# Invariance in ecological pattern

## TL;DR

This paper introduces an invariance-based framework to explain ecological patterns, showing how simple distributions like the log series and lognormal emerge from fundamental symmetry principles in ecological processes.

## Contribution

It applies the concept of invariance from physics to ecology, providing a unified, fundamental explanation for common ecological abundance patterns.

## Key findings

- Log series pattern arises from invariance to additive or multiplicative abundance transformations.
- Lognormal pattern results from rotational invariance in species growth processes.
- Invariance offers a simpler derivation of maximum entropy and neutral theory results.

## Abstract

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, we show that the log series pattern dominates when the consequences of processes acting on abundance are invariant to the addition or multiplication of abundance by a constant. The lognormal pattern dominates when the processes acting on net species growth rate obey rotational invariance (symmetry) with respect to the summing up of the individual component processes. 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 are easy to understand because they uniquely describe the basic underlying components that shape pattern.