demographic pvas
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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
Reproduction
Discrete but not binary
Reproduction or growth
Continuous or so
Age or size
Continuous
Logistic regression
Generalized linear models
Linear, polynomial or non-linear regression
Stage
Discrete
Log-linear models
Log-linear models
ANOVAs
P (survival)
0
0.2
0.4
0.6
0.8
1
0 4000 8000 12000
Area of Longest Leaf
P(s
urv
ival
)...
P(survival) (i,t+1)=exp (ßo +ß1*area (i,t) ) /(1+ exp (ßo +ß1*area (i,t)))
Growth
-2
-1
0
1
2
3
4
5
0 2000 4000 6000 8000 10000 12000
Area of longest leaf
gro
wth
rate
..
Area (i,t+1) =Area (i,t)*(1+(exp(ßo +ß1*ln(Area (i,t) ))))
P (flowering)
0
0.2
0.4
0.6
0.8
1
0 4000 8000 12000Area of the longest leaf
P(flo
wer
ing)...
P (flowering) (i,t+1) =exp (ßo +ß1*area (i,t) ) /(1+ exp (ßo +ß1*area (i,t)))
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
height
100806040200-20
1400
1200
1000
800
600
400
200
0
-200
Observed
Linear
Quadratic
Cubic
CUBIC r2 =.701, n= 642 P < .0001 y= 2.8500 -1.5481 x + .0577 x2 + .0010 x3
Classifying individuals
2245 20224 173662N =
AGE
432
HE
IGT
H
60
50
40
30
20
10
0
-10
STAGE
1
2
3
4
7065594429188
Hypericum cumulicola
Age 2-3 different years
STAGE2 * YEAR Crosstabulation
36 1 37
57.1% 11.1% 51.4%
22 4 26
34.9% 44.4% 36.1%
5 4 9
7.9% 44.4% 12.5%
63 9 72
100.0% 100.0% 100.0%
Count
% within YEAR
Count
% within YEAR
Count
% within YEAR
Count
% within YEAR
1
2
3
STAGE2
Total
1998 2000
YEAR
Total
Stage different years same cohortSTAGE3 * STAGE Crosstabulation
35 0 35
60.3% .0% 56.5%
20 2 22
34.5% 50.0% 35.5%
3 2 5
5.2% 50.0% 8.1%
58 4 62
100.0% 100.0% 100.0%
Count
% within STAGE
Count
% within STAGE
Count
% within STAGE
Count
% within STAGE
1.00
2.00
3.00
STAGE3
Total
1 2
STAGE 2
Total
STAGE4 * STAGE3 Crosstabulation
16 1 0 17
44.4% 4.5% .0% 27.0%
13 7 0 20
36.1% 31.8% .0% 31.7%
7 13 4 24
19.4% 59.1% 80.0% 38.1%
0 1 1 2
.0% 4.5% 20.0% 3.2%
36 22 5 63
100.0% 100.0% 100.0% 100.0%
Count
% within STAGE3
Count
% within STAGE3
Count
% within STAGE3
Count
% within STAGE3
Count
% within STAGE3
1.00
2.00
3.00
4.00
STAGE4
Total
1.00 2.00 3.00
STAGE3
Total
Stage different cohorts and yearsSTAGE2 * STAGE1 Crosstabulation
36 0 0 0 36
61.0% .0% .0% .0% 50.7%
20 4 2 0 26
33.9% 66.7% 50.0% .0% 36.6%
3 2 2 2 9
5.1% 33.3% 50.0% 100.0% 12.7%
59 6 4 2 71
100.0% 100.0% 100.0% 100.0% 100.0%
Count
% within STAGE1
Count
% within STAGE1
Count
% within STAGE1
Count
% within STAGE1
1
2
3
STAGE2
Total
1 2 3 4
STAGE1
Total
STAGE4 * STAGE3 Crosstabulation
16 1 0 17
44.4% 4.5% .0% 27.0%
13 7 0 20
36.1% 31.8% .0% 31.7%
7 13 4 24
19.4% 59.1% 80.0% 38.1%
0 1 1 2
.0% 4.5% 20.0% 3.2%
36 22 5 63
100.0% 100.0% 100.0% 100.0%
Count
% within STAGE3
Count
% within STAGE3
Count
% within STAGE3
Count
% within STAGE3
Count
% within STAGE3
1.00
2.00
3.00
4.00
STAGE4
Total
1.00 2.00 3.00
STAGE3
Total
survival Aug98 * Classes 97 Crosstabulation
26 11 15 16 14 82
49.1% 55.0% 37.5% 21.9% 29.8% 35.2%
27 9 25 57 33 151
50.9% 45.0% 62.5% 78.1% 70.2% 64.8%
53 20 40 73 47 233
100.0% 100.0% 100.0% 100.0% 100.0% 100.0%
Count
% within Classes 97
Count
% within Classes 97
Count
% within Classes 97
dead
alive
survivalAug98
Total
seedling vegetative rep <= 33 rep > 33<=50 rep > 50
Classes 97
Total
Chi-Square Tests
14.243a 4 .007
14.331 4 .006
10.043 1 .002
233
Pearson Chi-Square
Likelihood Ratio
Linear-by-LinearAssociation
N of Valid Cases
Value dfAsymp. Sig.
(2-sided)
0 cells (.0%) have expected count less than 5. Theminimum expected count is 7.04.
a.
1031941441582541N =
Classes 99
rep>50
rep> 33<=50
rep <=33
vegetative
seedlings
dead
#rep
ro s
truc
ture
s
3000
2000
1000
0
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 median3. 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
Classes 00 * Classes 99 Crosstabulation
249 0 0 1 8 8 4 270
100.0% .0% .0% 33.3% 18.2% 8.1% 19.0% 55.9%
0 5 0 0 1 1 0 7
.0% 27.8% .0% .0% 2.3% 1.0% .0% 1.4%
0 10 5 1 13 13 0 42
.0% 55.6% 10.2% 33.3% 29.5% 13.1% .0% 8.7%
0 3 41 1 20 57 7 129
.0% 16.7% 83.7% 33.3% 45.5% 57.6% 33.3% 26.7%
0 0 3 0 2 20 10 35
.0% .0% 6.1% .0% 4.5% 20.2% 47.6% 7.2%
249 18 49 3 44 99 21 483
100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0%
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
veg
<= 33
>33 <= 50
> 50
Classes00
Total
dead < 12 cm >=12 veg <= 33 >33 <= 50 > 50
Classes 99
Total
State transition rates
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
a11
a12
a13
a21
a22
a23
a31
a32
a33
A =
A typical projection matrix
0 F2
F3
P21
0
0
0 P32
0
A =
A matrix classified by age
P11
F2 + P12
F3
P21
P22
0
0 P32
P33
A =
A matrix classified by stage
Birth pulse, pre breeding
Census t Census t +1
fi
so
fi*so
Birth pulse, post breeding
Census t Census t +1
sj
sj*fi
Birth flow
Census t Census t +1
√sj
√sj*fi *√so
√so
Actual fertility
Average fertility