Review of METE.
At the core of METE is the time-independent structure functionR (n , ε |S , N , E ).R is a joint conditional distribution over abundance, n , of a species, and metabolic rate, ε of an individual;R ·dε is the probability that a species picked at random from the species pool has abundance n , and an individual picked at random from the species with abundance n has a metabolic rate in the interval (ε , ε + dε ).
We use a discrete notation, with summation, not integral, signs for all variables n , ε , S , N , E , and adopt the units convention that the smallest value for the metabolic rate isε = 1, the metabolic rate of the smallest observed organism in the community (e.g., a germinant tree in a forest, or a tree with 10 mm dbh if that is the smallest tree censused). The constraints on the static structure function are:
\(\frac{N}{S}\ =\ \sum_{n,\varepsilon}nR\left(n,\varepsilon\middle|S,N,E\right)\)(1)
and
\(\frac{E}{S}=\ \sum_{n,\varepsilon}\text{nε}R\left(n,\varepsilon\middle|S,N,E\right).\)(2)
The MaxEnt solution (Harte et al., 2008) for R (n ,ε |S ,N ,E ) subject these constraints is:
\(R\left(n,\varepsilon\middle|S,N,E\right)=\frac{e^{-\lambda_{1}n}e^{{-\lambda}_{2}\text{nε}}}{Z}\)(3)
where Z -1 is a normalization constant (see below). The λ ’s are Lagrange multipliers that are determined from the values of S , N , E . The quantity β =λ 1 + λ 2 is determined from
\(\frac{S}{N}\sum_{n=1}^{N}{x^{n}=}\)\(\sum_{n=1}^{N}\frac{x^{n}}{n}\) (4)
where \(x=e^{-\beta}\text{.\ }\) The Lagrange multiplier,\(\lambda_{2},\) is given by
\(\lambda_{2}=\frac{S}{E-N}.\) (5)
In many ecosystems, the state variables obey the inequalities S << N<< E , in which case\(\beta\ll 1\) and the solution to Eq. 4 is, to a good approximation,
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\frac{S}{N}\)\(\approx\beta\ln\left(\frac{1}{1-e^{-\beta}}\right)\approx\)\(\beta\ln\left(\frac{1}{\beta}\right)\) (6)
Moreover, in that approximation,
\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Z\approx\frac{ln(\frac{1}{\beta)}}{\lambda_{2}}\), (7)
In all that follows, we assume the validity of these approximations, and use =, not \(\approx,\) signs in the equations.
We note for later comparison with DynaMETE that the species abundance distribution (SAD), \(\phi\left(n\right),\) obtained by summingR over metabolic rate, \(\varepsilon,\) is the log-series distribution:
\(\phi\left(n\right)=\frac{e^{-\beta n}}{n\bullet ln(\frac{1}{\beta})}\)(8)
and the distribution of metabolic rates over all individuals (the MRDI), obtained by summing nR S/N over abundance, n , is:
\(\Psi\left(\varepsilon\right)=\frac{\beta\lambda_{2}e^{-\gamma(\varepsilon)}}{{(1-e^{-\gamma(\varepsilon)})}^{2}}\)(9)
where
\(\gamma\left(\varepsilon\right)=\ \lambda_{1}+\lambda_{2}\varepsilon\). (10)
METE also contains a spatially-explicit component but to focus on the essential concepts underlying DynaMETE, we ignore the spatial dimension here.
For further description of MaxEnt and METE, and software for deriving predictions, see Harte 2011; Bremmer and Newman 2019; Kitzes and Wilbur 2016; Rominger and Merow 2016.