differential equations with Markovian switching. This paper further treats more complex cases, namely, stochastic differential equations with Markovian switching. Our goal is to stabilize the systems under consideration. One of the difficulties is that the systems grow much faster than the allowable rates in the literature of stochastic differential equations. As a result, the underlying systems have finite explosion time. To overcome the difficulties, we develop feedback controls to extend the local solutions to global solutions and to stabilize the resulting systems. The feedback controls are Brownian type of perturbations. We establish the existence of global solution, prove the stability of the resulting systems, obtain boundedness in probability as $t\to\infty$, and provide sufficient conditions for almost sure stability. Then we present numerical examples to illustrate the main results.
keywords: stabilization feedback control. regularity stability Regime-switching diffusion
DCDS
A discrete stochastic Razumikhin-type theorem is established to investigate whether the Euler--Maruyama (EM) scheme can reproduce the moment exponential stability of exact solutions of stochastic functional differential equations (SFDEs). In addition, the Chebyshev inequality and the Borel-Cantelli lemma are applied to show the almost sure stability of the EM approximate solutions of SFDEs. To show our idea clearly, these results are used to discuss stability of numerical solutions of two classes of special SFDEs, including stochastic delay differential equations (SDDEs) with variable delay and stochastically perturbed equations.
keywords: stochastic functional differential equations stochastically perturbed equations. Moment exponential stability Razumikhin-type theorem Euler--Maruyama method
MCRF
This work develops moment exponential stability of functional differential equations (FDEs) with Markovian switching, in which a two-time-scale (real time $t$ and fast time $t/\epsilon$ with a small parameter $\epsilon>0$) continuous-time and finite-state Markov chain is used to represent the switching process. The essence is that there is a time-scale separation, which is motivated by the consideration of networked control systems and manufacturing systems. Under suitable conditions, we establish a Razumikhin-type theorem on the $p$th moment exponential $\epsilon$-stability for the small parameter $\epsilon$. By virtue of the Razumikhin-type theorem, we further deduce mean-square exponential stability results for delay differential equations (DDEs) and ordinary differential equations (ODEs) with two-time-scale Markovian switching. These stability results show that the overall system may be stabilized by the Markov switching even when some of the underlying subsystems are unstable. It is noted that in the presence of the Markovian switching, the stationary distribution of the fast changing part of the Markov chain plays an important role. Explicit conditions for the mean-square exponential stability of linear equations are derived and illustrative examples are provided to demonstrate our results.
keywords: moment exponential stability. Razumikhin-type theorem Functional differential equation two-time-scale Markov chain stationary distribution
DCDS-B
Population systems are often subject to various different types of environmental noises. This paper considers a class of Kolmogorov-type systems perturbed by three different types of noise including Brownian motions, Markovian switching processes, and Poisson jumps, which is described by a regime-switching jump diffusion process. This paper examines these three different types of noises and determines their effects on the properties of the systems. The properties to be studied include existence and uniqueness of global positive solutions, boundedness of this positive solution, and asymptotic growth property, and extinction in the senses of the almost sure and the $p$th moment. Finally, this paper also considers a stochastic Lotka-Volterra system with regime-switching jump diffusion processes as a special case.
keywords: extinction. jump process regime switching global solution Kolmogorov-type system bounedeness Brownian motion