Analysis of nonlinear dynamics via asymptotic and perturbation methods

我的研究将集中在以下几个相当相关的领域：首先，我将研究用数学方程模拟的单个或多个物种的空间运动和进化。最近对空间扩散和时间延迟的综合效应的研究进展导致了一些具有时间延迟和空间相互作用的有趣的数学方程。对这些数学方程的非线性动力学(如行波、其在一个区域内运动的稳定性、预测的难度)进行定性和定量的分析是我未来研究的主要目的。除了研究物种的空间运动外，我还将研究生态学与传染病的相互作用。当传染病在单个物种或多个物种中表现出来时，就会形成新的进化行为。纵观有记录的历史，在新的地区到达、建立和传播的非本土媒介已经煽动了疟疾、黄热病、斑疹伤寒、狂犬病、兔出血症、鼠疫和西尼罗河等人(或动物)疾病的流行。为了更好地了解疾病在这些物种中的传播，我们需要调查生态环境和疾病传播之间的相互作用。对这些疾病的动态传播进行系统研究是我感兴趣的另一个重要课题。对于短期目标，我将调查由赤狐引起的狂犬病在欧洲的传播情况。将通过考虑幼狐的传播行为来构建一个新的数学模型。我将把我们的新方法和研究经验扩展到研究西尼罗河、兔出血症等传染病的空间传播。对神经网络信号传递领域和血液病干细胞群周期振荡领域出现的实际问题的数学建模和分析也将在我的研究范围内。

My research will concentrate on the following rather related fields:     First, I will study the  spatial movement and evolution of   single or multiple species modeled by mathematical equations. The recent progress towards the combined effect of spatial diffusion and temporal delay leads to some interesting mathematical equations with time delay and spatial interaction. The qualitative and quantitative analyses of nonlinear dynamics (e.g. traveling wave, stability of their movement in an area, difficulty in prediction) of these mathematical equations are the main purpose of my research in the future.     Besides the investigation of spatial movement of species, I will also study the interaction between Ecology and infectious diseases. When infectious diseases are exhibited in a single species or multiple species, new evolution behaviors are formed. Throughout recorded history, non-indigenous vectors that arrive, establish and spread in new areas have fomented epidemics of human (or animal) disease such as malaria, yellow fever, typhus, rabies, rabbit hemorrhagic disease, plague and west Nile.  In order to better understand the transmission of the diseases in  species, we need to investigate the interaction between ecology  and the spread of the diseases. A systematic study of dynamical transmission of these diseases is another  subject of my  interest. For the short-term goal, I will investigate the spread of rabies in Europe caused by red foxes. A new mathematical model will be constructed by taking into account  the dispersal behavior of juvenile foxes.  I will extend our novel methods and research experiences to study the spatial spread of infectious disease such as west Nile, rabbit hemorrhagic disease and so on.     Mathematical modeling and analysis of real problems arising in the fields of signal transmission of neural networks and in the field of periodic oscillation of stem cell population in hematological disease will also be in my research scope.