On equivalent forms of Whittaker's similarity index

For two N-species assemblages A, B with specific proportionate abundances of the ith species a_i, b_l respectively, we consider the equality

$\text{∑}\text{t=1}\text{N}\text{c}{}_{\mathrm{i}}\text{= 1?}\text{1}\text{2}\text{∑}\text{t=1}\text{N}\text{|a}{}_{\mathrm{i}}\text{?b}{}_{\mathrm{i}}\text{|, c}{}_{\mathrm{i}}\text{=}\text{a}{}_{\mathrm{i}}\text{a}{}_{\mathrm{i}}\text{? b}{}_{\mathrm{i}}\text{b}{}_{\mathrm{i}}\text{a}{}_{\mathrm{i}}\text{> b}{}_{\mathrm{i}}\text{, 0?a,b,c?1}$

. The left-hand term is known as Sanders' minimum faunal abundance value, while the right side is referred to as Whittaker's similarity index. Both measures are commonly used in community studies. Equality between these two measures obtains only when proportionate abundances are utilized. We develop equivalent formulation which is valid for absolute abundance data, reduces to the Sanders-Whittaker equality when proportionate abundance data are employed, and is more sensitive to differences in species abundance distributions. Namely, we show that

$\text{2}\text{α+β}\text{∑}\text{t=1}\text{N}\text{c}{}_{\mathrm{i}}\text{= 1 ?}\text{1}\text{α+β}\text{∑}\text{t=1}\text{N}\text{|a}{}_{\mathrm{i}}\text{?b}{}_{\mathrm{i}}\text{|}$

, where

$\text{α =}\text{∑}\text{t=1}\text{N}\text{a}{}_{\mathrm{i}}\text{, β =}\text{∑}\text{t=1}\text{N}\text{b}{}_{\mathrm{i}}$

, and the a's, b's c's are as defined above.