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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

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Structured populations
Structured populations

  • Populations in which individuals differ in their contributions to population growth



Population projection matrix model1
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
States

  • Different variables can describe the “state” of an individual

  • Size

  • Age

  • Stage


Advantages
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
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
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
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
The construction of the stochastic projection matrix

  • Conduct a detailed demographic study

  • Determine the best state variable upon which to classify individuals, as well the number and boundaries of classes

  • Use the class-specific vital rate estimates to build a deterministic or stochastic projection matrix model


Conducting a demographic study
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
Ideally

  • The mark should be permanent but should not alter any of the organism’s vital rates


Determine the state of each individual
Determine the state of each individual

  • Measuring size (weight, height, girth, number of leaves, etc)

  • Determining age


Sampling
Sampling

  • Individuals included in the demographic study should be representative of the population as a whole

  • Stratified sampling


Census at regular intervals
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
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
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
Special procedures

  • Experiments

  • Seed Banks

  • Juvenile dispersal


Data collection should be repeated
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
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


P survival
P (survival) between vital rates and potential classifying variables

P(survival) (i,t+1)=exp (ßo +ß1*area (i,t) ) /(1+ exp (ßo +ß1*area (i,t)))


Growth
Growth between vital rates and potential classifying variables

Area (i,t+1) =Area (i,t)*(1+(exp(ßo +ß1*ln(Area (i,t) ))))


P flowering
P (flowering) between vital rates and potential classifying variables

P (flowering) (i,t+1) =

exp (ßo +ß1*area (i,t) ) /(1+ exp (ßo +ß1*area (i,t)))


Choosing a state variable
Choosing a state variable between vital rates and potential classifying variables

  • 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
Number of flowers and fruits between vital rates and potential classifying variables

CUBIC r2 =.701, n= 642 P < .0001 y= 2.8500 -1.5481 x + .0577 x2 + .0010 x3


Classifying individuals
Classifying individuals between vital rates and potential classifying variables

Hypericum cumulicola


Age 2 3 different years
Age 2-3 different years between vital rates and potential classifying variables


Stage different years same cohort
Stage different years same cohort between vital rates and potential classifying variables


Stage different cohorts and years
Stage different cohorts and years between vital rates and potential classifying variables


An old friend
An old friend between vital rates and potential classifying variables

  • AICc = -2(lnLmax,s + lnLmax,f)+

    + (2psns)/(ns-ps-1) + (2pfnf)/(nf-pf-1)

  • Growth is omitted for two reasons

  • State transitions are idiosyncratic to the state variable used

  • We can only use AIC to compare models fit to the same data


Setting class boundaries
Setting class boundaries between vital rates and potential classifying variables

  • Two considerations

  • We want the number of classes be large enough that reflect the real differences in vital rates

  • They should reflect the time individuals require to advance from birth to reproduction


Early wedding
Early wedding?!! between vital rates and potential classifying variables

Do not use too few classes

More formal procedures to make these decisions exist:

Vandermeer 1978,

Moloney 1986


Estimating vital rates
Estimating vital rates between vital rates and potential classifying variables

  • 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
Survival rates between vital rates and potential classifying variables

  • 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 rates1
Survival rates between vital rates and potential classifying variables

  • 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
A solution between vital rates and potential classifying variables

  • 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

  • Take the midpoint of each size class for the estimate

  • Use the median

  • Use the actual sizes


State transition rates
State transition rates between vital rates and potential classifying variables

  • 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 rates1
State transition rates between vital rates and potential classifying variables


Fertility rates
Fertility rates between vital rates and potential classifying variables

  • 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
Building the projection matrix between vital rates and potential classifying variables


A typical projection matrix

a between vital rates and potential classifying variables13

a11

a12

a21

a22

a23

a31

a32

a33

A typical projection matrix

A =


A matrix classified by age

F between vital rates and potential classifying variables3

0

F2

P21

0

0

0

P32

0

A matrix classified by age

A =


A matrix classified by stage
A matrix classified by stage between vital rates and potential classifying variables

F3

P11

F2 + P12

A =

P21

P22

0

0

P32

P33


Birth pulse pre breeding
Birth pulse, pre breeding between vital rates and potential classifying variables

fi

fi*so

so

Census t

Census t +1


Birth pulse post breeding
Birth pulse, post breeding between vital rates and potential classifying variables

sj*fi

sj

Census t

Census t +1


Birth flow1
Birth flow between vital rates and potential classifying variables

√sj*fi *√so

Average fertility

√sj

√so

Actual fertility

Census t

Census t +1


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