Math 121 Calculus for Biology
Spring Semester, 2007
Lab Help
15-Feb-06
San Diego State University
Laboratory Help Page for Lab 5
This lab examines discrete dynamical
systems and the Malthusian growth models that were studied in lecture on U. S.
population. To save you some work, you can download an Excel spreadsheet with
the U. S. census data.
Question:
This problem uses an applet
similar to the one in the lecture notes. You begin by finding the best fit of
the Discrete Malthusian growth model to the data by first adjusting the range
of the dates, then adjusting the growth rate. After finding the smallest sum of
squares error, you write down this number and the best value for the growth
rate. Next you create a table in Excel with the appropriate range of dates and
their corresponding populations. (Say these are in columns A and B with the
first entries in A2 and B2. I like to label my columns so the labels are in A1
and B1.) Be sure to include the dates for the decades following the data you
analyzed with the applet, as you will be extending your model to these dates.
You will put the model in the third column with its initial population agreeing
with the actual data, so in C2 write =B2. Say you got a growth rate of 0.263
from the applet, then in C3, you type = 1.263*C2. From there you fill down to
complete the simulation of the Malthusian growth model. Create your graph using
Chart Wizard, highlighting the first 3 columns. Convert the data points to
points and the model to a line, then add the appropriate labels.
In question 2a, to find the table for the solution
of the model, you should proceed as follows: In a different column of your
spreadsheet, say column D, you can find the value of “n” for D2 by typing
=(A2-(1790))/10, supposing 1790 is the initial year of your data table (type
the value according to your version). From there you fill down to get the
column for the rest of the independent values “n”. Then in the next column,
column E in this case, for the first value, type the formula of the general
solution for the Discrete MalthusianModel, say in cell E2 type,
=(1.263)^D1*Po, where Po is the actual value in cell B2 (do not type B2 but the
value inside the cell which is the size of the population at the beginning of
your data table). Then fill down to find the rest of the column. Note that in
this case, your “simulation” will be with respect to the variable “n” and not
the value of the previous population size. Copy the two tables for the two
different equations, the one with the year and the equation Pn+1=(1.263)Pn, and
the table with the values of n and the values obtained with the equation
Pn=(1.263)nP0.
Discuss your results.
