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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.
|First Test||Second Test|
|Population Growth Factor||Population Growth Factor|
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
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.