Adapted by Sarah J. Greenwald from the Middle Atlantic Consortium for Mathematics and its Applications Throughout the Curriculum

Roaches Take Over the World

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 that deaths decrease the population at a rate proportional to the population. If the birth rate proportionality constant is greater than the death rate, then the population increases, otherwise it decreases. And in this simple situation, the population increases or decreases exponentially.

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 2^t, where t is measured in minutes.

    A. Answer the following questions. Justify your answers!
    1. What is the differential equation which corresponds to the model of P=2^t for this cockroach population?
    2. How many cockroaches are there after 10 minutes?
    3.What total area do they occupy (after 10 minutes)?
    4.How many cockroaches are there after 15 minutes?
    5.What total area (in square FEET) do they occupy (after 15 minutes)?
    6.How long until you have 1,000,000 cockroaches?
    7.How long until there are enough to cover the floor of a more-or-less standard 8' by 12' kitchen?
    8.How about an area the size of the United States? (3,539,289 square miles including all 50 states plus DC)
    9.How about all of North America (as shown at the end of the video with sound)? (area=9,400,000 square miles)
    10.How about covering the world? (57,900,000 square miles)
    11.Use a watch to get timings from the video mpeg version without sound which stops at the kitchen or the BIG complete file with sound avi version on how long it takes for the population to double (at the beginning of the video) and how long for the entire video to run. Compare with question 8 or 9 (depending on which video you watch/time). Is the video "realistic" in this sense? Explain?

    B. Write a letter to the x-files creater Chris Carter detailing your ideas on how he can make a realistic "Cockroach takeover" movie. Include your reasoning and the answers to the above questions. Be sure to explain the math (Chris Carter is no mathematician!) in your letter. (You can, if you wish, give an overview of the math in your letter and refer to appendices for details.) You may work with one other person, if you wish, and turn in one report for your group.

    Please print out and attach this checklist to your paper (I will use this when grading and for revisions).