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Population Growth Of Various Population Growth Of Various CountriesCountries

Jose Henson

Sam Choi

Natalie Wagner

Alex Kang

Origins of Population Growth Models

• Thomas Malthus proposed that human growth would precede exponentially– An essay on the principle of population as it

affects the future improvement of society (1798)– Gave formula:– Growth Rate:

– Predicted the world would soon outgrow its resources, with the ultimate destruction of mankind

rtif eP=P

rP=dt

dP

Exponential GrowthExponential GrowthExponential Population Growth:

-Used for rapidly growing population

- Bacteria, insects, rabbits

-Assumptions

-All individuals reproduce constantly and equally

-Not valid for human population growth

- Valid over a short period of time

rtif eP=P

Growth of BacteriaGrowth of Bacteria

• Reproductive rate, r, is found from data by finding ratio of 2

• Exponential growth equation can now be used to calculate future populations

• At t=100

Time (Min)

Bacteria Cells

0 1000

10 2000

20 4000

30 8000

40 16000

50 32000

60 48000

2re 69314.r

rtif eP=P

8069314.1000e=P2710208.1 P

Revision of the Population Revision of the Population ModelModel

• Pierre-Francois Verhulst of Belgium - Suggested that there was a limit to population growth in 1838- Growth Formula:

- mu represents mortality caused by high population

-as population increases growth is overshadowed

2PrP=dt

dP

Modern NotationModern Notation

M

PrP=

dt

dP1 PMP

M

r=

dt

dP

r

M

2PrP=dt

dP

Factor from both termsrP

Verhulst’s Logistic growth model:

P

rrP=

dt

dP 1

Substitute

or

Population ModelPopulation Model

PMPM

r=

dt

dP

rdt

dP

PMP

11

CrtPMP lnln

Population equation can be found by solving the differential equation

Algebraic manipulation:

Integrate with respect to t:

CrtP

PM

lnSimplify:

Derivation ContinuedDerivation Continued

Crt eeP

PM

CeP

PM rt

AeP

M rt 1

rtAe

MP

1

Exponentiate:

Let :CeA

Simplify:

Solve for P:

Logistic Population GrowthLogistic Population Growth

Logistic Population Growth :

M = Population Capacity

A = Initial Population Constant

r = Reproductive rate constant

Assumptions:

- Found via logistic regression line

- Only valid over long periods of time

rtAe+

M=P 1

Calculating Regression Calculating Regression EquationsEquations

Linear Regression:

- Commonly called “best fit” line

Logistic Regression

- Creates a “best fit” S-shaped curve

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 1 2 3 4 5

Population of the WorldPopulation of the World

Current Population:

6,442,658,699 people

Growth Rate:

76,570,430 people per year

Estimates:

2020: 7.6 Billion people

2030: 8.2 Billion people

2050: 9.1 Billion People

United StatesUnited States

Y 9E8 ln x 7E9

te+=P

.01402.1751.000

854.1

Population of United States

0

50

100

150

200

250

300

350

0 10 20 30 40

Year

Po

pu

lati

on

(M

illio

ns

)

Analysis of Population LevelsAnalysis of Population Levels

154.3022.1751.000

706.135.0140

e+=P

100734.295

154.302734.295%100

Actual

alExperimentActualError

te+=P

.01402.1751.000

706.1

Calculated Current Population at t=35 (In Millions)

Actual Current Population = 295.734 Million People

%36.2Error

Analysis of Population GrowthAnalysis of Population Growth

22.1751.014

1.01421.501

)+(

)(=

dt

dPt

t

%100P

dtdP

10092.

8187.92.%100

Actual


Current Population Growth at t=35

Growth Rate:

%01.11Error

4211.22.1751.014

1.01421.501235

35

)+(

)(=

dt

dP

8187.%100734.295

4211.2

South KoreaSouth Korea

te+=P

.0454.7421.00

55.9

Population Of South Korea

0

10

20

30

40

50

60

0 10 20 30 40

Year

Po

pu

lati

on

(M

illio

ns

)

Analysis of Population Levels

te+=P

.0454.7421.00

55.9


546.48.7421.00

55.935.0454

e+=P


100546.48

546.48422.48%100

Actual


%256.Error


2932..7421.046

1.0461.883235

35

)+(

)(=

dt

dP

235

35

.7421.046

1.0461.883

)+(

)(=

dt

dP

%100P

dtdP


Growth Rate:

%0606.%100422.48

932.2

10038.

606.38.%100

Actual


%34.59Error

JapanJapan

te+=P

.083.2441.000

129.331

Population Of Japan

0

20

40

60

80

100

120

140

0 10 20 30 40

Year

Po

pu

lati

on

(M

illio

ns

)


100417.127

626.127417.127%100

Actual


626.127.2441.000

129.33135.083

e+=P

te+=P

.083.2441.000

129.331



%164.Error


2.2441.087

1.0872.619

)+(

)(=

dt

dPt

t

%100P

dtdP

100107.

107.05.%100

Actual



Growth Rate:

1376..2441.087

1.0872.619235

35

)+(

)(=

dt

dP

%1078.%100626.127

1376.

%69.115Error

IndiaIndia

te+=P

.03502.6731.000

1952

Population Of India

0

200

400

600

800

1000

1200

0 10 20 30 40

Year

Po

pu

lati

on

(M

illio

ns

)


100264.1080

426.1093264.1080%100

Actual


426.10932.6731.000

195235.0350

e+=P

te+=P

.03502.6731.000

1952



%218.1Error


22.6731.036

1.036182.619

)+(

)(=

dt

dPt

t

%100P

dtdP

1004.1

556.14.1%100

Actual



Growth Rate:

806.162.6731.036

1.036182.619235

35

)+(

)(=

dt

dP

%556.1%100264.1080

806.16

%143.11Error

Population Of South Korea

0

10

20

30

40

50

60

0 10 20 30 40

Year

Po

pu

lati

on

(M

illio

ns

)

Population Of The United States

0

50

100

150

200

250

300

350

0 10 20 30 40

Year

Po

pu

lati

on

(M

illio

ns

)

Population Of Japan

0

20

40

60

80

100

120

140

0 10 20 30 40

Year

Po

pu

lati

on

(M

illio

ns

)

Population Of India

0

200

400

600

800

1000

1200

0 10 20 30 40

Year

Po

pu

lati

on

(M

illio

ns

)

Carrying CapacityCarrying Capacity

kt

kt

Ae+

M=M

Ae+

M=P

1

.00001

1

Year country will reach the carrying capacity:

U.S.A: 2543

Korea: 2305

Japan: 2150

India: 2543

Set P=M - .00001

Solve for t

ConclusionsConclusions

There are TOO many people in the world…..Thomas

Malthus was right….

Works CitedWorks Cited

http://www.library.uu.nl/wesp/populstat/Asia/skoreac.htm

http://geography.about.com/gi/dynamic/offsite.htm?zi=1/XJ&sdn=geography&zu=http%3A%2F%2Fwww.census.gov%2Fftp%2Fpub%2Fipc%2Fwww%2Fidbsum.html

http://www.library.uu.nl/wesp/populstat/Asia/japanc.htm

http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/mordifeqs/logistic.html

http://en.wikipedia.org/wiki/Global_Population#Forecast_of_world_population

http://www.mnforsustain.org/united_states_population_growth_graph.htm

http://www.library.uu.nl/wesp/populstat/Asia/indiac.htm

http://www.cia.gov/cia/publications/factbook/geos/in.html#People

http://www.cia.gov/cia/publications/factbook/geos/us.html

http://www.cia.gov/cia/publications/factbook/geos/ja.html

http://www.cia.gov/cia/publications/factbook/geos/ks.html

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