  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.
