Xiaoyue Li

Population dynamical behavior of Lotka-Volterra system under regime switching

In this paper, we investigate a Lotka-Volterra system under regime switching dx(t)=diag(x1(t),...,xn(t))[(b(r(t))+A(r(t))x(t))dt+@s(r(t))dB(t)], where B(t) is a standard Brownian motion. The aim here is to find out what happens under regime switching. We first obtain the sufficient conditions for the existence of global positive solutions, stochastic permanence and extinction. We find out that both stochastic permanence and extinction have close relationships with the stationary probability distribution of the Markov chain. The limit of the average in time of the sample path of the solution is then estimated by two constants related to the stationary distribution and the coefficients. Finally, the main results are illustrated by several examples.

Qualitative analysis of a stochastic ratio-dependent predator-prey system

A stochastic ratio-dependent predator-prey model is investigated in this paper. By the comparison theorem of stochastic equations and Itos formula, we obtain the global existence of a positive unique solution of the ratio-dependent model. Besides, a condition for species to be extinct is given and a persistent condition is established. We also conclude that both the prey population and the ratio-dependent function are stable in time average. In the end, numerical simulations are carried out to confirm our findings.

A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching

In this paper, we consider neutral stochastic delay differential equations with Markovian switching. Our key aim is to establish LaSalle-type stability theorems for the underlying equations. The key techniques used in this paper are the method of Lyapunov functions and the convergence theorem of nonnegative semi-martingales. The key advantage of our new results lies in the fact that our results can be applied to more general non-autonomous equations.

Existence and uniqueness of solutions for singular (k, n - k ) conjugate boundary value problems

By mixed monotone method, the existence and uniqueness are established for singular (k, n - k) conjugate boundary value problems. The theorems obtained are very general and complement previous known results.

A new existence theory for positive periodic solutions to functional differential equations with impulse effects

The principle of this paper is to deal with a new existence theory for positive periodic solutions to a kind of nonautonomous functional differential equations with impulse actions at fixed moments. Easily verifiable sufficient criteria are established. The approach is based on the fixed-point theorem in cones. The paper extends some previous results and obtains some new results.

The Improved LaSalle-Type Theorems for Stochastic Differential Delay Equations

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.

Approximate solutions of stochastic differential delay equations with Markovian switching

Our main aim is to develop the existence theory for the solutions to stochastic differential delay equations with Markovian switching and to establish the convergence theory for the Euler–Maruyama approximate solutions under the local Lipschitz condition. As an application, our results are used to discuss a stochastic delay population system with Markovian switching.

A new existence theory for single and multiple positive periodic solutions to volterra integro-differential equations with impulse effects

This paper presents a new existence theory for single and multiple positive periodic solutions to a kind of nonautonomous Volterra integro-differential equations with impulse effects. Existence is established by using Krasnoselskii fixed-point theorem in cones. This paper extends some previous results and reports some new results about impulsive differential equation

Switching diffusion logistic models involving singularly perturbed Markov chains: Weak convergence and stochastic permanence

ABSTRACT Focusing on stochastic dynamics involve continuous states as well as discrete events, this article investigates stochastic logistic model with regime switching modulated by a singular Markov chain involving a small parameter. This Markov chain undergoes weak and strong interactions, where the small parameter is used to reflect rapid rate of regime switching among each state class. Two-time-scale formulation is used to reduce the complexity. We obtain weak convergence of the underlying system so that the limit has much simpler structure. Then we utilize the structure of limit system as a bridge, to invest stochastic permanence of original system driving by a singular Markov chain with a large number of states. Sufficient conditions for stochastic permanence are obtained. A couple of examples and numerical simulations are given to illustrate our results.

Convergence rate and stability of the truncated Euler-Maruyama method for stochastic differential equations

Abstract Recently, Mao (2015) developed a new explicit method, called the truncated Euler–Maruyama (EM) method, for the nonlinear SDE and established the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. In his another follow-up paper (Mao, 2016), he discussed the rates of L q -convergence of the truncated EM method for q ≥ 2 and showed that the order of L q -convergence can be arbitrarily close to q ∕ 2 under some additional conditions. However, there are some restrictions on the truncation functions and these restrictions sometimes might force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is to establish the convergence rate without these restrictions. The other aim is to study the stability of the truncated EM method. The advantages of our new results will be highlighted by the comparisons with the results in Mao (2015, 2016) as well as others on the tamed EM and implicit methods.