The scaled odds distribution
The second model assumes high variance across species and through time in counts (Figs. 1C, D). In this case, not all individuals are replaced at each time step, so there is a non-trivial death process. Both births and deaths depend on a random uniform variate τ that is reset at each time step. The number of deaths per time step is a random binomial draw from the n individuals of each species based on its own τ probability of death. The birth process is independent of population size, and random draws for each species are geometrically distributed with a success probability of exactly τ. Thus, birth and death rates are positively correlated, and an equilibrium is maintained because birth rates are steady and not per-capita.
The key biological assumption is that replacement probabilities at a given time are fixed for each given species but highly variable across species. Because the probabilities change completely at each time step, species have no innate properties and the model is neutral.
The PMF is derived as follows. First, relative abundances on a continuous scale are defined as tracking an odds distribution. The odds are multiplied by a constant µ and specifically taken to predict abundance values x on a continuous scale:
x = µ (1 – U )/U (2)
U is solved for by rearranging the odds:
x = µ/U – µ (3)
U = µ/(x + µ) (4)
Note that µ/U – µ still an odds ratio, just a scaled one.
Equation 4 defines 1 minus the cumulative distribution function (CDF) of the scaled odds distribution. Any CDF starts at zero and reaches 1 asymptotically, and eqn. 4 does in fact start with a value of 1 wherex is 0 and decline to zero where x reaches infinity. A monotonically trending function that has limits of 1 and 0 also has an integral of 1, as in this case. Any PMF can be derived from a monotonic CDF by taking first differences and rounding down to integer values. This is the exact procedure used to derive the PMF of the geometric series from the CDF of the exponential distribution (see Cohen, 1968 for an early example of using the exponential in the context of constructing SADs). Thus, the scaled odds PMF is:
P(X = x ) = µ/(xi + µ) – µ/(xi + 1 + µ) (5)
which can be rearranged algebraically as:
P(X = x ) = µ/[(xi + µ) (xi + 1 + µ)] (6)
The PMF’s value for the zero class is just:
P(X = 0) = µ/[µ (1 + µ)] = 1/(1 + µ) (7)
The sum of the values for the non-zero classes is therefore:
1 – 1/(1 + µ) = µ/(1 + µ) (8)
The PMF discounting the zero class is then easily defined by dividing eqn. 6 by eqn. 8:
P(X = x , X > 0) = (1 + µ)/[(xi + µ) (xi+ 1 + µ)] (9)
A valid estimator of the total species richness of the communityR is then trivially derived by dividing observed richnessS by eqn. 8:
R = S (1 + µ)/µ (10)