Last modified: 2014-03-28

#### Abstract

The lecture reports progress in developing the mathematical theory of ecological niche. Niche theory suggests that different species occupy different “niches” to avoid competition with each others. The original version of it was based on biological intuition mixed with elementary analysis of the Lotka-Volterra model. Earlier we developed a precise formulation which was based on the observation that the Lotka-Volterra model can be seen as a linearization of the general dynamics: Any model is LV-ish for small parameter-perturbations. We showed that coexisting species must differ both in environmental impact and in environmental sensitivity; otherwise the community is prohibitively unrobust against perturbations.

This understanding was later generalized for structured populations, where the additional problem arose from the perturbation of the population structure. Such generalization is useful for understanding ontogenetic niche shift as well as spatial niche structure. We also developed generalizations for different kinds of environmental fluctuations. In such cases temporal niche segregation could lead to fluctuation-maintained coexistence. Here the mathematical issue is the handling of different time-scales. This formalism reduces to Chesson's one in the proper approximation.

Lack of a widely accepted mathematical theory of niche causes serious confusions even in the theoretical literature. There are model-based suggestions for coexistence without niche-segregation, “self organized similarity”, or “emergent neutrality”. These controversies about validity of niche theory completely destroyed the intuitive understanding that already existed in the '60s. We showed that the models in question are either structurally unstable, or based on unmodeled niche segregation.

We conclude that over-generalization of specific models is dangerous; one has to attempt to develop robust mathematical theories. In particular, our mathematical niche theory precisely establishes the original niche expectation in a probabilistic sense: coexistence without significant niche segregation is restricted to narrow parameter range, therefore unlikely to occur in nature.