This can replace any Exam I problem.

Extra Credit  A population of mice has been accidentally introduced onto a fairly large island in the Pacific.  Researchers have estimated that the island is capable of supporting a population of up to 100,000 mice and that the growth of the mouse population would be accurately represented by a logistic growth model.  The researchers have also estimated that the growth factor (b) in the logistic growth model will be 1 if time (t) is measured in years.  Thus they are predicting that the rate of change (dP/dt) of the mouse population on the island will be

             (1)

where b is the growth factor, C is the largest sustainable mouse population, and P is the mouse population (as a function of time).  In this example

One hundred mice (about equal numbers of male and female mice) were initially introduced onto the island.  Assign t = 0 as the time the 100 mice were initially introduced onto the island.  In our logistic population growth model

and thus we have

Solve differential equation (1) above to show that

                   Hint

Substitute in the appropriate values for a, b, and C to construct the logistic growth function P(t) for this example.  From your P(t) function estimate the mouse population on the island (to the nearest whole mouse) one year after the 100 mice were first first introduced onto the island (i.e., find P(1)).  Also estimate the mouse population on the island 2 years, 5 years, 10 years, and 20 years after the original 100 mice were introduced onto the island.

Draw a graph of the mouse population (P) as a function of time (t in years) for the first 20 years after the mouse population is introduced onto the island.

How long (to the nearest tenth of a year) does it take for the mouse population on the island to grow from 100 to 50,000?

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        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats