Biology Reference
In-Depth Information
unrealistic. We then look at the long-term growth of a yeast culture to
build a more believable model.
The first models we construct are of exponential growth. Later in the
chapter, we study related models describing exponential decrease in the
concentration of drugs in the bloodstream. These exponential growth/
decay models are derived from the hypothesis that the time rate of
change (i.e., the derivative with respect to time) of a quantity is
proportional to the amount present.
We begin with a problem popularized in the late eighteenth century by
Thomas Robert Malthus—the growth of human populations.
I. USING DATA TO FORMULATE A MODEL
Contemporary research is hypothesis-driven and is based on
experimental evidence. A properly designed experiment can corroborate
a hypothesis, prove it false, or produce inconclusive data. An experiment
can also suggest new hypotheses that, in turn, will need to be tested.
This leads to an ever-repeating cycle of collecting data, formulating
hypotheses, designing new experiments to attempt to corroborate them,
and collecting new data. It should be emphasized, however, that
ultimately the validity of a hypothesis can never be proved. Karl Popper
gives the following very instructive example: If somebody sees one, two,
or three white swans, he or she may hypothesize, ''All swans are white.''
Each white swan seen corroborates the hypothesis but does not prove it,
because the first black swan would invalidate it completely. This
demonstrates the necessarily close interdependence between hypothesis
and experiment.
In this section, we explore the process of creating mathematical models
that describe the growth (or decline) in the size of populations of living
organisms. We would like to express the size as a mathematical function
of time. Although one model will not work for all species, there are
certain fundamental principles that apply almost universally. Our first
goal is to identify some of these principles and determine the best way to
express them mathematically. We begin by considering U.S. census data
for 1800-1860 (U.S. Census Bureau [1993]). Table 1-1 presents the figures
for the population of the United States over these 6 decades.
Examining the data plot is always a good idea, as it may suggest certain
relationships. Letting t
¼
0 be the year 1800 and one unit of time
¼
10
years, we present the data plot in Figure 1-1. Unfortunately, the
conventional plot of the data is not very illuminating. It is evident that
the growth is nonlinear, but it is not possible to determine the type of
nonlinear dependence by mere observation. There are many
mathematical functions that exhibit similar growth patterns. For
example, if P(t) represents the U.S. population as a function of the
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