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

进化论和生态学中的“竞争-排斥”原则最早是由达尔文提出的。近年来，认识“生物多样性”已成为植物生态学研究的中心主题，“空间异质性”在植物多样性中发挥着重要作用。另一方面，扩散已被广泛而成功地用于模拟自然界和科学中的各种现象。在这个项目中，首席研究员和他在生态学和生物学方面的合作者将继续探索在异质环境中分散的影响，这导致了对种群动态中“承载能力”的基本概念及其与“内在增长率”的关系的重新审视。将使用数学理论和涉及酵母的严格的生物学实验。其他有关问题，如时间变化，也将进行调查。在这个项目中，主要的主题是了解扩散（物种的分散）和空间以及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。这些问题将首先通过生物实验（受到数学理论的启发）进行研究，然后通过为描述这些实验而创建的新数学模型进行研究。这两个量之间的关系的凹凸性在种群动态中起着重要作用。从数学上讲，生物学实验的建议是用一个包含控制“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）逻辑方程。这似乎更接近现实，并将把两个竞争物种的著名的Lotka-Volterra系统改变为一个更具挑战性的系统，一个具有可再生资源的3个方程的新系统。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.