The principles of species coexistence have mostly been studied in uniform environments. However, natural environments are usually characterized by gradients and non-homogeneous distribution of resources. To date, we are largely unaware what conditions can provide maximum of the species richness and stability of the community, optimal stoichiometry and efficient use of resources.
The classical approach based on differential equation is a very convenient tool for the analysis of uniform systems. However, its straight forward application for competition in spatial systems encounters many mathematical difficulties. In order to investigate competition in spatially extended systems we have developed a graphical approach based on the analysis of invasion thresholds, - curves in space of species parameters which separate species that can survive in given conditions, from the species that cannot. Applying this analysis we have already discovered new fundamental results, which show a principle difference between competition in uniform and spatial system. See Ecology Letters (2011) 14: 220–228 and Alexey Ryabov's home page for further information.
Currently we are developing this approach further. Motivated students are welcome to proceed this research.