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Exponential Functions - Population Growth - The Malthusian Model

Larry Gladney and Dennis DeTurck

Read about Thomas Malthus, for whom the Malthusian Model is named.

The simplest model of population growth assumes essentially that the adult (female) members of a population reproduce at a steady rate, usually as fast as they can. This implies that births increase the population at a rate proportional to the population. Similarly, a certain proportion of the population dies off every year -- so deaths decrease the population at a rate proportional to the population. If the proportionality constant for the birth rate is greater than that for the death rate, then the population increases, otherwise it decreases. In this simple situation, the population either increases or decreases exponentially.

In this project, we examine the exponential growth of a population of fast-breeding organisms -- cockroaches. Before moving on to the problems below, view the movie clip that illustrates the situation. [This clip was found to be inoperable on 3/20/2012. The link was removed. Ed.]

Cockroaches are pretty large bugs that breed very quickly. For the purposes of this project, we will assume the following:

  • Cockroaches breed quickly enough that their population doubles every minute.
  • It takes 10 cockroaches to cover one square inch of the ground.

Assume that you start with a population of 1 cockroach (don't ask how it reproduces). Then the cockroach population is 2t, where t is measured in minutes.


  1. How many cockroaches are there after 10 minutes?
  2. What total area do they occupy after 10 minutes?
  3. How many cockroaches are there after 15 minutes?
  4. What total area (in square feet) do they occupy after 15 minutes?
  5. How long until you have 1,000,000 cockroaches?
  6. How long until there are enough to cover the floor of an 8' by 12' kitchen?
  7. How long until they cover the floor of a typical 2000 square foot house?
  8. How long unti they cover an area the size of the city of Philadelphia (according to the 1993 World Almanac, 136 square miles)?
  9. How about an area the size of Pennsylvania (44,820 square miles)?
  10. How about an area the size of the United States (3,539,289 square miles including all 50 states plus DC)?
  11. How about all of North America (as shown at the end of the video -- 9,400,000 square miles)?
  12. How about covering the world (57,900,000 square miles)?
  13. Use a watch to get timings from the video on how long it takes for the population to double (at the beginning of the video) and how long until North America is covered. Is the video realistic in this regard?


Larry Gladney and Dennis DeTurck, "Exponential Functions - Population Growth - The Malthusian Model," Convergence (November 2004)