Title: Model%20of%20the%20Population%20of%20the%20United%20States
1Model of the Population of the United States
- Simple and Natural Disaster Models
- Presenters
- Heidi Chesak, Luke Reves, Brian Crow, and Ryan
Wilson
2Growth rate population model.
- The growth rate population model is
- Where A is the growth coefficient, p0 is the
initial population, p is the population at time
t, and p1 is the population limit. - Here A0.00012, p03.73 and p1 251.7812.
3Growth rate population model with natural
disaster term
- The growth rate population model is
- Where A is the growth coefficient, p0 is the
initial population, p is the population at time
t, p1 is the population limit, and a is the
natural disaster coefficient. - Here A0.00012, p03.73 and p1 251.7812.
4Population model in 20 years.
Note Population is in millions. The population
model in the first 20 years shows almost linear
growth, topping the population around 7.2
million. The U.S. census data indicates that
the population was 7.23 million. So the model is
fairly close to the actual data with initial data
values.
5Population model in the long term
To observe the long term behavior of the
population model, the time of t was set to 220
years, which is the time since the U.S. census
began. The model shows exponential growth, then
to linear and then logarithmic growth. The growth
appears to slow down at around 250 million.
6U.S. Census graph
This is a table plot from Scilab with values of
the U.S. Census taken from http//www.census.gov/.
The growth is entirely exponential, save for a
few hiccups. The population since 2010 is 308
million people.
7Population model vs. census data
This is a side by side comparison to the
population growth model and the U.S. census data
set. For about the first 125 years, the model
seems to be reasonably accurate, then the two
start to drift apart. The population model seems
to reach an equilibrium at p1251 million.
8Population growth with p1308
To get a better understanding of the behavior of
the model, we can change some of the variables
and observe on the changes. This model has the
limiting variable, p1 set to the current U.S.
population of 308 million. The result is that
the model and the actual growth reach similar
ending points, but show vastly different growth
rates.
9Conclusions about the simple model.
- The simple model is good for estimating the
population growth for up to 120 years. - The p1 variable is equivalent to the equilibrium
value of the growth model. - Since U.S. population growth is almost always
exponential, the simple model can be used to
projected population growth on short time scales.
10Code for Simple Model
- This is the code I used in scilab to graph the
population growth by U.S. census data - t0,10,20,30,40,50,60,70,80,90,100,110,120,130,14
0,150,160,170,180,190,200,210,220 p3.93,
5.31, 7.24, 9.64,12.86,17.06, 23.19,
31.44,38.56,50.19,62.98,76.21,92.23,106.02,123.20,
132.16,151.33,179.32, 203.30, 226.54,
248.71,281.42,308.74plot(t,p,)for
i123,end
11Population Growth Behavior, 0 T 20 Simple vs.
Natural Disaster
P(20) 6.743 M
P1(20) 6.546 M
12Population Long-Term Behavior, 0 T 400
Simple vs. Natural Disaster
P(400) 251.687 M
P1(400) 239.198 M
13A closer look at P1(400)
- Note that as the Population curve increases, a
sinusoid reduces the rate of population growth
with a period T1 year. - This affects the long-term behavior of the model
as well, by both taking longer to reach
equilibrium and by having a maximum population at
equilibrium which will decrease with each
disaster, again with a period of T1 year. - Also, as the population gets closer to Peq
(Population at Equilibrium P1) the magnitude of
natural disasters effect on population increases
i.e. the distance from peak to trough increases.
14A Comparison with actual data
As population neared equilibrium, the growth rate
slowed, which is not reflected in the census
data. This indicates that the model is effective
up until approximately 1950, when U.S. Population
growth entered a linear period.