The exponential-to-e distribution
This quite different but equally simple distribution can be motivated by a very similar population dynamics model (Figs. 1E, F). Here, the turnover of each species is fixed and intrinsic, not changing through time. It is based on a random number ε with an exponential distribution. In the illustrated trial, the distribution’s rate λ is 0.5. Variable ε is converted to a proportion π using the ratio 1/(ε + 1), which scales π between 0 and 1. The binomial death probability is π2and the geometric success probability is π.
A similar set of manipulations can be used to derive the PMF and richness estimator of this exponential-to-e(expe ) distribution. The x values on a continuous scale are assumed to come not from an odds ratio but from an exponential random variate with a rate of λ that is raised to the power of e , keeping in mind that the negative logarithm of a uniform variate is exponentially distributed:
x = [–log(U )/λ]e (11)
The PMF follows quickly:
U = exp(–λ x 1/e ) (12)
P(X = x ) = exp(–λxi 1/e ) – exp(–λxi + 11/e ) (13)
which does not reduce algebraically into anything more simple. However, the size of the zero class, size of the non-zero class, and richness estimator are all trivial:
P(X = 0) = exp(–λ 01/e ) – exp(–λ 11/e ) = 1 – exp(–λ) (14)
P(X = x , X > 0) = exp(–λ) (15)
R = S /exp(–λ) (16)
The expe distribution is deeply related to the Weibull distribution, which has been of interest to ecologists seeking to model SADs (Ulrich et al., 2018). Specifically, the e power term in eqn. 11 is a direct function of the k variable in the Weibull probability density function. As a result, eqn. 13 is a special case of the discrete Weibull distribution (Nakagawa & Osaki, 1975). The fact that eqn. 13 exactly follows from a basic population dynamics model (Figs. 1E, F) justifies the choice of k used here.
As noted, the odds and expe population models differ only in how year-to-year variation works. This fact raises the question of whether the two might be unified by adding a single parameter to express the rate at which relative abundances are scrambled through time: rapidly in an odds world, and not at all in an expe world. The answer is no. In a potential unified model, a scaling variable κ could be introduced. Variable µ in the odds eqn. 3 is then multiplied by κ, and U is raised to the power 1/κ. As κ increases, simple calculations show that eqn. 3 rapidly converges on a scaled exponential function:
κ µ/U 1/κ – κ µ ~ –ln(U )/µ (17)
The right-hand side is just a root of the function defining expe . Therefore, bridging the gap between the distributions would require defining a model with at least three parameters: the equivalents of κ and µ plus a power term.