非局部随机微分系统的动力学行为及其应用

We plan to research the dynamic behaviours and applications of nonlocal stochastic differential systems with variable delays in this project. First，the theory analysis of general nonlocal stochastic differential systems will be established by innovative approaches, and then the results will be applied in stochastic population models and stochastic neural networks models. For the stochastic population dynamic systems, we will focus on how to control the population trends, specially the control conditions of the growth limitation,the persistence and the exctinction, and find out what will happened after introducing expectations into coefficients of systems。 For the stochastic neural networks with expectations in coefficients, we will study their stabilities. One of the innovative theory is that, we will try to renovate the semi-martingale convergence theorem which can not be applied in nonlocal models anymore, or we may find new approaches that can replace the semi-martingale convergence theorem. The topic of this project is based on one of the newest research directions of stochastic differential systems and the renewed models can give more accurate description of real problems. Hope that our research achivements will be applied in the ecology, the finance, engineering technology and other fields.

本项目研究含变动时滞的非局部随机微分系统的动力学行为及其应用。对于一般性的非局部随机微分系统，利用创新的研究方法，构建其动力学行为的理论体系，并用于随机生态模型和随机神经网络模型的研究.对于随机种群动力系统，重点研究在期望进入模型的情况下，如何实现对种群规模变化趋势的控制。特别研究生物种群的增长极限、持久性和灭绝性的控制条件；对于随机神经网络模型，重点研究在期望进入模型后对其稳定性的影响。在研究方法上，最重要的半鞅收敛定理并不能直接用于这一类系统，本项目将力图对其进行改造，使之能发挥作用，或者开发某种替代工具。探索解决该类问题的新途径是本研究的特色之一。本项目的选题基于近两年来随机微分系统理论研究中的某些新趋向，依此研究而更新的几种随机模型对于实际问题的模拟更加完善准确，在生态学，金融学以及工程应用等领域具有广泛的应用前景。

本项目主要研究非局部随机微分系统的性质及其应用，在项目截止时完成了对随机微分方程稳定性理论的推广、获得了非局部随机神经网络模型的指数稳定性、研究了随机种群动力系统的灭绝性，并对随机微分方程数值解的稳定性问题做了深入探讨。