Demographic PVAs

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


sj

sj*fi

Birth flow


√sj

√sj*fi *√so

√so

Actual fertility

Average fertility

Documents

Demographic PVAs