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

研究者进行研究，旨在获得 了解解的全局定性行为 一些具有时滞或扩散的生态模型。 这些落在 三部分：（1）有限维常微分方程组 微分方程，（2）延迟微分方程，（3） 时滞扩散系统 会议的主要主题 研究是找到各种标准的基本问题 对于这些考虑的系统的一致持久性， 稳定状态的全局稳定性和周期解的存在性 解决方案和空间模式。 特别是，调查员 我将研究一些人口扩散的数学模型 在异构环境中的补丁之间， 互动将被考虑。对于系统， 两个群体，主要目标之一是找到具体的 所有种群长期共存的条件 term. 对于两种群相互作用模型， 稳定状态，周期解的存在性或空间 模式将是主要关注点。 时间延迟和空间 变化构成了大多数生态， 生物的、生理的、化学的、物理的、经济的和 大气过程。 地球作为人类的栖息地， 其他物种已经变得太小， 开发其资源。 损害是不可避免的， 通常在异构环境中发生时间延迟;不 实验可以完全模拟这些效应。 在 为了更好地理解和控制这些过程， 数学上研究它们是必要的和重要的。 不幸的是，目前的数学技术还远没有 足以处理这些众多的真实的问题。 拟议 研究将导致更好地了解动态 时间延迟和扩散效应的相互作用。 在 特别是，最终的结果可以是有用的，以广泛的 像生态学家、工程师、系统科学家和其他人 他们计划更好地管理地球和生命。

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.