On-line Math 21

1.1  Modeling

Modeling is the way we adapt a ``real'' situation to mathematics. Much of the time, a particular problem you will have to solve, no matter where it comes from, has a number of particular details. Some of these details are irrelevancies, others matter only in a limited way, and still others are fundamental to the general way in which situations like that behave. The essence of modeling (or modelling) is the translation of that situation to its mathematical essence. Although certainly ``reducing'' the problem to mathematics loses the particulars of the problem, which is often what is most interesting, it also separates out what is relevant or irrelevant to answering the question.

The best example of this is the question of population growth:

Though undoubtedly important to those involved, the details of how a species procreates does not affect the mathematical underpinning of how the population changes with time.

The rate of gestation, the frequency of, shall we say, contact, and the life expectancy are relevant parameters that go into the particulars of the modeling of the population of a particular species, but are not central to the model itself.

What is central to the model of population is the simple fact that, the more members of a particular species there are, the more new members can be created, and the more that die. That central fact about populations is translated to mathematics by saying that, if P(t) is the population at time t , the rate of change of the population at time t , P¢(t) , is proportional to the population itself, or

P¢(t) = kP(t).

That differential equation models the dynamics of any population of creatures. One then applies the mathematics of differential equations (we will be studying this somewhat in this course) to see that, necessarily, the population has a very specific form,

P(t) = A ekt.

the ``raw'' calculus that goes into these problems is in that step, starting with the model and drawing from it the mathematical conclusion.

Other factors which affect population growth, in general, such as crowding and limitations of the food supply, can refine the model to improve its accuracy. But that model of population growth can then be applied to any population problem and give fairly accurate analysis of a wide variety of situations. That same model can also be applied in analogous situations, which share the feature of population problems that their rate of change depends upon the amount present, such as problems of radioactive decay, cooling, and the spread of epidemics.

Copyright (c) 2000 by David L. Johnson.