Study of applied dynamical systems in physical/biological sciences via asymptotic methods

拟议方案的总体目标是通过数学方法进一步研究物理和生物科学中的各种数学模型。 预期的结果不仅将解决各种应用动力系统中一些长期存在的问题，而且有助于建立时滞微分方程渐近分析的基本理论。具体地说，我们的目标是在以下方面推动这一领域的发展：(A)发展一种全新的方法来构造行波前；(B)从理论上建立周期介质中行波前的稳定性；(C)解决或部分解决该领域中的一些公开问题；(D)从理论上给出图案形成肯定发生的充分条件。 这些数学分析将有实际应用：(A)帮助我们更好地了解传染病(例如狂犬病)的传播模式；(B)更好地确定潜伏期对疾病传播的影响；(C)这一结果可用于评估过去几年使用的不同方法控制传染病的有效性。

The general objectives of the proposed program are to further the study of various mathematical models in physical and biological sciences via mathematical methods. The expected outcome will not only solve some open and long-standing problems in various applied dynamical systems, but also help to establish a fundamental theory about asymptotic analysis of delay differential equations. Specifically, we aim to to advance the field in the following ways: (a) develop a completely new method to construct traveling wave fronts; (b) theoretically establish the stability of traveling wavefronts in periodic media; (c) solve or partially solve some open problems in this field; (d) theoretically give sufficient conditions under which pattern formations are certain to happen. These mathematical analyses will have practical applications: (a) help us better understand transmission patterns in infectious diseases (e.g. rabies); (b) better identify the impact of the period of incubation in the spreading of diseases; (c) this result can be used to assess the effectiveness of different methods used during past years for controlling the infectious disease.