Daniel J. McGlinn and Michael W. Palmer. 2009. Modeling the sampling effect in the species–time–area relationship. Ecology 90:836–846.

Appendix A. Derivation of the expected slope of the log–log species–area relationship, z_E, and the expected slope of the log–log species–time relationship, w_E.

Proof:

From Eq. 3 in the main text the expected number of species is equal to:

.

(A.1)

The Arrhenius definition of z is the slope of the SAR in log-log space; therefore, we can define z_E for the sampling model as the partial derivative of the natural logarithm (ln) of the expected richness as a function of the ln of area. For notational simplicity we will define this as:

.

(A.2)

Because Eq. A.1 is given with respect to area and not ln(area) and defined for S_E and not ln(S_E) we must use the chain rule to see that Eq. A.2 is actually:

.

(A.3)

And using the rules of differentiation for exponential functions and the chain rule once more we find that:

.

(A.4)

Combining this equation with Eq. A.3 we can see that the equation for z_E is:

.

(A.5)

A similar process can be used to find w_E.

.

(A.6)

And using the rules of differentiation for exponential functions and the chain rule we find that:

.

(A.7)

Combining this equation with Eq. A.6 we can see that the equation for w_E is:

.

(A.8)