Unformatted text preview: Age Structures
Age Growth in populations with complex life
Growth
histories and long lives, such as humans
and whitetailed deer, is determined by a
number of processes:
number
1. Agespecific survivorship
2. Agespecific fertility
3. Generation time
4. Actual numbers of individuals in each age
Actual
class
class Age Structures
Age The following example might fix in your mind
The the potential importance of age distribution on
population growth. Imagine a group of several thousand young
Imagine
people attending a concert by Daddy Yankee
Daddy
on an island off the coast of California.
on Daddy Yankee!
Daddy Suddenly a catastrophic series of earthquakes eliminates the entire local population, leaving only the concertgoers! Age Structures
Age Simultaneously, California splits
Simultaneously, off from the rest of North
America. Assume further that a few of the
Assume concertgoers are able to
colonize this California island,
and future population growth is
now based on this group! Age Structures
Age Most of the
Most concertgoers would
obviously have been
teenage girls (assume
teenage
a few teenage boys
also were dragged
along). Assuming an
Assuming abundance of food, this
California island would
rapidly be repopulated. Age Structures
Age Growth would, however, be irregular, since all
Growth of the girls would be the same age and reach
menopause more or less simultaneously in
the future. Growth would then slow until their daughters
Growth
began to reproduce.
began Age Structures
Age The Future:
The Rapid,
Rapid, but irregular population
growth.
growth. Age Structures
Age Now imagine the same scenario, except that
Now survivors of the disaster were individuals
attending an AARP (American Association of
Retired Persons) convention. Presumably, all or almost all, of the females
Presumably, at the convention are over 55. AARP Convention Concert Starring:
Tony Bennett
Tony Age Structures
Age What would the future of the California island
What population be in this case? There is no future! And they don’t want to be
There young again!
young Age Structures
Age Take home lesson: Age
Age distribution can contribute to both
rapid growth and extinction of a
population!
population! Life Tables
Life Information on agespecific survivorship and
Information fertility is put together into a life table.
life The probability of living to different age
The
classes (x) is known as lx.
classes The number of female offspring per female in
The the population is known as mx.
the Survivorship
Survivorship Survivorship Type Curves:
1. Death at Senescence (Type I)
2. Equal Probability of Death at All Ages (Type II)
3. High Juvenile Mortality (Type III) Type I
Type II, Constant Number
Type II, Constant Probability
Type III 1 Survivorship, lx 0.75 0.5 0.25 0
0 2 4 6 Age classes 8 10 Type I
Type II, Constant Number
Type II, Constant Probability Log survivorship, Sx 1000 Type III 100 10 1
0 2 4 6 Age classes 8 10 Arithmetic Scale
1
0.8 lx 0.6
0.4
0.2
0
0 10 20 30 40 50
Age Clases 60 70 80 90 100 Log lx
3
2.5 Log lx 2
1.5
1
0.5
0
0 20 40 60
Age Classes 80 100 Number surviving per 1000 (Sx) US Males 1985
US Males 1910
Carlise, UK 1782
Northhampton, UK 1780 1000
750
500
250
0
0 25 50
75
Age classes 100 US Males 1985
US Males 1920
Carlisle, UK 1782
Northhampton, UK 1780 3
Log Survivorship 2.5
2
1.5
1
0.5
0
0 20 40 60 Age Classes 80 100 Age Class
95 85 75 65 55 45 35 25 15 9 9
9 8
9 7
9 6
9 5
9 4
9 3
9 2
9 1
9 5 <1 Log of Survivorship (lx) H uman Survivorship 2010 3.5 3 2.5 2
Males 1.5
Females 1 0.5 0 Proportion alive at beginning of age interval
(lx) Dall Sheep
Dall
1.000 0.800 0.600 0.400 0.200 0.000
0 2 4 6 8 Age classes 10 12 14 Dall Sheep
Dall
3 Log survivorship 2.5
2
1.5
1
0.5
0
0 2 4 6
8
Age classes 10 12 14 WhiteCrowned Sparrows
WhiteCrowned
3
2.5
Log survivorship July 1966
2 August 1966 1.5
1
0.5
0
0 1 2 3
Age Classes 4 5 6 Gray Squirrel
Gray
1 Survivorship (l
x) 0.75 0.5 0.25 0
0 1 2 3 4
5
Age classes 6 7 8 Golden Lion Tamarins
Golden Survivorship, lx 1 0.75 0.5 0.25 0
0 2 4 6
Age classes 8 10 12 Phlox
Phlox
1 Survivorship, l x 0.75 0.5 0.25 0
1 2 3 4 5 6 Age in days 7 8 9 10 Fertility
Fertility The other half of the life table is the fertility
The column, mx.
column Here each value represents the average
Here number of female offspring produced per
female of a given age. Again, gathering accurate data on fertility in
Again,
the field is problematic for many populations. Fertility
Fertility In order to simplify calculations, we count only
In the number of females. Fertility, like survivorship, can be graphed as
Fertility, a function of age and the resultant fertility
curve is usually triangular or rectangular in
shape. Gray Squirrel
Gray
2.5 Fertility mx 2 1.5
1 0.5 0
0 1 2 3 4
Age Class 5 6 7 8 Mean number of daughters per female Normal Human Fertility Curve
Normal
0.6
0.5
0.4
0.3
0.2
0.1
Reproductive 0
0 10 20 30 PreReproductive Age classes 40 50 60 PostReproductive 70 80 Population Parameters
Population The Gross Reproductive Rate is the
The Gross average number of female offspring per
female who lives an entire lifespan. This
figure is 14.96 for the gray squirrel.
figure GRR = ∑m x Conditional Survivorship, px
Conditional From the lx column we can develop two parallel columns, which provide information
on how survivorship and mortality rates
change with age. The lx column is based on the probability, at
birth, of surviving to a given age class.
birth The px column, by contrast, is the ageagespecific probability of surviving to the next
of
age class. Conditional Survivorship, px
Conditional That is, p2 tells us the probability that an individual, who has survived to the age of
two, will survive to be three years old. These px values are critically important when
we want to project future population growth
as will become clear later. The Formula for px
The l x +1
px =
lx Conditional Probability of Death
Conditional While px is the conditional probability of surviving to another age class, its opposite, qx
surviving
is the conditional probability of death in the
next age interval. Therefore qx is simply 1 – px Mortality Curves
Mortality Just as there are typical survivorship curves,
Just there are also typical mortality curves.
there One such curve for mammals is the Ushaped
One
curve, which emphasizes that death occurs
mainly in the very young and the very old age
classes.
classes. Human Mortality Curve for Flu of 1918 Male versus Female Mortality
Male There are often some important differences
There between male and female qx curves.
between
curves. Males have a “bump” up in the subadult to
Males early adult age classes.
early Why? Male versus Female Mortality
Male In many species, young males are forced
In from their homes.
from If they wish to breed they must find and join a
If
new group (male dispersal).
new Once they find a new group, in order to breed
Once
they must often engage in real or ritualized
combat with established males. Male versus Female Mortality
Male Both dispersal and combat with other males leads to
Both an increase in mortality in the subadult and early
adult age classes.
adult Mortality Curves for Impalas
Mortality Mortality rate (1000x)
q 1000
Females
Males 800
600
400
200
0
0 1 3 6 8 12 Age in years 17 23 30 qx for Humans
for Mortality rate (1000qx) 3
2.5
2
1.5
Males
1 Females 0.5
0
0 10 25 40
Age classes 55 70 85 Mortality versus age
1.2 Age specific Mortality, qx 1
0.8
0.6
0.4
0.2
0
0 20 40 60
Age Classes 80 100 H uman Mortality 2010
1.200 1.000 Males 0.600 Females 0.400 0.200 Age Class 99
95  89
85  79
75 65  69 59
55  49
45  39
35 25 15  19 9
5  1 29 0.000
< Mortal i ty 0.800 Doubling Time
Doubling Derivation: Given: Nt = N0ert We have: Nt/N0 = ert For the population to double we have:
2 = Nt/N0 = ert
Take the natural log of 2 and ert. Doubling Time
Doubling We have: 0.693 = rt Solve for t, which is now doubling time, and
Solve we have:
we
Doubling time = 0.693/r Doubling Time
Doubling Doubling Time = 0.693/r Or if r is negative we have the time for the
Or population to reach half its present size
population
= 0.693/r Population Parameters
Population The Net Reproductive Rate equals the
The Net mean number of female offspring produced
per female per generation. This is 1.19 for
the squirrel population.
the R0 = ∑ l x m x Life Tables
Life As above, the finite rate of increase is the
As ratio of populations at time t and time t+1.
ratio N t +1
λ=
Nt Life Tables
Life Over the long term, the survivorship and
Over fertility values determine the generation
time, G, the net reproductive rate and the
time,
the
intrinsic rate of increase, r. We can also show that: λ = er Life Tables
Life Given: N t +1
λ=
Nt and for time zero to time 1 λ = N1/N0
t And given: Nt = N0errt and
and From t =0 to t =1, we have
N1 = N0er
which becomes N1/N0 = er We have λ = er
We Life Tables
Life However, over the short term, λ is determined
However, by the present number of individuals in the
age classes.
age Thus λ fluctuates over time until a population
Thus reaches a stable age distribution. At that
stable
At
point it equals er.
point Life Tables
Life A stable age distribution is reached if the
stable mortality and fertility values in the life table
remain constant for several generations.
remain In a stable age distribution, the proportion
In
proportion (not the numbers) of the population belonging
to the different age classes becomes
constant.
constant. Life Tables
Life That is, for example 20% of the population
That belongs to oneyear old individuals. This proportion remains a constant, though
This the actual numbers can change if the
population is growing.
population Review: Relationships among Growth
Parameters
Parameters Ignoring age distributions:
1. Nt+1/Nt = λ 2. Nt+1/Nt = er 3. λ = er
lln λ = r
n 4. Finding the Intrinsic Rate of Increase
(r) from a Life Table
(r) To find r from life history data we use the
To Euler equation
Euler 1 = ∑lxmxerx The problem is that r is an exponent. With more than two age classes, this is
With difficult to solve explicitly.
difficult Continuous Growth Models
Continuous Recall that in the continuous growth model
Recall we use a differential equation:
we
dN/dt = rN Where r = the intrinsic rate of increase and
Where
the r=bd Solved form:
Solved Nt = N0 ert Finding the Intrinsic Rate of Increase
(r) from a Life Table
(r) Where G is generation time we have:
G
NG = N0errG This is growth per generation. Thus:
NG/N0 = erG Recall that R0 is also growth per generation.
is
NG/N0 = erG = R0 . Take natural logs:
Take
lnR0 = rG
lnR
and
r = lnRo/G
and Finding the Intrinsic Rate of Increase
(r) from a Life Table
(r) There is a simple estimation for generation
There time, from which r can be estimated.
time, There is also an optimization function in Excel
There that allows a direct solution for r.
that Age Structure
Age The age distribution is based on the
The proportion belonging to each age class.
proportion If we have data on the number of individuals
If per age class, nx, we can calculate the
per
we
proportions, symbolized as cx
proportions,
Cx = nx/N where N = ∑nx Age Structure
Age
Age Class nx cx 0 120 0.48 1 60 0.24 2 40 0.16 3 30 0.12 4 0 0 ∑ 250 1.00 Stable Age Distribution
Stable Whenever survivorship and fertility are
Whenever survivorship
fertility constant for a sufficient period of time, the
population moves to a unique stable age
distribution, based on the particular
distribution based
combination of survivorship and fertility
values of the population. Stable Age Distribution
Stable No matter what the original age distribution,
No the population moves to its unique stable age
distribution unless the survivorship or fertility
distribution
schedules change.
schedules Stable Age Distribution
Stable This stable age distribution is predictable, based on
This specific equations with r and lx the independent
specific
variables. This is an age structure diagram.
This Age Structure Diagrams Stable Age Distribution
Stable Again: When a population is in the stable age
When distribution, the entire population and all of
the age classes grow or decline at the same
rate :
rate
λ = er And:
r = lnλ Population Projections
Population In this exercise we will
In learn how to project a
population with an age
structure and also
demonstrate how
quickly a population
moves to its stable age
distribution.
distribution. Population Projections
Age, x
Age, lx mx px qx lxmx 0 1.00 0 0.50 0.50 0 1 0.50 2.0 0.40 0.60 1.0 2 0.20 1.0 0.50 0.50 0.2 3
4 0.10
0 1.0
0 0
 1.00
 0.1
 Sums GRR = 4.0 R0 = 1.3
1.3 Population Projections
Age, x
Age, nx
t=0 Cx
t=0 nx
t=1 Cx
t=1 0 200 1.00 200 200/300 =
200/300
0.67
0.67 1 0 0 100 100/300 =
0.33 2 0 0 0 0 3
4 0
0 0
0 0
0 0
0 Sums 200 1.00 300 1.00 Population Projections
Population
λ = N1/N0 = 300/200 = 1.50 Population Projections
Age, x
Age, nx
t=1 Cx
t=1 nx
t=2 Cx
t=2 0 200 0.67 240 0.63 1 100 0.33 100 0.26 2 0 0 40 0.11 3
4 0
0 0
 0
0 0
 Sums 300 1.00 380 1.00 Population Projections
Population N0 = (100*2) +(40*1) = 240
(100*2) λ = N1/N0 = 300/200 = 1.50 λ = N2/N1 = 380/300 = 1.27 Population Projections
Age, x
Age, nx
t=2 Cx
t=2 nx
t=3 Cx
t=3 0 240 0.63 300 0.625 1 100 0.26 120 0.25 2 40 0.11 40 0.083 3
4 0
0 0
0 20
0 0.042
0 Sums 380 1.00 480 1.00 Population Projections
Population N0 = (120*2) +(40*1) + (20*1) = 300
(120*2) λ = N1/N0 = 300/200 = 1.50 λ = N2/N1 = 380/300 = 1.27 λ = N3/N2 = 480/380 = 1.26
480/380 λ = N4/N3 = 586/480 = 1.22
586/480 λ = 1.23 and r = 0.206 at stable age distribution
0.206 Population Projections
Age, x
Age, nx
t=3 Cx
t=3 Cx
At Stable
At
Age Distribution 0 300 0.625 0.625 1 120 0.250 0.258 2
3 40
20 0.083
0.042 0.082
0.035 4
Sums 0 0
1.00 0
1.00 480 Sample Problem
Sample
Age, x lx mx px qx nx
t=0 0 1.000 0 1000 1 0.400 1.0 300 2 0.200 3.0 100 3 0.150 1.0 60 4 0 0 0 nx
t=1 Sample Problem
Sample Find the gross reproductive rate, the net
Find
gross
the reproductive rate, px and qx. Would r be
reproductive
predicted to be positive, negative or zero?
predicted Given the nx column at age zero, project the population for one year.
population Sample Problem
Sample
Age, x lx mx px qx nx
t=0 0 1.000 0 0.40 0.60 1000 1 0.400 1.0 0.50 0.50 300 2 0.200 3.0 0.75 0.25 100 3 0.150 1.0 0 1.00 60 4 0 0   0 nx
t=1 Sample Problem
Age,
Age,
x lx mx px qx nx
t=0 nx
t=1 lxmx 0 1.000 0 0.40 0.60 1000 925 0 1 0.400 1.0 0.50 0.50 300 400 0.40 2 0.200 3.0 0.75 0.25 100 150 0.60 3 0.150 1.0 0 1.00 60 75 0.15 4 0 0   0 0 0 Sums GRR
=
5.0 1460 1550 R0 =
1.15 Got it?? ...
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This note was uploaded on 01/23/2012 for the course BIOL/EVPP 307 taught by Professor Crerar during the Summer '11 term at George Mason.
 Summer '11
 Crerar

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