Mathematical Sciences: Global Qualitative Analysis of Ecological Models with Delays and Diffusions

小行星9306239 研究了一类具有时滞或扩散的生态模型解的全局定性行为。 这项工作的主要主题是找到这些考虑系统的一致持久性的各种标准，稳定状态的全局稳定性，周期解和空间模式的存在性，以及混沌行为的存在性的基本问题。 这些主题确实涵盖了非线性系统的所谓全局定性分析的大部分方面。 特别地，研究者研究了在异质环境中种群在斑块间分散的数学模型，其中考虑了各种相互作用。除了进行传统的局部稳定性或分岔分析，他认为全球稳定性，分岔和混沌方面的非线性系统。 时间延迟和空间变化是大多数生态、生物、生理、化学、物理、经济和大气过程的组成部分。 几乎所有这些现实生活中的过程都是非线性的。 它们通常过于复杂，无法在实验室中模拟，或者仅从实验工作中难以理解。 为了更好地管理或控制这些过程，有必要对它们进行数学研究。 以前解决这些应用数学问题的大多数方法都是局部的（假设系统在一个小区域内表现得好像是线性的而不是非线性的），因此不能描绘出真实的系统的全貌。 本项目试图通过关注所考虑系统的动力学的非线性效应来获得真实的画面的全貌。 这项工作提供了一个更好的理解的动态相互作用的影响的时间延迟和扩散。 特别是，最终的结果可以是有用的生态学家，工程师和系统科学家。 ***

9306239 Kuang The investigator studies the global qualitative behavior of solutions of some ecological models with delays or diffusions. The principal themes of this work are the fundamental problems of finding various criteria for the uniform persistence of these considered systems, the global stability of steady states, the existence of periodic solutions and spatial patterns, and the existence of chaotic behaviors. These topics indeed cover most aspects of the so-called global qualitative analysis of nonlinear systems. In particular, the investigator studies some mathematical models of populations dispersing among patches in heterogeneous environments, where various kind of interactions are considered. In addition to performing the traditional local stability or bifurcation analysis, he considers global stability, bifurcation, and chaos aspects of the nonlinear systems. Time delays and space variations constitute integral components of most ecological, biological, physiological, chemical, physical, economic and atmospheric processes. Almost all these real-life processes are nonlinear in nature. They are normally too complicated to simulate in a lab, or too difficult to understand from experimental work alone. In order to better manage or control such processes, it is necessary to study them mathematically. Most of the previous approaches to these applied mathematical problems have been local (pretending the system behaves in a small region as if it were linear rather than nonlinear) and thus do not paint the whole picture of the real systems. This project tries to gain a full view of the real picture by focusing on the nonlinear effects of the dynamics of the considered systems. This work offers a better understanding of the dynamic interplay of the effects of time delays and diffusions. In particular, the final results can be useful to ecologists, engineers, and system scientists. ***