From Sample Diversity to Information Fields
The
traditional notion of biodiversity considers a sample taken by a field
ecologist, using a fixed area (a quadrat). Within the area there is a
sample of the whole community being studied and by fully quantifying
some aspect of diversity in this area, if it is representative of the
whole, an estimate of the community diversity is obtained. Of course
this estimate is made more accurate and precise by increasing the area
of the sample, or equivalently, adding more samples to form a pooled
sample. The simplest measure is a count and the count of species
present in the sample is called the species richness - this is still
the most common measure of biodiversity found in the literature. A
single sample (perhaps composed of multiple quadrats, pooled) gives a
localised measure of species richness, termed the alpha-diversity. If
there are two or more alpha-diversity estimates, then the diversity
among them is termed the beta-diversity. Unfortunately, there are many
ways to define precisely what this means, but all amount to quantifying
the difference among alpha-samples and expressing it as a proportion of
the average diversity within them. The combined diversity of all the
quadrats used to calculate beta-diversity, when pooled, is termed the
gamma-diversity: it is equivalent to the alpha-diversity of the whole
area sampled. It is important to note at this point that all these
traditional measures describe samples of the biological system, not the
system itself (though they do imply something about it). For example,
if we think of an ecological community lying over an environmental
gradient (perhaps running from the valley floor to the top of a
mountain), then the alpha-diversity at one end will in general be
different from the other end. Further, because each end contains a
different (but maybe overlapping) set of organisms, there will be a
more or less high beta-diversity among samples taken from each end.
This means, in turn, that the gamma-diversity of the whole system is
relatively high, but notice that the beta-diversity among adjacent
alpha-samples will be small: in fact beta-diversity is a continuous
variable of the physical distance among alpha-samples.
All this suggests a more general outlook is needed - one which describes the diversity ‘density’ over the surface.
This notion is analogous to the potential difference which can be
calculated over an electrical field. At each point in the field, there
is an electrical potential and variation in space of the potential
gives a local gradient which can causes an electro-motive force.
Strictly at a singular point, there is no gradient, but it can be
defined by the local gradient via differential vector calculus - the
‘div’, ‘grad’ and ‘curl’ (we find these doing the same job in flow
equations such as for heat and fluid mechanics). The local
diversity around a point in space gives us a gradient measure in
diversity. That is hard to envisage, but if we recall that diversity is
a measure of difference and that difference is fundamentally
information content, then we have a local gradient in information,
which in turn can be described as a field (analogous to that of the
electrical potential). This more general view, then describes
biodiversity which extends over space, as a field of information
density. Alpha-diversity then becomes an estimate of the diversity in a
small region, obtained from integrating over the region (surface area,
or for aquatic and soil systems, perhaps a volume). In practice the
integration can be replaced with a summation. The beta-diversity is
then re-interpreted as a gradient measure, which in general will have a
direction, hence a ‘div’, ‘grad’ and ‘curl’ are needed to describe it.
What is more, this information-density interpretation enables us to
freely move from counts of species, or functional groups of organisms,
to estimates of the information content of organisms. In this way, the
alpha-diversity of a monoculture of simple bacteria will be less than
that of a single vertebrate, or a complex community of bacteria (which
may anyway be living inside the vertebrate). On the mountainside, the
information density will probably have a larger gradient up the slope
of the mountain, than around any contour line on the map: the
beta-diversity will be quantified with a direction, over the whole
space and independently of scale.