The goal of this project is to develop mathematical theory regarding biodiversity in the context of resource competition. Recent numerical work suggests that competition for multiple resources can generate periodic oscillations or even chaotic oscillations, and a large number of species can coexist on very few limiting resources. However, both mathematicians and ecologists still lack analytic understanding of mechanisms that generate diversity in ecological communities. The investigator studies several resource competition models, including standard competition models with multiple resources and competition models within food webs; he gives rigorous analysis for the models and provides insight into diversity of competing species. Theory of differential equations and theory of dynamical systems are used to determine the long-term behavior of solutions of the models in the form of attracting steady states, limiting cycles or strange attractors. Important issues are what causes competing species to coexist, and locating parameter ranges in which coexistence occurs. The question of the maintenance of species diversity in natural systems has intrigued mathematicians and ecologists for a very long time. This work provides a general understanding of factors affecting abundance and biodiversity. The results have important implications for determining the patterns of species composition in aquatic and terrestrial systems, and for managing natural resources such as forests and lakes.
