# Frank, S. A. and Bascompte, J. 2019. Invariance in ecological pattern. bioRxiv doi:10.1101/673590.

The abundance of different species in a community often follows the log series distribution. Other ecological patterns also have simple forms. Why does the complexity and variability of ecological systems reduce to such simplicity? Common answers include maximum entropy, neutrality, and convergent outcome from different underlying biological processes. This article proposes a more general answer based on the concept of invariance, the property by which a pattern remains the same after transformation. Invariance has a long tradition in physics. For example, general relativity emphasizes the need for the equations describing the laws of physics to have the same form in all frames of reference. By bringing this unifying invariance approach into ecology, one can see that the log series pattern of species abundances dominates when the consequences of density dependent processes are invariant to addition or multiplication. The lognormal pattern dominates when the total of the density independent processes obeys rotational invariance (symmetry) with respect to the summing up of the individual processes acting on abundance. Recognizing how these invariances connect pattern to process leads to a synthesis of previous approaches. First, invariance provides a simpler and more fundamental maximum entropy derivation of the log series distribution. Second, invariance provides a simple derivation of the key result from neutral theory: the log series at the metacommunity scale and a clearer form of the skewed lognormal at the local community scale. The invariance expressions of those patterns are easy to understand because they uniquely describe the basic underlying components that shape pattern.