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) |