Population Growthand Population Projections

Birth Intervals

Menarche

Marriage

1st Birth

2nd Birth

3rd Birth

Menopause

Issues-Events out of order (births then marry)-IVF-Right censoring-Multiple births

Post partum ovulation time to conceive birthAmenorrhea conception

Population Growth

P(t+1) = P(t) + B(t) – D(t) + I(t) – E(t)For now ignore migrationFocus on births and deaths together

Fertility and Population Growth

TFR– number of children born into a population

– 2.1 is considered replacement level fertility

Sex ratio of births – Number of females in the population

– Gross reproduction rate (GRR)• Same as TFR except counts only female births

• Usually male births outnumber female births

• TFR = GRR x 2.05 (approx)

Fertility and Population Growth

Mortality– To sustain a population, need to know how

many females survive to reproductive age– Evolutionary biologists often refer to this is

reproductive fitness (measured as – Net reproduction rate (NRR)

• Number of daughters born to a women that controls for mortality

Net Reproduction Rate

xx

dx Lf

49

15

Total number of daughters bornbetween 15 and 49 that takesinto account survival of mothers.

Lx refers to number of person years lived by a cohort of womenfx refers to the age – specific fertility rate

womanperdaughtersAvglLfNRR xx

dx #0

49

15

NRR

NRR < GRR because of mortalityIf NRR = GRR, then women are immortal

while in their reproductive yearsIf fertility and mortality rates are constant

for many years, have a constant growth rate.

Using the NRR to Predict Growth

NRR is a cohort measure of growth since it assumes a number of daughters over the reproductive years

Most calculate NRR for a single year Japan in 1968 –

– TFR = 1.6– GRR = 0.8– NRR < 0.8– Would predict population decline; didn’t happen

Geometric and Exponential Growth

declinepopulationthenrIf

growthpopulationthenrIf

rPP

PPPrchangeofRate

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0

01

01

Geometric Growth

Assumes additions/deletions happen once a year

Growth Rates

Population Growth Rates in Urban and Rural Areas, Less and More Developed Countries, 1975 to 2000 and 2000 to 2025. Derived from United Nations, World Urbanization Prospects: The 1999 Revision (2000).

Geometric and Exponential Growth

declinepopulationthenrIf

growthpopulationthenrIf

ePP

PPPrchangeofRate

PPChangeAbosulte

rtt

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Exponential Growth

Assumes additions/deletions happen throughout the year

changeofrateannualTimeDoubling

70

In some urban areas in developing countries, growth rate is .07 or 7% so 70/7 means a population doubling in 10 years

Many developed countries have very low growth rates and, as a result, the equation shows doubling times of hundreds or thousands of years. But these countries are not expected to ever double again. Most, in fact, likely have population declines in their future.

Many less developed countries have high growth rates that are associated with short doubling times, but are expected to grow more slowly as birth rates are expected to continue to decline

Excel example of geometric growth

R=6.9%

R=3.5%

R=4.2%

Annual Growth Rate and NRR

Link NRR to growth rateNRR is a comparison of one generation (mothers)

to another (daughters)Measure the population size after the length of

one generation (g)

Note: book does not mention gender per se

rg

o

rgo

o

g eP

ePNRRor

P

PNRR

Population Structure

Population pyramids– Age/sex histograms– http://www.census.gov/ipc/www/idbpyr.html

Shapes of Population Pyramids

HIGHLIGHTS IN WORLD POPULATION GROWTH 1 billion in 1804 2  billion in 1927 (123 years later)3  billion in 1960 (33 years later) 4  billion in 1974 (14 years later)5  billion in 1987 (13 years later) 6  billion in 1999 (12 years later) 

Dependency Ratio

Another measure of the age distributionDefined as the number of non-working age

persons per 100 working age persons

100

)()(

6418

65170

Pop

PopPop

Stable Population

Lotka's concept of a stable population (circa 1907). If any population has:

– No migration,

– Mortality & fertility age-specific rates remain constant for a long period

Then a fixed age structure will develop (called stable age structure) which does not depend of the initial age structure.

Population will also increase in size at a constant rate. Stationary population (which has a zero rate of increase) is a

special case of stable population

Stationary Population

A population with– No migration– Constant age specific mortality– Has birth and death rates that yield a growth

rate of ZEROThis is known as a stationary population.Its size is constant and its age structure

(% in each age category) is also constant.

How Many Are There in A Stable Population?

xxxx

x

x

Llllwhere

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Objective: Calculate Ax

How Many Are There in A Stable Population?

xx

x

xxx

x

x

xxx

cfbSo

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sizepopulationtotalPxageatsizepopulationP

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.

Objective: Calculate b, crude birth rate, in a stable population

How Many Are There in A Stable Population?

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xr

x PeP

The number of births in a stable population that occurred x + 1/2 years ago is simply the crude birth rate b (which does not change) times the population x + 1/2 years ago.

Now, not all those people survived – need to calculate proportion survived to age x:

Objective: Calculate population x+1/2 years ago

changenotdoeswhichpopulationstableainLx

How Many Are There in A Stable Population?

(Number born x+1/2 years ago) x (survived at age x+1/2) tells you how many people there will be at age x today in a stable population

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Excel Example

Population Projections

Why Do It?How To Do It?

– Mathematical models• Simple

• Works for some circumstances

– Component Method• Harder

• More extensive data requirements

Mathematical Models

)ln(,,where

1

1

GrowthLogistic

GrowthlExponentia

)(

0

kCrr

k

rK

e

KP

Cre

rP

ePP

tt

rtt

rtt

Implies perpetual growth or ultimate extinction

Assumes there are upper and lower bounds to population size

Exponential Growth

.

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:

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regressionsimpleaEstimate

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t

rtt

GO TO EXCEL

Logistic Growth

t

KP

KP

asymptoteupperistprojectiontheby

specifiedsizepopulationMaximumk

rK

e

KP

GrowthLogistic

t

t

tt

1ln

)(

1 )(

Last line is specified as a regression and can be estimated as oneGO TO EXCEL

Component Method

Needs a great deal of data, often at the level of detail of single ages.

The number of components can vary depending upon the type of projection needed

All projections reflect the assumptions you make about which components you use, their stability/change over time, and how far into the future you project

Component Method

x

ftx

ftxtxt

mt

mt

mt

mt

mtx

mx

mtx

mtx

PPfB

newbornsfor

MqBP

newbornsexceptagesfor

MqPP

1,,21

,

1,0,021

1,0

1,1,1,1

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Equations 1 and 2 show m(ale) superscripts; comparable equations for females

Europe has just entered a critical phase of its demographic evolution.

Around the year 2000, the population began to generate "negative momentum": a tendency to decline owing to shrinking cohorts of young people that was brought on by low fertility (birthrate) over the past three decades.

Currently, the effect of negative momentum on future population is small. However, each additional decade that fertility remains at its present low level will imply a further decline in the European Union (EU) of 25 to 40 million people, in the absence of offsetting effects from immigration or rising life expectancy.

Population Momentum

The tendency for population growth to continue beyond the time that replacement-level fertility has been achieved because of a relatively high concentration of people in the childbearing years.

For example, the absolute numbers of people in developing countries will continue to increase over the next several decades even as the rates of population growth will decline. This phenomenon is due to past high fertility rates which results in a large number of young people. As these youth grow older and move through reproductive ages, the greater number of births will exceed the number of deaths in the older populations

Projection methods and assumptions. The alternative population projections were carried out using standard cohort component population projection methods using software developed by the authors. Since this analysis aims at isolating the impacts of alternative fertility assumptions, in all scenarios only the fertility component was modified as described in Table 1, while we assumed that mortality stayed constant at life expectancies of 81.5 years for women and 75.5 years for men. We also assumed a closed population without migration.

Doing Component Methods of Population Projections