population growth models and resource consumption

POPULATION GROWTH MODELS AND RESOURCE CONSUMPTIONCaitlin Thomas November 12, 2009

Presentation Overview

Population Growth Models General Population Growth Model Mathematical Models

Exponential Growth Model Logistic Growth Model

Cohort-Component Model Systems Models

Why Population Growth Matters Resource Consumption/Management

Non-Renewable Resources Renewable Resource

General Population Growth Model

Destruction Rate

Production Rate

Rate of Change of Quantity

In general, the rate of change for any quantity can be modeled as:

For biological populations specifically, the model looks like:

Death RateBirth RatePopulation

Growth Rate

General Population Growth

For mobile populations that can move from one place to another (primary example: humans), we must also take into consideration net migration

So our population growth rate equation becomes:

Net Migration = Immigration – Out Migration

Death RateNet MigrationBirth Rate

Population Growth

Rate

General Population Growth ModelBirth Rate and Death Rate are fixed and normalized (divided

by the total population size):

B, D, M can be: constants, functions of time, or functions of population size

B = birth rate = number of births per unit time per unit population P

D = death rate = number of deaths per unit time per unit population P

M = net migration = number of migrations per unit time per unit population P

General Population Growth ModelBy substitution into

We have:

Death RateNet MigrationBirth Rate

Population Growth

Rate

Exponential Growth Model

Consider if a population has birth, death, and migration rates that remain constant over time. Birth Rate = B Death Rate = D Migration Rate = M

Let , for k a constant Then the population growth rate

equation becomes:


We can now solve for the total population as a function of time, P(t).


Now, consider the Initial Value Problem:For t = 0:

Where P0 is the initial size of the population.


Question 1: At the start of an experiment, there are 100 bacteria. If the bacteria follow an exponential growth pattern with rate k = 0.02:

(a) What will be the size of the population after 5 hours?


Question 1: At the start of an experiment, there are 100 bacteria. If the bacteria follow an exponential growth pattern with rate k = 0.02:

(b) How long will it take for the population to double?


Question 1: But what if k = 0.03?

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350f(x) = 100 exp( 0.03 x )f(x) = 100 exp( 0.0199999999999999 x )

Exponential Growth Model of Bacteria Population

Population Size k=0.02Exponential (Population Size k=0.02)Population Size k=0.03Exponential (Population Size k=0.03)

Time (in hours)

Popula

tion S

ize


Consider exponential model of the population of the United States:

Consider if k = 0.01


Now, consider if k = 0.02

k = 0.02 predicts an extra 13 MILLION people!


But does an exponential growth equation give us an accurate model for population growth?

No. Why?

Populations cannot continue to grow at an exponential rate forever: Availability of resources

Carrying Capacity

Carrying Capacity: “the maximum population size that can be supported by the available resources of the environment”

The logistic model of population growth takes the carrying capacity of an environment into account

Logistic Growth Model

In the logistic growth model:

And the annual increase rP is decreased by a factor of how close the population size is to the carrying capacity (1– P/M)

P = population size; P(t)

r = annual rate of population increase

M = carrying capacity of the environment


The sign of dP/dt changes for different values of P

Value of P dP/dt Behavior of Population

P < 0 dP/dt <0 But does not make sense to analyze for negative population

P = 0 dP/dt = 0 No population growth

0 < P < M dP/dt > 0 Population size is increasing

P = M dP/dt = 0 No population growth

P > M dP/dt < 0 Population size is decreasing


So, when we solve for P(t), we get:

And:


Consider for different values of P0


Question 3: Draw a rough sketch of the graph of P(t) for:

And label the lines of equilibrium on the graph.

0 100 200 300 400 500 600 700 8000

20000

40000

60000

80000

100000

120000

140000

160000

Time (years)

Popula

tion S

ize P(t)

Equilibrium P = 150,000

Equilibrium P = 0P(0) =

500

Comparing Logistic and Exponential Consider for:

P0 = 100 M = 2,000 r = 0.02; k = 0.02 (for each equation

respectively) We have:

Comparing Logistic and Exponential

0 100 200 300 400 500 600 700 800 900 10000

500

1000

1500

2000

2500

3000

Comparing Exponential and Logistic Popu-lation Growth Models

Population Size (Ex-ponential)

Population Size (Lo-gisitc)

Time (years)

Popula

tion S

ize

Comparing Logistic and Exponential So the Logistic Model gives us a more

realistic representation of population growth under the constraint of limited resources

Problems with Mathematical Models

But all mathematical population growth models share one fatal flaw: Assume that future population size is

determined by present and past population sizes only

Mathematical models have no visible connection to the observable measures of human population growth

Ignores the age and sex composition of current populations


Population A =100 million people aged 60 and older

Population B = 100 million people aged 20-45

Birth Rate A < Birth Rate BDeath Rate A > Death Rate B

We would NOT expect Population A and Population B to grow at the same rate


Population C =200,000 men in an isolated gold-rush mining town

Population D = 100,000 men + 100,000 women

Birth Rate C < Birth Rate D

We would NOT expect Population C and Population D to grow at the same rate

Cohort Component Growth Model The demographic composition of a

population matters The Cohort-Component method predicts

future size of each subgroup of a population individually

Used by the U.S. Census Bureau to predict future population size

Cohort Component Growth Model U.S. Census Bureau

1. Base Population2. Plus births to U.S. resident women3. Minus deaths to U.S. residents4. Plus net international migrants

In its most simple form: Pt+1 = Pt+ Bt,t+1 – Dt,t+1+ Mt,t+1

Pt = population at time t;Pt+1 = population at time t+1;Bt,t+1 = births, in the interval from time t+1 to time t;Dt,t+1 = deaths, in the interval from time t+1 to time t; andMt,t+1 = net migration, in the interval from time t+1 to time t

Cohort Component Growth Model

Total Female Population 2005

Total Female Population 2010

P(+75, 2005) P(+75, 2010)

P(70-74, 2005) P(70-74, 2010)

… …

P(25-29, 2005) P(25-29, 2010)

P(20-24, 2005) P(20-24, 2010)

P(15-19, 2005) P(15-19, 2010)

P(10-14, 2005) P(10-14, 2010)

P(5-9, 2005) P(5-9, 2010)

P(0-4, 2005) P(0-4, 2010)Surviving population New births

Female Population


Total Male Population 2005

Total Male Population 2010

P(+75, 2005) P(+75, 2010)

P(70-74, 2005) P(70-74, 2010)

… …

P(25-29, 2005) P(25-29, 2010)

P(20-24, 2005) P(20-24, 2010)

P(15-19, 2005) P(15-19, 2010)

P(10-14, 2005) P(10-14, 2010)

P(5-9, 2005) P(5-9, 2010)

P(0-4, 2005) P(0-4, 2010)

Male Population

Surviving population New births


Surviving population

New births

Si, t+1 = Surviving Population in age group i at time t+1Bi-1, t = Base population in age group i-1 at time tDi-1 = Death rate for age group i-1Mi-1 = Net migration of individuals in age group i-1 between t and t+1

bi = Birth rate for age group i


The total fertility rate for each age group is not the same

1990

2007

http://www.cdc.gov/nchs/data/nvsr/nvsr57/nvsr57_12.pdf

10--14 15--19 20--24 25--29 30--34 35--39 40--44 45--490

20406080

100120140

Total Fertility Rate by Age in 2007

Series1

Series2

Age

Bir

ths p

er

1000 w

om

en

2007



Furthermore, the total fertility rate for each age group is different across races

10--14 15--19 20--24 25--29 30--34 35--39 40--44 45--490

20406080

100120140160

Total Fertility Rate by Age and Race in 2007

Series1

Series2

Age

Bir

ths p

er

1000 w

om

en

White

Black

http://www.cdc.gov/nchs/data/nvsr/nvsr57/nvsr57_12.pdfhttp://www.cdc.gov/nchs/data/statab/t991x07.pdf


http://www.cdc.gov/nchs/data/statab/t991x07.pdf

Cohort Component Growth Model The U.S. Census Bureau calculates the

population for each age, sex, race, and Hispanic origin subgroup

P female, white, non-Hispanic origin, 15-19

P male, black, non-Hispanic origin, 40-44

So the predicted total population at time t+1 is the sum of the predicted populations of all subsets at time t+1

So, as you can imagine, to actually calculate this for a real population gets very complicated!

System Models of Population Growth

System models For the large part, ignore detailed age and sex

composition of population Capture quantitative interactions between population

growth and size and non-demographic factors like industrialization, agriculture, pollution, natural resources

Economic, political, environmental, and cultural factors But even most ambitious efforts so far show that we

don’t have the capabilities/our knowledge is not yet up to the task of modeling population growth in this way

Resource Consumption

Non-Renewable Resources If the fixed amount, S, is consumed at a

constant rate, U, per year The supply will last for S/U years

S/U is called the static reserve Important for:

Gasoline, coal, natural gas


Non-Renewable Resources But what if the fixed amount, S, is not

consumed at a constant rate? The rate of resource consumption is

dependent upon: Population growth Increasing standards of living


Let Uk be the consumption in year k Let consumption increase by a fixed rate.

Let r = 0.05 Then:

So, in year k:

Total usage over the next five years would be:


Let total amount of the resource that has been used by the end of n years

And now set S = A

And after solving for n


Increase

Population

Growth Rate

Increase in U

Decrease in n

Population growth makes a difference!


But what about renewable resources? If:

Natural

Rate of

Replaceme

nt

Rate of

Consumptio

n

The Resource will not run out

But are there too many people?

Malthus Agreed

Thomas Malthus (1766-1834)

“The power of population is indefinitely greater than the power in the earth to produce subsistence for man. Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio. A slight acquaintance with numbers will show the immensity of the first power in comparison with the second."Malthus T.R. 1798. An essay on the principle of population.

Is this guy right?

Developing countries Faster population growth rates than

more developed/industrialized nations 2.8% in Nigeria

135 million in 2007 – 231 million in 2030 Concerns for providing sufficient food

and resources for everyone

Is that guy right?

Social Security

Baby Boom Baby Boom

Is that guy right?

Social Security Dependency Ratio

Trends not just in US: also in Europe and Japan

Aged Dependency Ratio = (Number of people aged 65+)

(Number of people aged 15-64)

Elderly

1950 13.8

1975 19.0

2000 20.8

2025E 31.2

2050E 38.0

Total Dependency Ratio per 100 people of “working age” (15-64)Source: Congressional Research Service, Age Dependency Ratios and Social Security Solvency (2006)

Is that guy right?

But maybe it’s not so bad: Total dependency ratio is what

matters

Elderly Children Total

1950 13.8 58.7 72.5

1975 19.0 63.8 82.8

2000 20.8 48.5 69.3

2025E 31.2 44.3 75.5

2050E 38.0 44.1 82.1

Total Dependency Ratio = (Number of people aged 0-14)+(Number of people aged 65+)

(Number of people aged 15-64)

Total Dependency Ratio per 100 people of “working age” (15-64)Source: Congressional Research Service, Age Dependency Ratios and Social Security Solvency (2006)

We can handle it; it’s been worse!

Is that guy right?

But maybe it’s not so bad: Change in composition of dependency

ratio from change in population structure:

Elderly Children Total

1950 13.8 58.7 72.5

2050E 38.0 44.1 82.1But does supporting one child = one retiree?

Is that guy right?

Social Security cannot pay full benefits in 2042 2018 first year that benefits are larger than payroll

taxes 2028 first year benefits are larger than payroll taxes

plus interest 2028-2042 start spending down surplus 2042 year trust fund assets are exhausted Social

Security not bankrupt (still have payroll taxes coming in) Social Security cannot pay full benefits after this point Can cover maybe 80% of benefits

Disability cannot pay full benefits in 2025 Medicare cannot pay full benefits in 2019

Has already started using interest on surplushttp://www.ssa.gov/OACT/TR/

http://www.ssa.gov/OACT/TR/

Discussion

What are some factors that lead to changes in population growth rates for humans? Urbanization (need fewer children) Education of women Contraception Liberalization of society (women’s role can be out of

the home) Decrease in religiosity (more religious women tend

to have more children) Medical/technological advancements (live longer;

lower infant mortality rates) Lower rates of fertility for men due to pollution

Discussion

What are some implications on the local, national, or international level of changing/high growth rates? WJCC school construction Housing development Environmental destruction (as a result of

increased construction) Overcrowding Resource provision (social welfare, food, water,

etc.) Politics

Sources

COMAP Book http://www.tmt.ugal.ro/crios/Support/AN

PT/Curs/deqn/a1/popmodels/popmodels.html

http://www.census.gov/population/www/projections/cohortcomponentmethod.html

http://en.wikipedia.org/wiki/Logistic_function

http://www.census.gov/Press-Release/www/releases/archives/population/013127.html


http://www.tmt.ugal.ro/crios/Support/ANPT/Curs/deqn/a1/popmodels/popmodels.html












Sources

http://sedac.ciesin.columbia.edu/tg/guide_glue.jsp?rd=pp&ds=5.1

http://www.census.gov/popest/topics/methodology/2008-nat-meth.pdf

http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/P/Populations.html







population growth models and resource consumption

Documents