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## PowerPoint Slideshow about ' Demographic PVAs' - arnie

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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)

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

Growth

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

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

Classifying individuals

Hypericum cumulicola

An old friend

- 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

- 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?!!

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
- Take the midpoint of each size class for the estimate
- Use the median
- 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

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

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