Fig. 3. Versatility of the NB distribution.

The NB distribution is a two-parameter distribution that shows self-similarity and can display both monotonic LS-like behavior (in the limit r→0, the NB tends to the LS distribution) and a unimodal shape, as a function of the scaling parameter ξ. The red curve represents the analytical threshold separating these two cases. The SAD, especially at large scales or with increasing sampling effort (27), often displays an interior mode that cannot be captured by the LS distribution but can be described by the NB. The NB distribution naturally arises as the steady-state SAD of an ecosystem undergoing generalized dynamics of birth, death, speciation, and migration processes (see Materials and Methods). Finally, any discrete probability distribution, such as the SAD, can be approximated to any degree of accuracy by a suitable linear combination of NBs that retains the self-similarity feature (see Materials and Methods). An example is shown of how the parameter ξ of the NB increases as the area of the forest doubles. Starting from ξ = 0.36, as the area doubles, the ξ value moves upward to the value corresponding to the successive (dashed) horizontal line in the upward direction.