## 2014-Bodine, Erin N. - Discrete Difference Equations.

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Bodine, Erin N. Discrete Difference Equations. Rhodes College. Notes. 37 pp.

From, An Introduction to Discrete Mathematical Modeling

"The field of mathematics provides many dierent means for modeling the world around us. Some mathematical tools

are more appropriate for modeling certain phenomenon and less appropriate for other phenomenon. In this course we focus on how to use mathematics to simulate biological situations in which it is reasonable to assume the underlying variable of the model is discrete, that is, it can be written as a sequence of values. In contrast, the real number line is continuous and for any two values a and b (no matter how close a and b are on the number line), we can always find another value that is between a and b. We may have to increase our precision to achieve this, but it can always be done. A sequence of discrete values is a set of points along the real number line. Thus, with a discrete model, we are assuming that we can ignore the space between the points.

"As one example, suppose we grow a cell culture in petri dish and measure the cell density (cells per unit area) every hour for 24 hours. This would create a sequence of 25 data points (if we include the initial cell density). We know that the cell culture is growing during the time between when we take our measurements, but measuring the population each hour over the course of a day will give us a reasonable representation of how the culture grows over the course of one day. Correspondingly, when we build a mathematical model to represent the growth of the cell culture, it is reasonable for the model to only predict how the culture grows from hour to hour. Thus, the underlying variable, time, is represented in the mathematical model as increasing in discrete, one hour increments.

For another example, consider a population which breeds seasonally, that is, only once a year. If we wanted to determine how the population size is changing over the course of many years, we could collect data to estimate the size of the population at the same time each year (say shortly after the breeding season ends). We know that between the times at which we estimate the population size some individuals will die and during the breeding season many new individuals will be born, however, we ignore the changes in population size from day to day, or week to week, and look only at how the population size changes from year to year. Thus, when we build a mathematical model of this population, it is reasonable for the model to only predict the population size for each year shortly after the breeding season. Here, the underlying variable, time, is represented in the mathematical model as increasing in discrete, one year increments."

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