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The ability to predict the population size of a group of individuals is extremely useful to the study of ecology. It allows for the estimation of the various effects imposed upon a group by internal and external *forces*. We note that the word *force* has a different meaning in population modeling than in physics. You can think of these forces as factors that impact the population – for example, availability of food, spread of disease, interactions with other species. These forces can then be divided into *density-independent forces* and *density-dependent forces*.

Brandon Hale is a senior undergraduate at Murray State University. Maeve McCarthy is an Associate Professor of Mathematics at Murray State.

With no forces acting upon a population, we expect the population to have simple exponentially increasing growth (Vandermeer & Goldberg, 2004). Mathematically, we expect a population function whose rate of growth increases with the population’s size, that is,

This differential equation says that the rate of change in population size over time (*dP/dt*) increases by a proportional rate of growth (*r*) multiplied by the current population size (*P*). The biological force modeled here is an example of a density-independent force, because it depends only on the population *P*, not on external forces such as crowding or food supply.

We know from calculus that the solution of this equation is *P*(*t*) = *P*_{0}*e ^{rt}*. The graph of this exponential function (Figure 1) shows the behavior of the population over time. Biologists call this a

The exponential curve is best used for microorganisms, especially bacteria. Figure 2 shows the growth of a population of *Escherichia coli* bacteria. Notice that the population begins with just three bacteria, and the population has doubled within 20 minutes. Within three hours, the population has increased dramatically! (To see the figure in full size and resolution, click on the reduced figure shown here.)

three neighboring cells, by James Shapiro and Clara Hsu.

Used by permission of James Shapiro.

We can use the doubling time to calculate the rate of growth for the *E. coli* population. If the initial population is *P*_{0}, then the doubling time *T* for the population is the time required for the population to reach 2*P*_{0}. This requires *e ^{rT}* = 2. Solving for

For microbes, this may not be a problem. The entire colony of *E. coli* in the bottom right of Figure 2 could easily fit on the head of a pin. However, for larger organisms, exponential growth can be maintained for an extended length of time only under very rigid (and often unrealistic) conditions (Molles, 2004). For such organisms, logistic growth (next page) is more realistic.

- If a population doubles every 5 hours, what is its rate of increase?
- If a population triples every 5 hours, what is its rate of increase?
- Suppose that the rate of increase for a particular bacterium is
*r*= 0.24, how long does the bacterium take to double its population?

Brandon M. Hale and Maeve L. McCarthy, "An Introduction to Population Ecology - Introduction to Population Modeling," *Convergence* (October 2005)

Journal of Online Mathematics and its Applications