Ecology is the science which deals with interactions between living
organisms and their environment. Historically it has focused on questions such as:
- Why do we observe certain organisms in certain places and not others?
- What limits the abundances of organisms and controls their dynamics?
- What causes the observed groupings of organisms of different species,
called the community, to
vary across the planet?
- What are the major pathways for movement of matter and energy within
and between natural
systems?

The above questions serve as the focus of several distinct fields
within ecology. Physiological
ecology deals with interactions between individual organisms and
external environmental forces,
such as temperature, with a focus on how individual physiology and
behavior varies across
different environment. Population ecology deals with the dynamics and
structure (age, size, sex,
etc.) of groups of organisms of the same species. Community ecology
deals with the biological
interactions (predator-prey, competition, mutualism, etc.) which
occur between species. Ecosystem
ecology deals with the movement of matter and energy between
communities and the physical
environment.

Mathematics, as the language of science, allows us to carefully
phrase questions concerning each
of the above areas of ecology. It is through mathematical
descriptions of ecological systems that
we abstract out the basic principles of these systems and determine
the implications of these.
Ecological systems are enormously complex. A major advantage of
mathematical ecology is the
capability to selectively ignore much of this complexity and
determine whether by doing so we can
still explain the major patterns of life on the planet. Thus simple
population models group together
all individuals of the same species and follow only the total number
in the population. By ignoring
the complexity of differences in physiology, size, and age between
individuals, the models attempt
to compare the basic dynamics obtained from the model with
observations on different species. As
a next step, additional complexity, associated with introducing
different age classes for example, is
included. How the inclusion of such additional complexity affects the
predictions of the model
determines whether this additional complexity is necessary to answer
the biological questions you
are interested in.

Mathematical models in physiological ecology are often compartmental
in form, in which the
organism is assumed to be composed of several different components.
For example, many plant
growth models consider leaves, stem and roots as different
compartments. The models then make
assumptions about how different environmental factors affect the rate
of change of biomass or
nutrients in different compartments. These models are typically
framed as systems of differential
equations with one equation for each compartment. Population models
are used to determine the
effects of different assumptions about the age, size, or spatial
structure of a population on the
dynamics of the population. Mathematical approaches include
differential equations (both ordinary
and partial), integral equations, and matrix theory. Models for
communities are often framed as
systems of ordinary differential equations, with separate equations
for each of the interacting
populations. Additional models apply graph theory to elucidate the
topological structure of food
webs, the links which determine who eats who in a particular
community.

The above has focused on the use of mathematics to formulate basic
theory in ecology. There are
also many applications of mathematical and computer models to very
practical questions arising
from environmental problems. This includes the entire field of
ecotoxicology, in which
mathematical models predict the effects of environmental pollutants
on populations and
communities. The field of natural resource management uses models to
help set harvest quotas for
fish and game, based upon population models similar in form to those
mentioned above.
Conservation ecology uses models to help determine the relative
effects of alternative recovery
plans for endangered species, as well as aid in the design of nature
preserves.

- More information about mathematical ecology may be found on-line in:
Mathematics Archives for the Life Sciences
and on my Home Page