Bernard Hugueny, Howard Cornell, and Susan Harrison. 2007. Metacommunity models predict the local–regional species richness relationship in a natural system. Ecology 88:1696–1706.

Appendix A. Identical parameterization of Levins-like and mainland–island models.

We use metacommunity models that are special cases of Hastings’ (1987) models, in that we assume a linear relationship between per species extinction rate and LSR (Eq. 1). This allows us to rearrange Hastings’ formulae so that they are expressed using the same parameters for both Levins-like and mainland-island models. First, note that for i > 0 the main components of Eqs. 7 and 9 can be rearranged as follows

.

(A.1)

Then, from Eqs. 13 and 14 we obtain

and given that M equals m_MI or y(n) m_LL for the mainland-island or the Levins-like models respectively we obtain

(A.4)

for the Levins-like model and

(A.5)

for the mainland–island model.

From Eq. 1 we have for j > 0

For computational purposes F₀ is set to 1. Substituting K for M and F_j for e_j, Eqs. 7, 9, and 11 become respectively

,	(A.7)
,	(A.8)
and .	(A.9)

Equations A.7–A.9 are now expressed as functions of y(1), which is part of the term K (Eq. A.4 or A.5), and c/e₁, which is part of the term F_j (Eq. A.6) instead of m_MI or m_LL, c and e₁. As before, in the Levins-like model E_LSR(n) is needed to calculate M and thus appears in both sides of Eq. A.9 (as it does in Eq. 11).