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