
Mathematical ModelingOne of the primary uses of mathematics is in making predictions or inferences about future or past events based upon data collected from natural processes or experiments. In order to do this effectively, the mathematician or scientist must determine a mathematical "model" which does a good job of simulating the results of these processes or experiments. Such a model may take the form of an equation or an inequality, a system of equations or inequalities, geometric patterns, or computer programs that emulate numerical or geometric patterns in the data, or any combination of these. We often say that we want our mathematical models to "capture" the trend of the data so that we may be more confident in predictions made from the models. In truth, we rarely, if ever, obtain a mathematical model which perfectly captures the data we are studying. In fact, a mathematical model should not be viewed as a fixed final construction, but as a work in progress, subject to evaluation and revision if it fails to account for new results in the system being studied. The two major types of mathematical models are discrete models and continuous models. The MMII/BC course at Polytechnic School gives almost equal weight to each type. The first half of the course concentrates on discrete dynamical systems, which in many cases may be modeled using difference equations, the discrete counterpart of the differential equations studied in calculus. The second half of the course is devoted to the continuous models of the calculus. Until the last third of the twentieth century, discrete mathematics remained in the shadow of the calculus of Isaac Newton and Gottfried Leibniz, which is the foundation upon which modern physics is built. The advent of the digital computer in the 1950s and 60s ushered in a new era in mathematical modeling using discrete mathematics. Most of the discoveries in chaos theory and fractal geometry would have remained beyond the reach of humans without the assistance of computers to simulate discrete dynamical systems. Throughout the MMII/BC course, we will stress the interplay between discrete
and continuous models, noting that often the choice between using a discrete
model and a continuous model is a matter of convenience. Nevertheless, our
study of the continuous logistic differential equation and its discrete
counterpart, the logistic difference equation, will warn us that sometimes
the model we choose can significantly effect what we see in the system we are studying.

