随机时滞微分方程解的矩稳定性和有界性

This project will study the stability and boundedness of moments of the solutions to linear stochastic delay differential equations through techniques of the Laplace transform. For the linear stochastic differential equation with discrete delay and the linear stochastic differential equation with distributed delay, although Lei et al and Wang et al established the characteristic equations for the second moments of their solutions and gave the sufficient conditions for boundedness and unboundedness of the moments, respectively, they did not analysize the distribution of the roots of the characteristic equations and the sufficient conditions were not applicable easily. Thus, in this project, for the special cases of two kinds of equations: the linear stochastic differential equations with discrete delay and with distributed delay, we will analysize the roots of the corresponding characteristic equations detailedly and will establish the sufficient conditions for the moments to be bounded and unbounded for application. As the application of theoretic results, we will investigate the moment stability and boundedness for the stochastic logistic model and the peripheral control of white blood cell production with stochastic perturbation.

本项目将利用Laplace变换研究线性随机时滞微分方程解的矩稳定性和有界性。对于具有离散时滞的线性随机微分方程和具有分布时滞的线性随机微分方程，Lei等和Wang等已经分别建立了其解的二阶矩对应的特征方程，并利用该特征方程给出了二阶矩有界和无界的充分条件，但是他们都没有详细分析特征方程根的分布情况，给出的二阶矩有界和无界的充分条件都不方便应用。为此，本项目将对具有离散时滞和分布时滞的这两类线性随机时滞微分方程的特殊情形，深入细致地分析其对应的特征方程，研究特征方程根的分布情况，建立便于应用的二阶矩有界和无界的充分条件。作为理论结果的应用，本项目还将研究随机时滞logistic模型和具有随机扰动的白细胞增殖的外围控制模型的矩稳定性和有界性。

年龄结构模型是描述生命系统的非常必要的模型，是结构种群动力学模型中一类非常重要的模型。如果考虑物种的成熟年龄，把物种分成幼年个体和成年个体；假设死亡系数和出生系数这两个函数在大于成熟年龄和小于成熟年龄时分别是常数，对年龄结构模型通过特征线法进行积分就可以得到时滞微分方程，即通过引入时滞（成熟年龄）将年龄结构模型与时滞微分方程之间的联系起来。在年龄结构模型中考虑随机因素，即得随机年龄结构模型。本项目研究了随机年龄结构模型解的稳定性。首先利用Lyapunov泛函方法研究了非稠定的半线性随机发展方程解的随机依概率稳定和二阶矩的稳定性，然后将所得理论结果应用到随机年龄结构模型。对于随机时滞logistic模型和具有随机扰动的白细胞增殖的外围控制模型，应用已知的理论结果得到这两个生物模型解的稳定性和矩有界性。