Year
|
Population
|
1950
|
41.83
|
1960
|
45.67
|
1970
|
50.79
|
1980
|
53.87
|
1990
|
56.74
|
2000
|
59.38
|
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).
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.
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.