Discrete Model for Census Prediction

This problem is a substitute for those who do not attend the Computational Science Seminar series on Sept. 9, 2000. Below is a table of census data (population in millions) for Canada. We want to repeat for this country the modeling effort that we performed in class for the U. S.

Year	Population	Year	Population	Year	Population
1861	3.23	1911	7.21	1961	18.24
1871	3.69	1921	8.79	1971	21.57
1881	4.33	1931	10.38	1981	24.82
1891	4.83	1941	11.51	1991	28.03
1901	5.37	1951	13.65

The steps below are to be repeated for both the data for Canada and India. (You are supposed to compare these to the work on the webpage for the analysis of the U.S. census data.)

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).

b. The Discrete Malthusian growth model is given by

P_n+1 = (1 + r)P_n.

where r is computed in Part a. and P₀ is the population in 1861 or 1901. Solve this model, not forgetting that n is in decades. Create a table showing the predicted values of the model through the year 2001.

c. Now consider the revised growth model

P_n+1 = (1 + k(t_n))P_n.

where k(t_n) is computed in Part a. and P₀ is again the population in 1861 or 1901. Solve this nonautonomous discrete dynamical system. (Note that t_n = 1861 + 10n.) Again create a table showing the predicted values of the model through the year 2001.

d. Use EXCEL to graph the data and the solutions to the models in Parts b. and c. for the period from 1861(or 1901) to 2001. 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.

f. Find the percent error between the actual census data and the models in 1881, 1921, and 1981. Also, use each of the models to predict the population in 2001 and 2051. Decide if the calculated value is most likely high or low.