Demographic PVAs
Structured populations
• Populations in which individuals differ in
their contributions to population growth
Population projection matrix model
Population projection matrix model
• Divides the population into discrete
classes
• Tracks the contribution of individuals in
each class at one census to all classes in
the following census
States
• Different variables can describe the “state”
of an individual
• Size
• Age
• Stage
Advantages
• Provide a more accurate portray of
populations in which individuals differ in
their contributions to population growth
• Help us to make more targeted
management decisions
Disadvantages
• These models contain more parameters
than do simpler models, and hence require
both more data and different kinds of data
Estimation of demographic rates
• Individuals may differ in any of three
general types of demographic processes,
the so-called vital rates
• Probability of survival
• Probability that it will be in a particular
state in the next census
• The number of offspring it produces
between one census and the next
Vital rates
• Survival rate
• State transition rate (growth rate)
• Fertility rate
The elements in a projection
matrix represent different
combinations of these vital rates
The construction of the stochastic
projection matrix
1. Conduct a detailed demographic study
2. Determine the best state variable upon
which to classify individuals, as well the
number and boundaries of classes
3. Use the class-specific vital rate estimates
to build a deterministic or stochastic
projection matrix model
Conducting a demographic study
• Typically follow the
states and fates of a
set of known
individuals over
several years
• Mark individuals in a
way that allows them
to be re-identified at
subsequent censuses
Ideally
• The mark should be
permanent but should
not alter any of the
organism’s vital rates
Determine the state of each
individual
• Measuring size (weight, height, girth,
number of leaves, etc)
• Determining age
Sampling
• Individuals included in the demographic
study should be representative of the
population as a whole
• Stratified sampling
Census at regular intervals
• Because seasonality is ubiquitous, for
most species a reasonable choice is to
census, and hence project, over one-year
intervals
Birth pulse
• Reproduction concentrated in a small
interval of time each year
• It make sense to conduct the census just
before the pulse, while the number of
“seeds” produced by each parent plant
can still be determined
Birth flow
• Reproduce continuously throughout the year
• Frequent checks of potentially reproductive
individuals at time points within an inter-census
intervals may be necessary to estimate annual
per-capita offspring production or more
sophisticated methods may be needed to
identify the parents
Special procedures
• Experiments
• Seed Banks
• Juvenile dispersal
Data collection should be repeated
• To estimate the variability in the vital rates
• It may be necessary to add new marked
individuals in other stages to maintain adequate
sample sizes
Establishing classes
• Because a projection model categorizes
individuals into discrete classes but some
state variables are often continuous…
• The first step in constructing the model is
to use the demographic data to decide
which state variable to use as the
classifying variable, and
• if it is continuous, how to break the state
variable into a set of discrete classes
Appropriate Statistical tools for testing associations
between vital rates and potential classifying variables
Vital rate
Classifying
variable
Survival or
reproduction
binary
Age or size Logistic
regression
Continuous
Stage
Discrete
Log-linear
models
Reproduction
Discrete but not
binary
Reproduction or
growth
Continuous or
so
Generalized Linear,
linear models polynomial or
non-linear
regression
Log-linear
models
ANOVAs
P (survival)
P(survival) (i,t+1)=exp (ßo +ß1*area (i,t) ) /(1+ exp (ßo +ß1*area (i,t)))
P(survival)...
1
0.8
0.6
0.4
0.2
0
0
4000
8000
Area of Longest Leaf
12000
Growth
Area (i,t+1) =Area (i,t)*(1+(exp(ßo +ß1*ln(Area (i,t) ))))
5
growth rate..
4
3
2
1
0
-1
-2
0
2000
4000
6000
8000
Area of longest leaf
10000 12000
P (flowering)
P (flowering) (i,t+1) =
exp (ßo +ß1*area (i,t) ) /(1+ exp (ßo +ß1*area (i,t)))
P(flowering)...
1
0.8
0.6
0.4
0.2
0
0
4000
8000
Area of the longest leaf
12000
Choosing a state variable
• Apart from practicalities and biological
rules-of-thumb
• An ideal state variable will be highly
correlated with all vital rates for a
population, allowing accurate prediction of
an individual’s reproductive rate, survival,
and growth
• Accuracy of measurement
Number of flowers and fruits
#repro structures
1400
1200
1000
800
600
400
Observed
200
Linear
Quadratic
0
-200
Cubic
-20
0
20
40
60
80
100
height
CUBIC r2 =.701, n= 642 P < .0001 y= 2.8500 -1.5481 x + .0577 x2 + .0010 x3
Classifying individuals
Hypericum cumulicola
60
50
40
30
STAGE
20
1
18
8
29
44
59
65
70
10
2
0
3
-10
4
N =
62
4
36
2
AGE
22
5
3
17
20
24
4
2
Age 2-3 different years
STAGE2 * YEAR Crosstabulation
YEAR
STAGE2
1
2
3
Total
Count
% within YEAR
Count
% within YEAR
Count
% within YEAR
Count
% within YEAR
1998
36
57.1%
22
34.9%
5
7.9%
63
100.0%
2000
1
11.1%
4
44.4%
4
44.4%
9
100.0%
Total
37
51.4%
26
36.1%
9
12.5%
72
100.0%
Stage different years same cohort
STAGE3 * STAGE Crosstabulation
STAGE 2
1
STAGE3
1.00
2.00
3.00
Total
Count
% within STAGE
Count
% within STAGE
Count
% within STAGE
Count
% within STAGE
35
60.3%
20
34.5%
3
5.2%
58
100.0%
2
0
.0%
2
50.0%
2
50.0%
4
100.0%
Total
35
56.5%
22
35.5%
5
8.1%
62
100.0%
STAGE4 * STAGE3 Crosstabulation
1.00
STAGE4
1.00
2.00
3.00
4.00
Total
Count
% within STAGE3
Count
% within STAGE3
Count
% within STAGE3
Count
% within STAGE3
Count
% within STAGE3
16
44.4%
13
36.1%
7
19.4%
0
.0%
36
100.0%
STAGE3
2.00
1
4.5%
7
31.8%
13
59.1%
1
4.5%
22
100.0%
3.00
0
.0%
0
.0%
4
80.0%
1
20.0%
5
100.0%
Total
17
27.0%
20
31.7%
24
38.1%
2
3.2%
63
100.0%
Stage different cohorts and years
STAGE2 * STAGE1 Crosstabulation
STAGE1
1
STAGE2
1
2
3
Total
Count
% within STAGE1
Count
% within STAGE1
Count
% within STAGE1
Count
% within STAGE1
36
61.0%
20
33.9%
3
5.1%
59
100.0%
2
0
.0%
4
66.7%
2
33.3%
6
100.0%
3
0
.0%
2
50.0%
2
50.0%
4
100.0%
4
0
.0%
0
.0%
2
100.0%
2
100.0%
STAGE4 * STAGE3 Crosstabulation
1.00
STAGE4
1.00
2.00
3.00
4.00
Total
Count
% within STAGE3
Count
% within STAGE3
Count
% within STAGE3
Count
% within STAGE3
Count
% within STAGE3
16
44.4%
13
36.1%
7
19.4%
0
.0%
36
100.0%
STAGE3
2.00
1
4.5%
7
31.8%
13
59.1%
1
4.5%
22
100.0%
3.00
0
.0%
0
.0%
4
80.0%
1
20.0%
5
100.0%
Total
17
27.0%
20
31.7%
24
38.1%
2
3.2%
63
100.0%
Total
36
50.7%
26
36.6%
9
12.7%
71
100.0%
Chi-Square Tests
Pears on Chi-Square
Likelihood Ratio
Linear-by-Linear
Ass ociation
N of Valid Cases
Value
14.243a
14.331
10.043
4
4
Asymp. Sig.
(2-s ided)
.007
.006
1
.002
df
233
a. 0 cells (.0%) have expected count les s than 5. The
minimum expected count is 7.04.
survival Aug98 * Classes 97 Crosstabulation
s urvival
Aug98
dead
alive
Total
Count
% within Class es 97
Count
% within Class es 97
Count
% within Class es 97
s eedling
26
49.1%
27
50.9%
53
100.0%
vegetative
11
55.0%
9
45.0%
20
100.0%
Class es 97
rep <= 33 rep > 33<=50
15
16
37.5%
21.9%
25
57
62.5%
78.1%
40
73
100.0%
100.0%
rep > 50
14
29.8%
33
70.2%
47
100.0%
Total
82
35.2%
151
64.8%
233
100.0%
#repro structures
3000
2000
1000
0
N =
1
254
dead
158
vegetative
seedlings
Classes 99
144
194
103
rep> 33<=50
rep <=33
rep>50
An old friend
• AICc = -2(lnLmax,s + lnLmax,f)+
+ (2psns)/(ns-ps-1) + (2pfnf)/(nf-pf-1)
Growth is omitted for two reasons
1. State transitions are idiosyncratic to
the state variable used
2. We can only use AIC to compare
models fit to the same data
Setting class boundaries
• Two considerations
1. We want the number of classes be large
enough that reflect the real differences in
vital rates
2. They should reflect the time individuals
require to advance from birth to
reproduction
Early wedding?!!
Do not use too few classes
More formal
procedures to make
these decisions exist:
Vandermeer 1978,
Moloney 1986
Estimating vital rates
• Once the number and boundaries of
classes have been determined, we can
use the demographic data to estimate the
three types of class-specific vital rates
Survival rates
• For stage:
• Determine the number of individuals that
are still alive at the current census
regardless of their state
• Dive the number of survivors by the initial
number of individuals
Survival rates
• For size or age :
• Determine the number of individuals that
are still alive at the current census
regardless of their size class
• Dive the number of survivors by the initial
number of individuals
• But… some estimates may be based on
small sample sizes and will be sensitive to
chance variation
A solution
•
Use the entire data set to perform a
logistic regression of survival against age
or size
• Use the fitted regression equation to
calculate survival for each class
1. Take the midpoint of each size class for
the estimate
2. Use the median
3. Use the actual sizes
State transition rates
• We must also estimate the probability that
a surviving individual undergoes a
transition from its original class to each of
the other potential classes
State transition rates
Classes 00 * Classes 99 Crosstabulation
Classes
00
dead
veg
<= 33
>33 <= 50
> 50
Total
Count
% within Classes 99
Count
% within Classes 99
Count
% within Classes 99
Count
% within Classes 99
Count
% within Classes 99
Count
% within Classes 99
dead
249
100.0%
0
.0%
0
.0%
0
.0%
0
.0%
249
100.0%
< 12 cm
0
.0%
5
27.8%
10
55.6%
3
16.7%
0
.0%
18
100.0%
>=12
0
.0%
0
.0%
5
10.2%
41
83.7%
3
6.1%
49
100.0%
Classes 99
veg
1
33.3%
0
.0%
1
33.3%
1
33.3%
0
.0%
3
100.0%
<= 33
8
18.2%
1
2.3%
13
29.5%
20
45.5%
2
4.5%
44
100.0%
>33 <= 50
8
8.1%
1
1.0%
13
13.1%
57
57.6%
20
20.2%
99
100.0%
> 50
4
19.0%
0
.0%
0
.0%
7
33.3%
10
47.6%
21
100.0%
Total
270
55.9%
7
1.4%
42
8.7%
129
26.7%
35
7.2%
483
100.0%
Fertility rates
• The average number of offspring that
individuals in each class produce during
the interval from one census to the next
• Stage: imply the arithmetic mean of the
number of offspring produced over the
year by all individuals in a given stage
• Size: use all individuals in the data set
Building the projection matrix
A typical projection matrix
A
=
a11
a12
a13
a21
a22
a23
a31
a32
a33
A matrix classified by age
A
=
0
F2
F3
P21
0
0
0
P32
0
A matrix classified by stage
A
=
P11
F2 + P12
F3
P21
P22
0
0
P32
P33
Birth pulse, pre breeding
fi
so
Census t
fi*so
Census t +1
Birth pulse, post breeding
sj*fi
sj
Census t
Census t +1
Birth flow
√sj*fi *√so
Average fertility
√sj
Census t
√so
Actual fertility
Census t +1