Logistic Growth

Step 1:  Go to the following website and choose a country whose population is growing, record the population data for the years 2011, 2012, 2013, 2014.
Website: http://data.worldbank.org/indicator/SP.POP.TOTL

Country:

Population Data (write data in “thousands” so, for example, 2,346,126 = 2345.125 thousands)
2011:
2012:
2013:
2014:

In your opinion, does this website gives you valid data?  Explain why or why not:

Step 2: From the data, approximate the rate of change of population (in thousands) per year,  , for the years 2012, 2013, and 2014.   Use t = 0 to represent the year 2011, and so forth.   Include units below.  (Show how you approximated these rates of change.)

Step 3:  Using Desmos, plot as x1 the population (in thousands) and as y1 the rate of change of population calculated in Step 2 above. (Remember, in our logistic differential equation, the rate of change of population is a function of the population, not time.)   From these data points, have Desmos calculate the curve of best fit of the logistic differential equation.
Recall, a logistic differential equation has the form:

3a:  Write the Desmos curve of best fit/Logistic Differential equation below:

3b: Attach a graph of your plotted points, you curve of best fit, and your curve of best fit statistics.  Label that as Graph 1.

Step 4:  Using separate of variable techniques, solve your logistic differential equation to get the particular solution, P(t), for your country.   [HINT 1:  This will require you to use Partial Fraction Decomposition.   HINT 2:  Again, use t = 0 to represent the year 2011, and so forth.]  SHOW ALL STEPS CLEARLY

Step 5:  Graph together, P(t) and your original data points from Step 1 (Again, use t = 0 to represent the year 2011, and so forth.)  Attach to this project as Graph 2.

Step 6:  Analysis ( Show  all work):

1) What is  ?

2) For what level of population, P, is the rate of change of population  greatest?  What is that greatest rate of change of population (include units)?

3) In what year is the rate of change of population greatest?

4) In what year will the population of the country first be at or above 90% of its carrying capacity?

5) Using calculus and the P(t) function, find the average population of your country for the years 2011 through 2014.
Step 7:  Thought/Analysis questions
1) Is your logistic growth model for this country a good model (i.e., is your specific logistic growth model a good fit for your country’s data)?  Explain why or why not.

2) Putting aside your specific logistic growth model, does it make sense to use any logistic growth model to analyze the population of individual countries?  Explain why or why not.

3) Use a country whose populations are increasing.  Would this project have worked with a country whose population was decreasing?  Explain why or why not.

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