depend on the length of time sampling is done, the intensity of the sampling procedure,
and the size of the area sampled. See, for example, Boulinier et al. (1998), Gotelli and
Colwell (2001), Soberon and Llorente (1993). The statistical analysis of the interplay
between species “richness” and the sampling procedure calls for new mathematical
work. There are already some interesting mathematical approaches to the connection
between time spent sampling and number of species detected; for instance, as noted by
Soberon and Llorente (1993), there is evidence that as time spent collecting increases,
the number of species identified asymptotically approaches some limit. Soberon and
Llorente also investigate different assumptions about the probability of detecting a new
species in a given time period given the number of species that have been detected so
far. Which model of species richness is most suitable depends on the collecting
experience/procedure. For example, does the probability of finding a new species
decrease linearly or become more and more difficult (e.g., exponentially) over time?
Much more work along these lines is needed.
Notions of evenness in the biological literature are frequently based on ideas
going back in the economic literature to the early 1900s, in particular on the work of Gini
(1909, 1912) on measures of even income or wealth distribution and the work of Dalton
(1920) on measures of inequality. This work is of interest in its own right with regard to
sustainability, as we study ways to characterize the human sense of well-being and the
extent to which we have achieved a stable degree of social or economic equity (see e.g.,
Firebaugh 1999, 2003). Other measures of biodiversity or of evenness go back to work
in communication theory, in particular work of Shannon (1948) on the concept of
entropy in information theory, though they are predated in the biological literature by
work of Boltzmann (1872). Still others, such as the well-known Simpson index (Simpson
1949), measure the probability that any two individuals drawn at random from an infinite
population will belong to the same species. There are many indices that have been
proposed over the years. How does one choose among these? One idea is to write
down some general principles (axioms) that a measure of evenness should satisfy and
see which of the suggested indices satisfy them. This approach goes back to the work of
Dalton (1920) in the economics literature and is widely discussed in the literature of
biodiversity (Egghe and Rousseau 1990, 1991 or Rousseau 1992). Some axiom
systems lead to theorems that limit the possible measures of evenness very greatly, but
much more is needed to isolate the appropriate axioms for different contexts and to
derive the evenness measures that follow from them. Another approach is to derive a
partial order on vectors giving number of individuals of each kind of species, so-called
abundance vectors. Then we require that a measure of evenness reflect this partial
order (Nijssen et al. 1998, Patil and Taillie 1982, Rousseau et al. 1999). While the
literature has several widely-used ways to define such partial orders, approaches to
define them axiomatically or derive them from fundamental theories about species
distributions are lacking. Also, the problem gets to be quite subtle if we compare two
abundance vectors with different numbers of species, which is often of interest. Another
challenge is to modify the classic approach to measurement of evenness when we
incorporate weights of importance for different species, e.g., indicator species or
invasive species.
Since biodiversity is more than just richness (number of species) and more than
just evenness, we can explore ways of combining both measures into one index. This
presents major challenges for mathematical analysis, including finding axiom systems
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