随机Kolmogorov型系统及其数值解的渐近性质分析

Stochastic Kolmogorov-type system and its special form—stochastic Lotka-Volterra system—have important application values not only in the field of biology and economics as well as many other fields, such as population dynamic systems and R&D model. Although there are a lot of literatures and studies in this area, it is can not be denied that most researches are unable to give analytical solutions. Therefore, numerical method has become an important technique and means of studying practical problems. While using the numerical solution substitute the real solution, whether it can reproduce asymptotic properties of the original system, which led the great attention of researchers . Nowadays, the most research results of numerical solution have been obtained under the linear growth condition or the global Lipschitz condition on the diffusion coefficient. However, stochastic Kolmogorov-type system does not satisfy the above conditions.. This subject attempts to study numerical trivial stability and stationary distribution of stochastic Kolmogorov-type system under stochastic sense of monotony conditions, which defy conventional conditions. And it explore sufficient conditions under which the numerical approximations can reproduce the above asymptotic properties of the original system. Meanwhile this subject aims to study the asymptotic stationary distribution of stochastic approximation deeply. And compare it with the stationary distribution of the analytical solution, in order to clarify the relationship between the above two distribution and the impact of the stepsize .

随机Kolmogorov型系统以及它的特殊形式随机Lotka-Volterra系统在生物学和经济学等领域有重要的应用价值，如种群动力系统和R&D模型。虽然此领域已有大量文献和研究，但不可否认，大部分是无法给出解析解的。因此数值方法就成为研究实际问题的重要技术和手段。而用数值解来替代真实解时，能否复现原系统的渐近性质，已经引发许多研究者的高度重视。目前大多数值解方面的研究成果是系统的扩散系数满足线性增长或全局Lipschitz条件下得到的。而随机Kolmogorov型系统并不满足上诉条件。. 本课题拟在突破常规条件的约束，仅凭借随机意义下的单调型条件，研究随机Kolmogorov型系统及其数值解的零解稳定性以及稳定分布性质。探索数值解能保留原系统上述渐近性质的充分条件。同时深入研究随机近似的渐近稳定分布，并与解析解的渐近稳定分布比较，阐明两者的关系以及步长对两者差距的影响。

随机Kolmogorov型系统通常并不满足线性增长或全局Lipschitz条件。本项目突破常规条件的约束，分别在Khasminskii型条件、单调条件和多项式条件下建立了带无界时滞的随机Kolmogorov型系统及其数值解的零解稳定性。. 本项目得出以下结论：（1）数值解稳定性的建立并不需要步长充分小。只要数值解本身的定义是合理的，在证明过程中不再需要对步长加以约束和限制。即：大步长算法仍可保留原系统的稳定性质。（2）数值解的稳定速度以原系统的稳定速度为极限。课题在研究过程中发现，数值解的稳定速度和步长有密切关系，且该速度通常是低于原系统的稳定速度。当步长趋向于0时，数值解的稳定速度以原系统的稳定速度为极限。（3）在无界时滞情形下，稳定的速度与衰减因子有直接关系。无论是解析解还是数值解的衰减速度都是以衰减因子作为参照的。即：不同的参照函数会导致不同的衰退速度。