events and health using national data sets; and then 2) predict the health impact of
future climate change scenarios.
Research Challenge for the Mathematical Sciences: Develop mathematical
models that characterize human susceptibility to and adaptability to changing ambient
temperatures; new statistical tools for public health surveillance of effects of changing
climate; and new theories of uncertainty quantification and propagation to enhance the
usefulness and applicability of mathematical models.
Example 4: Measurement of Biodiversity
Biodiversity is a term that is used to describe certain aspects of the health of an
ecosystem. The Convention on Biological Diversity (CBD) (http://www.biodiv.org) defines
biodiversity as: “the variability among living organisms from all sources including, inter
alia, terrestrial, marine and other aquatic ecosystems and the ecological complexes of
which they are part; this includes diversity within species, between species and of
ecosystems” (CBD 1992). Loss of biodiversity is considered an indicator of declining
health of an ecosystem and there is great concern that climate change and other
environmental stressors – natural and man-made – are leading to such a loss. One way
of measuring progress in controlling the unwanted environmental effects of human
activities -- effects of human systems on natural systems -- is to determine the extent to
which the loss of biodiversity has been controlled. CBD set the goal that, by 2010, we
should achieve a significant reduction of the current state of biodiversity loss at the
global, regional, and national level (UNEP 2002). But how can we tell if we have
achieved this goal? We need to be able to measure biodiversity. There are some
fundamental mathematical challenges arising from the need to do so. Only by putting the
measurement of biodiversity on a firm mathematical foundation can we be confident that
we are capturing the true diversity in nature.
There is a long history of defining biodiversity and it is a multidimensional
concept. The term was coined by Walter G. Rosen during the 1986 National Forum on
BioDiversity (Takacs 1996). It was first used in the literature in the proceedings of that
meeting, edited by E.O. Wilson and F.M. Peters (1988). Since then, there have been
hundreds of papers attempting to define it precisely. Traditional approaches consider
two basic determinants of biodiversity: Richness is the number of species and Evenness
is the extent to which species are equally distributed (Magurran 1991). However, these
concepts assume that all species are equal, that all individuals are equal (we disregard
differences in size, health, etc.), and that spatial distribution is irrelevant. These may not
be appropriate assumptions. Some species are highly “visible” or considered centrally
important for conservation biology purposes (e.g., lions, elephants). Moreover, some
species are indicator species of the health of an ecosystem. For example, lichens
respond to changes in forest structure (air quality, climate) and the disappearance of
lichens may indicate environmental stress (high levels of sulfur dioxide, nitrogen oxides,
etc.). Thus, we may want to give the presence or absence of such indicator species
higher priority.
Richness is usually interpreted as the number of different species in an
ecosystem. This has some major disadvantages. It doesn’t pay attention to
presence/absence of “important” or “indicator” species. Also, richness defined this way
could increase with the presence of species we don’t want to have, e.g., invasive
species (Lamb et al. 2009, Magurran 2004). Finally, richness may be dependent on the
sampling process to detect species and that sampling process could be biased or could