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Exponential Functions

A Look Towards Applications

Mathematical Models in Biology

Mathematical models are devised to describe a phenomenon of interest. This is usually done through equation(s) that have been derived from experimental data that capture the phenomenon to some degree. In practice, models are not perfect descriptors and may only accurately describe a phenomenon under a restricted set of conditions.

For example, exponential growth models only accurately describe population growth under ideal conditions. As a populations grows, nutrients and space eventually become limited. Therefore, a population cannot grow exponentially indefinitely. Instead, populations often exhibit periods of exponential growth followed by periods of slower growth or even decline.

Modeling Exponential Growth and Decay

Exponential functions are commonly used in the biological sciences to model the amount of a particular quantity being modeled, such as population size, over time. Graphs of experimental data are usually drawn with time on the x-axis and the quantity on the y-axis.

How do you decide which base to use when modeling a biological phenomena? For some applications you may intuitively know which base to use. Bacterial growth by binary fission, for instance, can be easily described using a base 2 exponential function, because each round of cell division doubles the population of bacteria. In such cases, we can use a general model for exponential growth/decay,

(t) = Ca t,

To represent the passage of time these models are often constructed with the variable t ≥ 0.Notice that when t = 0,

y (0) = Ca0 = C,

thus we can interpret the constant C as the initial value of the quantity being modeled (e.g. initial population size). The base of the exponent, a, depends upon the phenomena being modeled. In our example of bacterial growth, the easiest base to use would be base 2 because we are studying a population that is doubling.

What if you don't know what base to use?

In such cases, you would reach for a natural exponential model of growth and decay, given by

y (t) = Ce kt

where the constant k is a real number. The natural exponential base (e ≈ 2.718) is convenient because of its simple differentiation rules in calculus, and because it is easy to access on most calculators.

Notice that our new function can be viewed as an exponential function that has been transformed. As we discussed in the graphical transformation section, the constant k horizontally stretches or shrinks the graph. Also, recall that if < 0, the transformed graph also involves a reflection across the y-axis.

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How do we know if a natural exponential function represents growth or decay?

Using the laws of exponents, we can rewrite our new model as follows,

y(t) = Ce^kt = C (e^k)^t

which is an exponential function with base e k . This simplification allows us to determine if the function exhibits exponential growth or decay using the following rules:

if k > 0, then > 1, and the function exhibits exponential growth,

if k < 0, then 0 < e k < 1, and the function exhibits exponential decay.


The next section provides problems to check your knowledge of exponential functions.


The Biology Project > Biomath > Exponential Functions > Look Toward Applications

The Biology Project
Department of Biochemistry and Molecular Biophysics
The University of Arizona

December 2005
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