Wow. The fact that there is an objective answer that is independent of any perspective on the importance of rare species is a rare gift, at least for this part of the problem.
Some questions and thoughts.
It seems that the result could vary based on how you construct the similarity matrix Z, e.g. is it purely taxonomic? or does it try to account for the ecological roles that a species is playing in the community, etc.
A seeming limitation is that the optimization works only for a fixed set of n species.
While it is useful for managing existing communities, it means that there is still a question of whether larger n is strictly better, and leaves open questions of how to deal with transient or migratory members (if the community is spatially bound).
The answer I think, is that it depends on how the similarity matrix is constructed. If every species is fully dissimilar then increasing n is always a good thing. If you use niche space to construct it and new species do not some add or enter new niches so they overlap with others, then they will be close to another species in the matrix and increasing n will not have much impact. On the other hand if you use a purely taxonomic approach then you wind up balancing the number of birds and mammals regardless of niche.
It is not clear to me whether it is possible to construct a similarity matrix that can account for the interaction between n, the carrying capacity of the ecosystem, and the number of available niches (or the ability of species to create new niches). By analogy if you have a stream (sunlight) powering water wheels, how many wheels and how many levels of gears (layers in the ecosystem) can be added, created, and/or sustained? At what point does adding an additional species mean that either two species are forced to be close together in the similarity matrix or both their populations must shrink in size because they must compete for the same energy sources?
Does the model sometimes produce impractical results, e.g. that it is good to have a single member of a sexually reproducing species (this is probably an orthogonal concern and you would want to scale to real population sizes such that the minimum corresponded to the smallest viable a self sustaining population)?
Is there evidence that maximizing diversity using this measure actually produces more robust and stable ecologies?
Some questions and thoughts.
It seems that the result could vary based on how you construct the similarity matrix Z, e.g. is it purely taxonomic? or does it try to account for the ecological roles that a species is playing in the community, etc.
A seeming limitation is that the optimization works only for a fixed set of n species. While it is useful for managing existing communities, it means that there is still a question of whether larger n is strictly better, and leaves open questions of how to deal with transient or migratory members (if the community is spatially bound).
The answer I think, is that it depends on how the similarity matrix is constructed. If every species is fully dissimilar then increasing n is always a good thing. If you use niche space to construct it and new species do not some add or enter new niches so they overlap with others, then they will be close to another species in the matrix and increasing n will not have much impact. On the other hand if you use a purely taxonomic approach then you wind up balancing the number of birds and mammals regardless of niche.
It is not clear to me whether it is possible to construct a similarity matrix that can account for the interaction between n, the carrying capacity of the ecosystem, and the number of available niches (or the ability of species to create new niches). By analogy if you have a stream (sunlight) powering water wheels, how many wheels and how many levels of gears (layers in the ecosystem) can be added, created, and/or sustained? At what point does adding an additional species mean that either two species are forced to be close together in the similarity matrix or both their populations must shrink in size because they must compete for the same energy sources?
Does the model sometimes produce impractical results, e.g. that it is good to have a single member of a sexually reproducing species (this is probably an orthogonal concern and you would want to scale to real population sizes such that the minimum corresponded to the smallest viable a self sustaining population)?
Is there evidence that maximizing diversity using this measure actually produces more robust and stable ecologies?