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Structured populations

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

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

- 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

- 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

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 between vital rates and potential classifying variables

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

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 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 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 between vital rates and potential classifying variables

Hypericum cumulicola

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

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

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

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 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?!! 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 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 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 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 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 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 rates between vital rates and potential classifying variables

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 between vital rates and potential classifying variables

a between vital rates and potential classifying variables13

a11

a12

a21

a22

a23

a31

a32

a33

A typical projection matrixA =

F between vital rates and potential classifying variables3

0

F2

P21

0

0

0

P32

0

A matrix classified by ageA =

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 between vital rates and potential classifying variables

fi

fi*so

so

Census t

Census t +1

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

sj*fi

sj

Census t

Census t +1

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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