that characterize a partial order on abundance vectors that reflects the combined effect
of richness and evenness (Rousseau and Van Hecke 1999, Rousseau et al. 1999).
Because different indices of biodiversity have different advantages and
disadvantages, we sometimes look to use several of them in addressing a question and
see if they yield a consistent conclusion. This raises both mathematical and statistical
challenges, for example studying families of biodiversity indices that depend upon some
parameter and giving conditions on the range of values of the parameter where the
indices will give consistent rankings of biodiversity (Buckland et al. 2005, Ricotta 2003).
It should be noted that making precise notions of richness or evenness or other
notions of biodiversity is an example of what is now being termed a hybrid mathematical
model. For, in many cases we can describe ecosystems in terms of number of
individuals of different kinds of species (a discrete variable) and other times we can
describe them in terms of the biomass of different kinds of species (a continuous
variable), and sometimes, however, we need hybrid models that include both discrete
and continuous counterparts.
Finally, a measure of biodiversity is applied to a particular ecosystem at a
particular instant of time. A goal of biodiversity preservation is to achieve ecosystems
that are sustainable, i.e., maintain relatively stable biodiversity into the future. A good
measure of biodiversity should be usable in mathematical models that help us predict
that under certain conditions of an evolving ecosystem, the biodiversity will remain
relatively stable. The development of such mathematical models is a key goal of
sustainability science, and it is intimately connected to finding precise definitions of
biodiversity.
Research Challenge for the Mathematical Sciences: Develop clear criteria for
how to measure biodiversity, derived from mathematically-precise assumptions; devise
methods for applying the criteria that take into account potential biases and problems in
data gathering to inform the measures and the multiple criteria for a biodiversity
measure; understand the uncertainty involved in claims about changes (positive or
negative) in biodiversity; and find ways to use the measures to understand how to
achieve ecosystems that are sustainable and maintain stable biodiversity into the future.
Example 5: Migration
Migration of animals, birds, fish, insects, and plants are key processes in the
balance of natural systems. These processes can be dramatically sped up by modern
transportation systems that move “stowaway” species from one part of the world to
another in a matter of days or even hours. All of these processes interact in a
fundamental way with human well-being. For example, fish contribute a great
percentage of our planet’s biomass, while animal migration/invasion affects agriculture
and disease. Yet, changing environmental conditions, often traceable to human
activities, threaten to impact these critical migration processes.
Fish populations are a key case in point. An International Symposium on Climate
Change, held on April 25-29, 2010 in Sendai, Japan, dealt with forecasting impacts,
assessing ecosystem response, and evaluating management strategies in ocean fishery.
Fishery conditions in the ocean are affected by global changes in temperature and by
acidification arising from increasingly dissolved CO 2. Some of the consequences are
changing migration patterns, and increased ratio of small-to-large fish populations; see,
e.g. Garcia and Moreno (2003), Yakubu and Fogarty (2006). There are many