We perform a full similarity analysis of an idealized ecosystem using Buckinghams $Pi$ 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.
Similarity analysis is used to identify the control parameter $R_A$ for the subset of avalanching systems that can exhibit Self- Organized Criticality (SOC). This parameter expresses the ratio of driving to dissipation. The transition to SOC, when th
e number of excited degrees of freedom is maximal, is found to occur when $R_A to 0$. This is in the opposite sense to (Kolmogorov) turbulence, thus identifying a deep distinction between turbulence and SOC and suggesting an observable property that could distinguish them. A corollary of this similarity analysis is that SOC phenomenology, that is, power law scaling of avalanches, can persist for finite $R_A$, with the same $R_A to 0$ exponent, if the system supports a sufficiently large range of lengthscales; necessary for SOC to be a candidate for physical ($R_A$ finite) systems.
From the starting point of the well known Reynolds number of fluid turbulence we propose a control parameter $R$ for a wider class of systems including avalanche models that show Self Organized Criticality (SOC) and ecosystems. $R$ is related to the
driving and dissipation rates and from similarity analysis we obtain a relationship $Rsim N^{beta_N}$ where $N$ is the number of degrees of freedom. The value of the exponent $beta_N$ is determined by detailed phenomenology but its sign follows from our similarity analysis. For SOC, $R=h/epsilon$ and we show that $beta_N<0$ hence we show independent of the details that the transition to SOC is when $R to 0$, in contrast to fluid turbulence, formalizing the relationship between turbulence (since $beta_N >0$, $R to infty$) and SOC ($R=h/epsilonto 0$). A corollary is that SOC phenomenology, that is, power law scaling of avalanches, can persist for finite $R$ with unchanged exponent if the system supports a sufficiently large range of lengthscales; necessary for SOC to be a candidate for physical systems. We propose a conceptual model ecosystem where $R$ is an observable parameter which depends on the rate of throughput of biomass or energy; we show this has $beta_N>0$, so that increasing $R$ increases the abundance of species, pointing to a critical value for species explosion.
