Combining richness and abundance into a single diversity index using matrix analogues of Shannon's and Simpson's indices

Shannon's and Simpson's indices have been the most widely accepted measures of ecological diversity for the past fifty years, even though neither statistic accounts for species abundances across geographic locales ("patches"). An abundant species that is endemic to a single patch can be as much of a conservation concern as a rare cosmopolitan species. I extend Shannon's and Simpson's indices to simultaneously account for species richness and relative abundances - i.e. extend them to multispecies metacommunities - by making the inputs to each index a matrix, rather than a vector. The Shannon's index analogue of diversity is mutual entropy of species and patches divided by marginal entropy of the individual geographic patches. The Simpson's index analogue of diversity is a modification of mutual entropy, with the logarithm moved to the outside of the summation, divided by Simpson's index of the patches. Both indices are normalized for number of patches, with the result being inversely proportional to biodiversity. These methods can be extended to account for time-series of such matrices and average age-classes of each species within each patch, as well as provide a measure of spatial coherence of communities.