随机时滞生物数学模型的渐近行为

In the natural world, the populations are inevitably affected by the environmental noises and delay exists universally in the stochastic differential equations, so the study of dynamics of the stochastic delay differential equations has great value in theory and practical significance. The thesis mainly focuses on the stochastic models with delays. Firstly, by using stochastic comparison theorem and stochastic Lyapunov method, we present the existence of the positive solution of stochastic systems with delays. Then we show the persistence, extinction and stochastic stability of systems. Moreover, for systems with degenerate and non degenerate diffusion terms, we investigate the stationary distribution, the ergodicity and the existence of the periodic solution of them through diffusion theory, ergodic theory of Has'minskii, Markov semigroup theory and the Has'minskii's theory of periodic solutions and so on. Based on the analysis, we tend to reveal the effect of stochastic perturbation and delays. And hence it can offer the theory support for predicting the change of population, forecasting the development trends of disease and analyzing causes of its outbreak.

在自然界中，生物个体不可避免地受到环境白噪声的影响，且时滞在随机微分方程中普遍存在，因此研究随机时滞微分方程的动力学行为有极强的实际意义和研究价值。本课题主要针对随机时滞模型，首先通过随机比较定理和随机Lyapunov函数方法给出系统正解的存在性。在此基础上研究系统的持久性、灭绝性和随机稳定性等。对于具有退化和非退化扩散项的随机时滞模型，通过扩散过程理论、Markov半群理论、Has'minskii的遍历性理论和周期解理论等探讨系统的平稳分布、遍历性以及周期解的存在性等。通过上述分析，目的在于揭示随机扰动和时滞项对模型渐近行为的影响，从而为预测生态各物种数量的变化和疾病的发展趋势以及分析疾病流行的原因提供理论依据。

在自然界中，生物个体不可避免地受到环境白噪声的影响，且时滞在随机微分方程中普遍.存在，因此研究随机时滞微分方程的动力学行为有极强的实际意义和研究价值。本课题主要针.对随机时滞模型，首先通过随机比较定理和随机Lyapunov函数方法给出系统正解的存在性。在.此基础上研究系统的持久性、灭绝性和随机稳定性等。对于具有退化和非退化扩散项的随机时.滞模型，通过扩散过程理论、Markov半群理论、Has'minskii的遍历性理论和周期解理论等探.讨系统的平稳分布、遍历性以及周期解的存在性等。通过上述分析，目的在于揭示随机扰动和.时滞项对模型渐近行为的影响，从而为预测生态各物种数量的变化和疾病的发展趋势以及分析.疾病流行的原因提供理论依据。