**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,

*y *(*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)
= *Ca*^{0} = *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 *k *< 0,
the transformed graph also involves a reflection across the *y*-axis.

**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,

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 *e *^{k }> 1,
and the function exhibits
exponential growth,

if *k* < 0,
then 0 < *e *^{k} < 1,
and the function exhibits exponential decay.

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