Urban Population Dynamics

1
Urban Population Dynamics

• Cliff Bowmen
• Scott Fulton
• Gary Reynolds
• Mike Wells

2
Population Modeling

• Planners at the city, state, and national level
  forecast population to plan for future needs
• Housing
• Schools
• Care for elderly
• Jobs
• Utilities
• Businesses predict how the portions of the
  population that use their product will be
  changing.

3
Population Modeling

• Ecologists study ecological systems, especially
  endangered species.
• Medical researchers study growth of microorganism
  and virus populations.

4
Human Population Dynamics

• Normal to measure in intervals of 10 years as the
  census is taken every 10 years.
• Age groups would be 0-9,10-19,11-20 etc.
• The survival fractions would then show the
  fraction of "newborns" (0-9) who survive to age
  10, the fraction of 10 to 19 year olds who
  survive to 20 etc.

5
Basic Equations

• The basic equations we begin with are
• x(k1) Ax(k) k0,1,2,. . . and x(0) given
• With solution found iteratively to be
• x(k) Akx(0)

6
Leslie Matrices

• First described by P. H. Leslie in 1945
• Used to describe the population dynamics of a
  wide variety of organisms including humans

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Leslie Matrix Assumptions

• Only females in the population are considered
• The population is divided into age groups. (0-9,
  10-19, 20-29, 30-39, 40-49, 50-59, 60).
• The survival rate Pt for each age group is known.
• The reproduction rate Ft for each age group is
  known.
• The initial age distribution is known.

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Part 1 Our Leslie Matrix
Average number of females Ft born to a single
female in that age group

• (0-9) (10-19) (20-29) (30-39) (40-49) (50-59) (6
  0-69)
• .2 1.2 1.1 .9 .1 0 0
• .7 0 0 0 0 0 0
• 0 .82 0 0 0 0 0
• A 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

Percent of females in one age class Pt who
survive to move on to the next age group Pt1
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Part 1 Effecting these Factors

• Affecting the birth rate
• Social (i.e. popularity of having children)
• Economical (i.e. ability to support offspring)
• Political (i.e. limiting number of births)
• Environmental (i.e. environmental estrogens)
• Effecting the survival rate
• War
• Disease
• Health Care (Accessibility and Quality)

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Part 2 Predictions
11
Part 2 Predictions
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Part 2 Largest Positive Eigenvalue

• Using Mat-Lab software, the largest positive
  eigenvalue of A 1.3658

13
Part 3 Determine Stability

• Although some age classes change at different
  rates than others, the overall population for
  this model grows by a constantly increasing
  margin at each simulation. Therefore, the
  population growth is unstable.
• However, if the calculations are performed long
  enough, the proportions of the population
  stabilize after twelve simulations. That is, the
  proportionality factors for years 2120 and 2130,
  (and for the following years as well ), are equal
  if they are calculated to one decimal place.

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Part 3 Unstable Population

• We have decided it is unstable
• We simulate it long enough that the column
  matrices for two successive populations are
  proportional to one another.
• Calculating that proportionality factor to one
  decimal place gives us

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Part 3 Growth Rate

• The growth rate stabilized at 1.366
• This also happens to be our eigenvalue that we
  found earlier (A 1.3658)

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Part 4 With Birth Rates For Second Age Class
Reduced by 25
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Part 4 Stability

• The system is still unstable.
• Proportions still stabilize after year 2120
• Our growth rate found here is 1.3037 which we
  know to also be the eigenvalue

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Part 5 Adding in Immigration

• We assume that a constant number of immigrants
  are added to each age group during each time
  interval
• The new basic equations are
• x(k1) Ax(k) B
• k0,1,2,. . . 
• x(0) given
• B is a 7 x 1 matrix where Mij i (the amount of
  immigration in each period), for all i and j
• With solution found iteratively to be
• x(k) (Ak) x(0) kB

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Part 5 20,000 Immigrants Entering Each Age Group
Each 10 Year Period
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Part 5 Comparisons