Model-based ordination for species with unequal niche widths

It is common practice for ecologists to examine species niches in the study of community composition. The response curve of a species in the fundamental niche is usually assumed to be quadratic. The center of a quadratic curve represents a species' optimal environmental conditions, and the width its ability to tolerate deviations from the optimum.

Most multivariate methods assume species respond linearly to the environment of the niche, or with a quadratic curve that is of equal width for all species. However, it is widely understood that some species are generalists that can tolerate deviations from their optimal environment better than others. Rare species often tolerate a smaller range of environments than more common species, corresponding to a narrow niche. 

We propose a new method, for ordination and fitting Joint Species Distribution Models, based on Generalized Linear Mixed-Effects Models, which relaxes the assumptions of equal tolerances.

By explicitly estimating species maxima, and species optima, and tolerances per ecological gradient, we can better understand how species relate to each other.

Methods
This elevation data was provided by the Spatial Ecology Group at the University of Lausanne, specifically by Manuela D'Amen, and originally comes from the Swiss Federal Office of Topography.