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Population Growth Models and Resource consumption. Caitlin ThomasNovember 12, 2009. Presentation Overview. Population Growth Models General Population Growth Model Mathematical Models Exponential Growth Model Logistic Growth Model Cohort-Component Model Systems Models - PowerPoint PPT Presentation
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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:
Exponential Growth Model
We can now solve for the total population as a function of time, P(t).
Exponential Growth Model
Now, consider the Initial Value Problem:For t = 0:
Where P0 is the initial size of the population.
Exponential Growth Model
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?
Exponential Growth Model
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?
Exponential Growth Model
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
Exponential Growth Model
Consider exponential model of the population of the United States:
Consider if k = 0.01
Exponential Growth Model
Now, consider if k = 0.02
k = 0.02 predicts an extra 13 MILLION people!
Exponential Growth Model
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
Logistic Growth Model
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
Logistic Growth Model
So, when we solve for P(t), we get:
And:
Logistic Growth Model
Consider for different values of P0
Logistic Growth Model
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
Problems with Mathematical Models
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
Problems with Mathematical Models
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
Cohort Component Growth Model
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
Cohort Component Growth Model
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
Cohort Component Growth Model
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
Cohort Component Growth Model
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
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
Resource Consumption
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
Resource Consumption
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:
Resource Consumption
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
Resource Consumption
Increase
Population
Growth Rate
Increase in U
Decrease in n
Population growth makes a difference!
Resource Consumption
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/
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.ssa.gov/OACT/TR/
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