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

研究者进行的研究旨在获得一些具有延迟或扩散的生态模型解的整体定性行为的理解。它们分为三部分：(1)有限维常微分方程系统；(2)时滞微分方程系统；(3)具有时滞的扩散系统。研究的主要主题是寻找这些被考虑系统的一致持久性的各种标准的基本问题，稳定状态的全局稳定性，以及周期解和空间模式的存在性。特别是，研究者将研究在异质环境中种群在斑块之间分散的一些数学模型，其中将考虑各种相互作用。对于两个以上种群的系统，主要目标之一是找到所有种群长期共存的具体条件。对于两种相互作用模型，稳态的全局稳定性、周期解的存在性或空间格局将是主要关注的问题。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。地球作为人类和其他物种的栖息地已经变得太小，无法支持对其资源的过度开发。在异构环境中，损伤是不可避免的，通常伴随着时间延迟而发生；没有任何实验可以完全模拟这些效应。因此，为了更好地理解和控制这些过程，对它们进行数学研究是必要和重要的。不幸的是，目前的数学技术远远不足以处理这些众多的实际问题。提出的研究将导致对时间延迟和扩散效应的动态相互作用的更好理解。特别是，最终的结果对生态学家、工程师、系统科学家和其他计划更好地管理地球和生命的人等广泛的人都很有用。

The investigator undertakes studies aimed at gaining an understanding of the global qualitative behavior of solutions of some ecological models with delays or diffusions. These fall in three parts: (1) finite-dimensional systems of ordinary differential equations, (2) delay differential equations, and (3) diffusive systems with time delays. The principal themes of the research are the fundamental problems of finding various criteria for the uniform persistence of these considered systems, the global stability of steady states, and the existence of periodic solutions and spatial patterns. In particular, the investigator will study some mathematical models of populations dispersing among patches in heterogeneous environments, where various kind of interactions will be considered. For systems with more than two populations, one of the main objectives is to find concrete conditions under which all populations will coexist in the long term. For two-species interaction models, global stability of steady states, existence of periodic solutions or spatial patterns will be main concerns. Time delays and space variations constitute integral components of most ecological, biological, physiological, chemical, physical, economic and atmospheric processes. The planet as a habitat of mankind and other species has become too small to support an extravagant exploitation of its resources. The damages are inevitable and usually occur with time delays in heterogeneous environments; no experiments are possible to fully simulate these effects. In order to better understand and control such processes, it is thus necessary and important to study them mathematically. Unfortunately, current mathematical techniques are far from adequate to handle these numerous real problems. The proposed research will lead to an improved understanding of the dynamical interplay of the effects of time delays and diffusions. In particular, the final results can be useful to a wide spectrum of people like ecologists, engineers, system scientists and others who plan for a better management of the planet and life.