Question

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 perpectives : 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 find 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) = A*x(0) (0.2)
NEED SOLUTION FOR THIS PROJECT STEP BY STEP WITH DETAILS. AS EARLY AS POSSIBLE.

Answer 1

Part One

The numbers below the diagonal say how many people survived from one group to another. For example of the people which was in the interval
6
in one previous census is
10 -19
of the current census.

The elements in the first row indicate how many children did a group have (i.e. birth rate) in a period of years. For example, people in
10 -19
had approximately children each.

Factors that can change this data are:
Social: If people receive more sexual education they may have less children in the period .
Economic: A better economy provides better conditions for living, then more children.

10 -19

Part Two

Since time zero is 1990, we need to calculate $Ax_0$ , $A^2x_0$ , $A^3x_0$ and $A^4x_0$ for the population of 2000, 2010, 2020 and 2030 respectively. We have that the populations $\times 10^5$ are

P2000 = 8.6300 2.1700 2.2960 1.9400 2.4250 1.8000 1.5660 P2010 = 8.8441 6.0410 1.7794 2.2271 1.8818 2.1825 1.5660 P2020 = 13.1679 6.1909 4.9536 1.7260 2.1603 1.6936 1.8988 P2030 = 17.2811 9.2176 5.0765 4.8050 1.6742 1.9443 1.4734

The total population is the sum of the vector components for each decade. We get,

S2000 = 20.8270 S2010 24.5219 — S2020 31.7912 S2030 = 41.4721

These are the fractions of how much did the population changed each decade,

F1900_2000 1.2180 F2000_2010 1.1774 F2010_2020 = 1.2964 F2020_2030 = 1.3045

The maximum eigenvalue of is . Population change fraction seems to converge to the value .

Part Three

According to this model, the population will increase indefinitely. That is, the population is unstable. The problem is that we are not having into consideration external factors, such as limited resources. In population 24, the two populations are approximately a multiple of the previous one:

>> A 24*x0 ans = 1.0e+03 9.0098 4.6178 2.7725 1.9691 1.3985 0.9216 0.5871

>> A 25*x0 ans = 1.0e+04 * 1.2305 0.6307 0.3787 0.2689 0.1910 0.1259 0.0802

The constant is , the largest eigenvalue of .

Part Four

The new birth rate of the second group is $0.75 \times 1.2 = 0.9$ . So, the new matrix is

A = 1.1000 0.9000 0.2000 0.7000 0 0.1000 0 0 0 0 0 0.9000 0 0.8200 0 0 0 0 0 0.9700 0 0 0 0 0.9700 0 0 0 0 0 0 0 0 0 0.9000 0 0 0 0 0.8700 0 0 0 0

These are the new values for the population

P2000 = 7.7900 2.1700 2.2960 1.9400 2.4250 1.8000 1.5660 P2010 = 8.0251 5.4530 1.7794 2.2271 1.8818 2.1825 1.5660 P2020 = 10.6626 5.6176 4.4715 1.7260 2.1603 1.6936 1.8988 P2030 = 13.8764 7.4639 4.6064 4.3373 1.6742 1.9443 1.4734

with a total of

.

S2000 19.9870 S2010 = 23.1149 S2020 28.2304 S2030 = 35.3759

The population is still unstable since people in group three are having more than one children each generation.