# Linear Algebra, Urban population dynamics Author: Bernard Kolman

1. Feb 15, 2012

### TheMechanic

1. The problem statement, all variables and given/known data

Linear Algebra Project : Urban Population Dynamics
This project is about population modeling and how linear algebra tools may be used to study it.
Background
Population modeling is useful from different perspectives :
1. planners at the city, state, and national level who look at human populations and need forecasts of
populations in order to do planning for future needs. These future needs include housing, schools, care
for elderly, jobs, and utilities such as electricity, water and transportation.
2. businesses do population planning so as to predict how the portions of the population that use their
product will be changing.
3. ecologists use population models to study ecological systems, especially those where endangered species
are involved so as to try to nd measures that will restore the population.
4. medical researchers treat microorganisms and viruses as populations and seek to understand the dy-
namics of their populations ; especially why some thrive in certain environments but don't in others.
In human situations, it is normal to take the intervals of 10 years as the census is taken every 10 years.
Thus the age groups would be 0-9, 10-19, 20-29 etc, so 8 or 9 age categories would probably be appropriate.
The survival fractions would then show the fraction of "newborns" (0-9) who survive to age 10, the fraction
of 10 to 19 years old who survive to 20 etc. This type of data is compiled, for example, by actuaries working
for insurance companies for life and medical insurance purposes.
The basic equations we begin with are
x(k + 1) = Ax(k) k = 0; 1; 2; ::: and x(0) given (0.1)
with solution found iteratively to be
x(k) = Akx(0) (0.2)
Suppose we are studying the population dynamics of Los Angeles for the purpose of making planning proposal.
As above, we take the unit of time to be 10 years, and take 7 age groups : 0-9, 10-19,..., 50-59, 60+. Suppose
further that the population distribution as of 1990 is
(3:1; 2:8; 2:0; 2:5; 2:0; 1:8; 2:9)(x10^5)

and that the Leslie matrix, A, for this model appears as
A :=[
:2 1:2 1:1 :9 :1 0 0
:7 0 0 0 0 0 0
0 :82 0 0 0 0 0
0 0 :97 0 0 0 0
0 0 0 :97 0 0 0
0 0 0 0 :90 0 0
0 0 0 0 0 :87 0 ]
Part One :
Interpret carefully each of the nonzero terms in the matrix. In addition, indicate what factors you think
might change those numbers (they might be social, economical, political or environmental).
Part Two :
Predict :
what the population distribution will look like in 2000, 2010, 2020, and 2030 ?
what the total population will be in each of these years ?
by what fraction the total population changed each year ?
Additionally, what does your software tell you the largest positive eigenvalue of A is ?
Part Three :
Decide if you believe the population is going to zero, becoming stable, or is unstable in the long run. Be
sure and describe in your write up how you arrived at your conclusion. If you have decided it is unstable,
simulate it long enough that the column matrices for two successive populations are proportional to one
another. Calculate that proportionality factor to one decimal place and report it.
Part Four :
Suppose the birth rates for the second age can be reduced by 25% by the year 2000. How does that change
your predictions for 2010, 2020, and 2030 ? Is the population still unstable ?

2. Relevant equations

x(k + 1) = Ax(k) k = 0; 1; 2; ::: and x(0) given (0.1)
with solution found iteratively to be
x(k) = Akx(0) (0.2)

3. The attempt at a solution

Confused on how to start this, I wanted to use Maple or Matlab.

2. Feb 17, 2012

### TheMechanic

Ok, so figured out I have to solve for k as the years needed for the matlab for loop.

3. Nov 27, 2013

### Ninja2013

Hi- I was assigned this project and I'm having a hard understanding and using this is in maple. Were you able to complete this assignment?