Using data from the U. S. census bureau, the table below presents the population (in millions) for France. This lab has you repeat for this country the modeling effort that we performed in class for the U. S.

Year

Population

1950

41.83

1960

45.67

1970

50.79

1980

53.87

1990

56.74

2000

59.38



a. Find the growth rate for each decade with the data above by dividing the population from one decade by the population of the previous decade and subtracting 1 from this ratio. Associate each growth rate with the earlier of the two census dates. Determine the average (mean) growth rate, r, from the data above. Associate t with the earlier of the dates in the growth ratio, and use EXCEL's trendline to find the best straight line
k(t) = a + bt

through the growth data. Graph the constant function r, k(t), and the data as a function of t over the period of the census data. It is very important that you click on the trendline equation and reformat the coefficient b so that it has more significant figures (obtain 4 significant figures for a and b).

b. The Discrete Malthusian growth model is given by
Pn+1 = (1 + r)Pn.

where r is computed in Part a. and P0 is the population in 1950. Write the general solution to this model, where n is in decades. Use the model to predict the population in 2020 and 2050.

c. The revised growth model is given by
Pn+1 = (1 + k(tn))Pn.

where k(tn) is computed in Part a. and P0 is again the population in 1950. Simulate this nonautonomous discrete dynamical model from 1950 to 2050. (Note that tn = 1950 + 10n.) Use the model to predict the population in 2020 and 2050.

d. Create a table listing the date, the population data, the predicted values from the Malthusian growth model, the Nonautonomous dynamical model, and the percent error between the actual population and each of the predicted populations from the models from 1950 to 2000. What is the maximum error for each model over this time interval? Use EXCEL to graph the data and the solutions to the each of the models above for the period from 1950 to 2050. Briefly discuss how well these models predict the population over this period. List some strengths and weaknesses of each of the models and how you might obtain a better means of predicting the population.

e. The growth rate of the Nonautonomous dynamical model goes to zero during this century for France. At this time, this model predicts that the population will reach its maximum and start declining. Use the growth rate k(t) to find when this model predicts a maximum population, then estimate what that maximum population will be.