Wenxue Li

Global stability analysis for stochastic coupled systems on networks

This paper considers the global stability problem for some stochastic coupled systems on networks (SCSNs). We provide a systematic method for constructing a global Lyapunov function for these SCSNs, by using graph theory and the Lyapunov second method. Consequently, a new global stability principle, which has a close relation to the topology property of the network, is given. As an application to the results, we employ the principle to two well-known coupled systems in physical and ecology, and then some easy-verified sufficient conditions which guarantee the global stability are obtained.

Global stability for discrete Cohen–Grossberg neural networks with finite and infinite delays

Abstract In this paper, the global stability problem for a general discrete Cohen–Grossberg neural network with finite and infinite delays is investigated. A simple criterion ensuring the global asymptotical stability is established, by applying the Lyapunov method and graph theory. Finally, an example showing the effectiveness of the provided criterion is given.

Optimal harvesting policy for stochastic Logistic population model

Abstract In this paper, we consider some optimal harvesting policies for single population models, in which the harvest effort and the intrinsic growth rate are disturbed by environment noises. We choose the maximum sustainable yield and the maximum retained profits as two management objectives, and obtain the optimal harvesting policies, respectively. For the two objectives, we give the optimal harvest effort that maximizes the sustainable yield (or retained profits), the maximum of expectation of sustainable yield (or retained profits) and the corresponding variance. Their explicit expressions are determined by the coefficients of equation and the disturbance intensity.

A graph-theoretic approach to boundedness of stochastic Cohen-Grossberg neural networks with Markovian switching

In this paper, a novel class of stochastic Cohen-Grossberg neural networks with Markovian switching (SCGNNMSs) is investigated, where the white noise and the color noise are taken into account. By utilizing Lyapunov method, some graph theory and M-matrix technique, several sufficient conditions are obtained to ensure the asymptotic boundedness of the SCGNNMSs. These criteria have a close relation to the topology property of the network and are easy to be verified in practice. Two numerical examples are also presented to substantiate the theoretical results.

Ergodic method on optimal harvesting for a stochastic Gompertz-type diffusion process

Abstract In this paper, the stochastic harvesting problem is regarded as a mathematical formulation of finding the maximum sustained yield and the corresponding best sustainable harvesting strategies under uncertainty. We use a new method to solve this problem, and prove the equivalency between this method and previous methods. This paper is the first attempt to apply the ergodic theory on the optimal harvesting problem, to the best of our knowledge.

Persistence and extinction of a stochastic delay Logistic equation under regime switching

Abstract A stochastic delay Logistic equation under regime switching is proposed and studied. Sufficient conditions for extinction, non-persistence in the mean and weak persistence of the solutions are established. The critical value between weak persistence and extinction is obtained.

Stability analysis for stochastic neural network with infinite delay

This paper considers a stochastic neural network (SNN) with infinite delay. Some sufficient conditions for stochastic stability, stochastic asymptotical stability and global stochastic asymptotical stability, respectively, are derived by means of Lyapunov method, Ito formula and some inequalities. As a corollary, we show that if the neural network with infinite delay is stable under some conditions, then the stochastic stability is maintained provided the environmental noises are small. Estimates on the allowable sizes of environmental noises are also given. Finally, a three-dimensional SNN with infinite delay is analyzed and some numerical simulations are illustrated to show our results.

Razumikhin-type theorems on exponential stability of stochastic functional differential equations on networks

In this paper, stability problem of stochastic functional differential equations on networks (SFDENs) is investigated. Based on the graph theory and Razumikhin technique, some conditions are captured to guarantee that SFDEN is p-th moment and almost sure exponentially stable. Furthermore, the criterion ensuring the exponential stability for stochastic coupled systems on networks with time-varying delay is established, by applying Razumikhin-type theorem. Finally, a numerical example is provided to demonstrate the effectiveness of the theoretical results.

A Generalization of Itô's Formula and the Stability of Stochastic Volterra Integral Equations

It is well known that Ito’s formula is an essential tool in stochastic analysis. But it cannot be used for general stochastic Volterra integral equations (SVIEs). In this paper, we first introduce the concept of quasi-Ito process which is a generalization of well-known Ito process. And then we extend Ito’s formula to a more general form applicable to some kinds of SVIEs. Furthermore, the stability in probability for some SVIEs is analyzed by the generalized Ito’s formula. Our work shows that the generalized Ito’s formula is powerful and flexible to use in many relevant fields.

Exponential Stability of Coupled Systems on Networks with Mixed Delays and Reaction-Diffusion Terms

This paper is concerned with the stability analysis issue for coupled systems on networks with mixed delays and reaction-diffusion terms (CSNMRs). By employing Lyapunov method and Kirchhoffs Theorem in graph theory, a systematic method is proposed to guarantee exponential stability of CSNMRs. Two different kinds nof sufficient criteria are derived in the form of Lyapunov function and coefficients of the system, respectively. Finally, a numerical example is given to show the effectiveness of the proposed criteria.