几类随机生物数学模型不变概率分布及数值模拟

In the development of the real ecological systems, there are kinds of noises everywhere, whether in the theoretical or the practical applications, the interference of the noises in the system is inevitable, to study the properties of stochastic system has important academic value. In the studying of the deterministic models, the existence and stability of the equilibrium is one of the most interesting research topics in academic fields, while for stochastic system, due to the interference of the noise, the properties of stochastic model is very complex, it is hard to obtain the specific states of the system at some an exact moment exactly, so it is very necessary to discuss the properties in some statistical sense of stochastic system.. The aim of this project is to investigate several important biological models, taking into account the white noise and color noise in the model, then establish stochastic biomathematical models with white and color noise. We will consider two kinds of problems, one is stability in distribution which is an important property corresponding to stability of the equilibrium of deterministic models, namely study the stability in the sense of distribution. The other is to investigate the existence of equilibrium for stochastic models corresponding to the equilibrium of deterministic models, the existence of the invariant distribution and numerical simulations. The studying of these two problems will provide an important method for the application of the knowledge of stochastic analysis to the practical model.

在现实生态系统的发展变化过程中，各种形式的随机干扰无处不在，无论是从理论上还是实际应用上, 系统的噪声干扰是不可避免的，研究随机系统的性质具有重要的学术意义。在确定性模型性质的研究中，均衡解的存在性和稳定性一直是学术界比较感兴趣的研究课题之一，而对随机模型而言，由于随机噪声干扰，模型的性质比较复杂，很难确切的得到系统在某个时刻的具体状态，因此研究随机系统在某种统计意义下的性质是很必要的。. 本项目拟对几类重要的生物模型考虑白噪声和有色噪声的干扰，建立具有白噪声和有色噪声干扰的随机生物数学模型。研究两类基本问题，其一，对应于确定性模型均衡解稳定性的一个重要性质，依分布稳定性，也即研究解过程在分布意义下的一种稳定性。二，对应于确定性模型均衡解的存在性，研究均衡解的随机对应物，不变分布的存在性及数值模拟。这两个问题的研究为随机分析的理论知识应用到实际模型中提供了重要的研究方法。

本项目主要研究了随机生物数学模型的依分布稳定性和不变概率分布的存在性，利用计算机模拟随机系统的不变概率分布及随机方程解过程样本轨道。在重要的生物数学模型中引入白噪声和有色噪声的干扰，建立具有白噪声和有色噪声干扰的随机生物数学模型。针对随机Gompertz模型，研究了两类基本问题，其一，依分布稳定性，也即研究解过程在分布意义下的一种稳定性。二，研究确定性Gompertz模型均衡解的随机对应物，不变分布的存在性及数值模拟。另外本项目对随机周期解问题做了初步的研究，并给出相应的结果。通过对这两个问题的研究，为随机分析的理论知识及应用提供了重要的研究方法。