具有时空结构和时滞的随机扩散和非局部扩散方程的动力学行为研究

In the course of the study of mathematical biology, population dynamics has become an important content, of which the study of population model with stochastic diffusion and nonlocal diffusion have been attracted much attention and study by many mathematicians and biologists. In addition, in order to reflect the real dynamical behaviors of models that relay on the past history of systems,it is reasonable to bring time delays into the systems. In particular, in mathematical biology, there are a lot of population dynamics models which are used to describe by the reaction diffusion equations with time delays. Due to the differences in the energy in the transmission and transformation of individual biological, the research on the dynamic properties of predator-prey systems with different functional response function and diffusion possesses a strong theoretical and practical significance, such as the existence of the steady state solutions, the stability of the equilibrium points, Hopf bifurcation problem caused by time delay. In nonlinear science, the study of bifurcation problems is more important, since it can reflect the subtle changes of one or some factors in the real world can cause a huge impact on the change of material. In study of the population dynamics, bifurcation research can help us understand that the change of some of the parameters (such as living space and the maturation period) causes the change of the population dynamics (such as equilibrium stable or oscillatory), through the regulation of these data can make the species toward the expected direction.

在生物数学的研究过程中,种群动力学性质已经成为了一个重要内容,而其中对具有随机扩散项和非局部扩散项的种群模型的研究受到了许多数学家和生物学家的广泛关注和研究.此外,为了使得系统更能准确的反映其动力学行为,我们将引入时滞.特别是在生物数学的研究中,大量的种群动力学模型是用时滞反应扩散方程来描述的.由于能量在生物个体中的传递、转化的差异,对具有不同功能反应函数以及扩散项的捕食-被捕食系统的动力学性质的研究,如稳态解的存在性,平衡点的稳定性,由时滞引起的Hopf分支等问题,具有很强的理论和实际意义.在非线性科学中,分支问题的研究是比较重要的,尤其在种群动力学的研究中,分支研究可以帮助我们了解某些参数(比如生物的生存空间和成熟期等)对种群动力产生的变化(比如平衡态的稳定或振荡),通过调控这些数据可以使得物种向着人们所期望的方向发展.

本课题在具有时空结构和时滞的随机扩散和非局部扩散方程的动力学行为的研究方面取得了一系列的研究成果，我们主要利用Lyapunov-Schmidt约化理论、Crandall–Rabinowitz分支理论、中心流形方法和规范型理论、单调动力系统理论以及其他分线性分析技巧研究了几类捕食系统的相关动力学研究具体包括一类带有非局部项的非局部扩散的LOGISTIC模型以及考察了几类带有时滞和一般功能捕食项的两种群反应扩散捕食模型的相关动力学行为研究。此外，还给出了几类带有对流项的反应扩散系统的相关动力学行为研究以及考察了某些非时滞复杂系统分支以及相关动力学行为。因此，基本完成了项目的预期目标，为今后进一步的深入研究打下了良好的铺垫。