# Population Growth and Exponential Functions

### INTRODUCTION:

• Populations can increase or decrease in number. Decreasing populations are due to death and emigration.
• What are the consequences of increases in population (or increases in population density)? [Think about resources (food, water, oxygen, shelter, etc.) pollution, and other factors.]
• Is there a limit to population growth? [What factors could influence “carrying capacity”?]
• What are some possible ways in which a population could grow – i.e. how can this growth be mathematically modeled? [Think about linear, exponential, and other models of how things can grow.]
OBJECTIVES:
• Differentiate between linear and exponential models based on qualitative shape of curve.
• Differentiate between the mathematical expression of linear and exponential models.
• Understand the consequences of exponential growth.
• Define the terms used in this lab, including exponential function, population (as used in biology), limited resources.

### MATERIALS:

Each group of 3-4 students will need:

• 10 paper cups or small beakers
• 1 large beaker (250 ml or 400 ml)
• 1 wax pencil

### ACTIVITY:

Bacteria in an ideal environment (warm, moist, plenty of room and nutrients) will divide every 20 minutes. Thus, if you begin with one bacterium, in twenty minutes you will have two, and so on. How many bacteria will you have at in 24 hours? (estimate without doing the calculation) Why don’t bacteria rule? (or do they?)

This exercise represents a model of bacterial population growth.

• Put one object (one bean) in the large beaker. The beaker represents the limits of the environment or ecosystem. …
• Place 10 small beakers in a row on the desk or lab bench.
• In the first beaker, place two objects (beans) (21=2).
• Record the number of objects (beans) on the outside of each beaker with a wax pencil.
• In the second beaker, place twice as many objects (beans) as in the first beaker (22=4).
• In the third beaker, place twice as many objects (beans) as in the second beaker (23=8).
• Continue this procedure by placing twice as many objects (beans) as in the previous beaker (doubling the number of beans), in cups 4 through 10. Record the numbers on the outside of each beaker.
• The two beans in the large beaker represent the two bacteria that exist after 20 minutes of incubation. (20 minutes is the “doubling time”)
• Add the contents of small beaker number 1 to the large beaker. These four beans represent the four bacteria that exist after 40 minutes of incubation.
• Continue adding the contents of cups 2 through 10 to the large beaker. Assume that (in our model) each up is added at 20 minute intervals.
• Graph the population in the beaker over time (assuming 20 minute intervals).
Would a similar curve also represent human population growth? Why or why not? How would the time scale change? What factors might affect human population growth? Estimate how long it will take for the human population to double.

What are some of the consequences of exponential growth: for bacteria in your body?
for field mice?
for humans on the globe?

This exponential model of growth rate applies to biologically important topics, including bacterial growth and the polymerase chain reaction (PCR), which is used to “expand” a small sample of DNA to lots (!!) of identical DNA.

This exercise is adapted from an activity available on the web at