appendixs1.algebraiccomparisonsofalternativespecies–.docVIP

Appendix S1. Algebraic comparisons of alternative species–area functions. In the following, a, b, c, and d are fitted parameters, with a an asymptote and c the coefficient of area. The area of isolate i is xi, for i = 1, … , n isolates; Si is the number of species in isolate i; and P = Smax is the number of species in the source pool. Natural logarithms are used throughout. 1. Comparison of the Morgan–Mercer–Flodin (MMF), Lomolino (2000), Archibald (1949b) logistic, and MONOD functions. The Morgan–Mercer–Flodin function (Tj?rve, 2003) is Si = axic / (b + xic), = axic / [ xic (b/xic + 1) ] = a / (1 + bxi–c) (xic ≠ 0, since xi ≠ 0). The Lomolino function (Lomolino, 2000) is Si = = = replacing b with b = exp(logb), and assuming b, c ≠ 0 Si = P / [ 1 + exp(logb logc) / exp(logb logxi) ] = P / (1 + clogb/xilogb) Setting b’ = clogb and c’ = logb Si = P / (1 + b’xi–c’) Hence, the Lomolino (2000) function is equivalent to the MMF, with a = P. The logistic function (Archibald, 1949b) is log ( Si / ( P – Si ) ) = k (logxi – logx50) , k and x50 are fitted parameters, with x50 = the area at which half of the species in the source pool are present (i.e. the value of xi such that the estimated value of Si = P/2). i.e. log ( Si / ( P – Si ) ) = log (xi / x50)k i.e. Si / ( P – Si ) = (xi / x50)k i.e. ( P – Si ) / Si = (x50 / xi)k i.e. P / Si = (x50 / xi)k + 1 i.e. Si / P = 1 / [ 1 + (x50k / xi)k ] i.e. Si = P / [ 1 + x50k / xik ] Setting b = x50 k and c = k Si = P / (1 + bxi–c) Hence the logistic function is identical to Lomolino’s (2000) function and equivalent to the MMF, with a = P. The MONOD function (Clench, 1979) is Si = a (xi / (b + xi)) = axi / [ xi (b/xi + 1) ] = a / (1 + bxi–1 ) ) (since xi ≠ 0). Hence the MONOD function is a special case of the MMF with c = 1. 2. Comparison of the EVF, cumulative Weibull function with 3 parameters, negative exponential and Chapman–Richards functions. The cumulative Weibull function with 3 parameters (Weibull-3) is