What is a common exampleof mathematical modeling?
One of the most common examples of a mathematical model has to do with the growth of
populations—human, animal, or otherwise. In the sciences, this type of mathematical
modeling often leads to intelligent hypotheses about the future and past of a population—
from a possible population explosion to even an extinction. (For more information about
the use of mathematical modeling in the sciences, see “Math in the Natural Sciences.”)
What does a priorirefer to in mathematical modeling?
The term a priorirefers to the amount of information available in a system. Based on
the amount of a priori information, mathematical modeling problems are often classi-
fied into white-box (a system in which all necessary information is available) or black-
box (a system in which there is no a priori information available) models. Practically
all systems are somewhere between the white-box and black-box models; thus, this
concept only works as an intuitive guide on how to approach a mathematical problem.
How can the accuracyof a mathematical modelbe determined?
There is usually one good way to determine the accuracy of a mathematical model:
Once a set of equations has been built and solved, if the data generated by the equa- 267
APPLIED MATHEMATICS
Can a mathematical model be too complex?
Y
es, mathematical models can be too complex for a number of reasons. For
example, if we wanted to model the development of a hurricane, we would
take data from a forming storm—water vapor, pressure, temperature, and so
on—and incorporate it into our model. In this way, we would try to develop as
close to a white-box model of the hurricane system as possible.But in reality a collection of such a huge amount of data—not to mention the
computational cost—would effectively inhibit the use of such a weather model.
There is also uncertainty because the development of a hurricane is an overly
complex system, mainly because each separate part of a hurricane and its develop-
ment causes some amount of variance in the model. For example, not only would
we have to know about the details of the hurricane’s development, but other fac-
tors would come into play, such as the ocean-interaction variables that contribute
to the hurricane, variability of solar radiation, and even how periodic events such
as El Niño—a periodic warming of the waters off the South America coast—affect
the hurricane. Thus, meteorologists usually use some approximations to make the
mathematical model more manageable, which is also why we still can’t predict
where and how much rain, wind, and tornadoes will occur during a hurricane.