Diffusion in Heterogeneous Environments

在本提案中，倪伟明教授打算以经典的扩散Lotka-Volterra竞争系统为基本例子之一，系统地研究扩散与空间变化之间的相互作用，特别强调不同扩散速率的影响——包括一般阴影系统；也就是说，当扩散速率趋近于无穷大时的极限系统。此外，为了处理生态上更现实的模型，将调查由空间异质性或种间人口压力造成的定向运动和交叉扩散系统。空间变化反映了我们环境中资源的不均匀或不均匀分布。在本论文中，倪伟明教授将首先探讨不同的扩散速率和空间变异及其相互作用在两个或多个竞争物种种群动态中的作用。然后，定向运动，由于空间异质性和/或种间人口压力，将进行研究，以数学理解所涉及的基本机制，从而提高生态学家在生态模拟更现实的情况的能力。

In this proposal, using the classical diffusive Lotka-Volterra competition systems as one of the basic examples for illustrations, Professor Wei-Ming Ni intends to systematically study the interactions between diffusion and spatial variations, with particular emphasis on the effects of varying diffusion rates - including general shadow systems; that is, the limiting systems when one of the diffusion rates tends to infinity. Moreover, to handle ecologically more realistic models, directed movements as well as cross-diffusion systems, created by either spatial heterogeneity or inter-specific population pressures, will be investigated. Spatial variations reflect inhomogeneous or non-uniform distributions of resources in our environments. In this proposal, Professor Wei-Ming Ni will first explore the roles of different dispersal rates and spatial variations, as well as their interactions, in population dynamics of two or more competing species. Then, directed movements, due to spatial heterogeneity and/or inter-specific population pressures, will be studied in order to understand mathematically the basic mechanisms involved, and thereby enhance the ability of ecologists in modeling ecologically more realistic situations.