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# Logistic Growth

This worksheet discusses 5 sample models that will give you some practice working with the ideas of
logistic growth .

The basic framework for each model is the same. A laboratory is developing a procedure to grow a
certain kind of mold that will be used to make a new antibiotic. The mold is grown in a vat with a
nutrient solution made up of sugars, water, and other ingredients. For each different nutrient solution,
the laboratory makes two tests by introducing a known amount of the mold and observing how the
mold grows over a 24 hour period. For each of the two test amounts, the growth factor for the 24 hour
period is recorded. The results of these tests are shown in the table below.

Test
Solution
First Test Second Test
Population Growth Factor Population Growth Factor
A
B
C
D
E
 100 100 100 100 100 0.8 0.9 2.0 3.0 4.0
 500 400 500 600 600 0.6 0.3 1.2 1.5 1.5

For each Test Solution (A - E), your job is to develop and analyze a logistic growth model. A special
worksheet has been provided for this purpose. Use a separate worksheet for each model, and follow the
outline below.

1. Develop a linear equation that relates the growth factor to the population size. There is room on
the work sheet to graph the line and work out the equation.

2. Formulate a difference equation in the form pn+1 = m (L - pn) pn: What are the constants m and
L for this model?

3. Make a graph using p n as the x value and pn+1 as the y value. Find the highest point on the
graph. Use this to decide whether the model will always produce values of p n+1 that are between
0 and L for your L. Relate your conclusion to the condition mL < 4:

4. Is there a value of the population that results in a growth factor equal to 1? What does that tell
you about the future population growth for this model? Relate your conclusion to the value of
L
- 1/m:

5. Verify your conclusions by computing and graphing the first several values of population model
for some starting population.

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