  We perform a full similarity analysis of an idealized ecosystem using
Buckingham's $\Pi$ theorem to obtain dimensionless similarity parameters given
that some (non- unique) method exists that can differentiate different
functional groups of individuals within an ecosystem. We then obtain the
relationship between the similarity parameters under the assumptions of (i)
that the ecosystem is in a dynamically balanced steady state and (ii) that
these functional groups are connected to each other by the flow of resource.
The expression that we obtain relates the level of complexity that the
ecosystem can support to intrinsic macroscopic variables such as density,
diversity and characteristic length scales for foraging or dispersal, and
extrinsic macroscopic variables such as habitat size and the rate of supply of
resource. This expression relates these macroscopic variables to each other,
generating commonly observed macroecological patterns; these broad trends
simply reflect the similarity property of ecosystems. We thus find that details
of the ecosystem function are not required to obtain these broad
macroecological patterns this may explain why they are ubiquitous. Departures
from our relationship may indicate that the ecosystem is in a state of rapid
change, i.e., abundance or diversity explosion or collapse. Our result provides
normalised variables that can be used to isolate the trend in one ecosystem
variable from another, providing a new method for isolating macroecological
patterns in data. A dimensionless control parameter for ecosystem complexity
emerges from our analysis and this will be a control parameter in dynamical
models for ecosystems based on energy flow and conservation and will order the
emergent behaviour of these models.
