Diffusion, Directed Movement, Spatial and Temporal Heterogeneity in Population Dynamics

进化论和生态学中的“竞争-排斥”原理是由达尔文首先提出的。现代，理解“生物多样性”已成为植物生态学的中心主题，人们观察到“空间异质性”在生物多样性中发挥着重要作用。另一方面，扩散已被广泛和成功地用于模拟自然和科学中的各种现象。在这个项目中，首席研究员和他的生态学和生物学合作者将继续探索在异质环境中扩散的影响，从而重新审视人口动态中“承载能力”的基本概念及其与“内在增长率”的关系。数学理论和严格的生物实验，涉及酵母将被使用。本项目的主题是了解扩散（物种的扩散）与环境的空间和时间变化之间相互作用的后果。这种理解的一个重要因素是“承载能力”和“内在增长率”这两个基本概念之间的复杂关系。这些将被研究-首先通过生物实验（这是由数学理论启发的），然后通过新的数学模型来描述这些实验。这两个量之间的关系的凸性或凸性将在种群动力学中起重要作用。从数学上讲，生物学实验表明，用一个包含管理“可再生”资源的额外方程的系统来取代单一物种种群的经典（单一）逻辑斯谛方程。这似乎更接近现实，并将著名的两个竞争物种的Lotka-Volterra系统改变为一个更具挑战性的系统，一个新的可再生资源3方程系统。在这个项目中，我们将深入研究这些新系统。

"Competition-exclusion" principle in evolution and ecology was first proposed by Darwin. In modern era, understanding "biodiversity" has become a central theme in plant ecology, and it has been observed that "spatial heterogeneity" plays an important role in biodiversity. On the other hand, diffusion has been used extensively and successfully in modeling various phenomena in nature and science. In this project, the Principal Investigator and his collaborators in ecology and biology, will continue to explore the effect of dispersal in heterogeneous environment which has led to a re-examination of the fundamental concept of "carrying capacity" and its relation to "intrinsic growth rates" in population dynamics. Both mathematical theories and rigorous biological experiments involving yeast will be used. Other related issues, such as temporal variations, will also be investigated.In this project, the main theme is to understand the consequences of interactions between diffusion (dispersal of species) and spatial, as well as temporal, variations in environment. An important element in this understanding is the intricate relation between the fundamental concepts of "carrying capacity" and "intrinsic growth rates". Those will be studied - first by biological experiments (which have been inspired by mathematical theories) and then by new mathematical models created to describe those experiments. The concavity or convexity of the relation between the two quantities would play an important role in population dynamics. Mathematically, what biological experiments suggest is to replace the classical (single) logistic equation for a single species population by a system containing an extra equation governing the "renewable" resources. This seems to be much closer to reality, and will change the well-known Lotka-Volterra system for two competing species to a much more challenging one, a new system of 3 equations with renewable resources. In this project, we will study these new systems thoroughly.