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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
  • Different variables can describe the “state” of an individual
  • Size
  • Age
  • Stage
  • 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
  • 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
  • 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
  • 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)

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


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

p flowering
P (flowering)

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
  • 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

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

classifying individuals
Classifying individuals

Hypericum cumulicola

an old friend
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
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
Early wedding?!!

Do not use too few classes

More formal procedures to make these decisions exist:

Vandermeer 1978,

Moloney 1986

estimating vital rates
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
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 rates1
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
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
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
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
a typical projection matrix










A typical projection matrix

A =

a matrix classified by stage
A matrix classified by stage



F2 + P12

A =







birth pulse pre breeding
Birth pulse, pre breeding




Census t

Census t +1

birth pulse post breeding
Birth pulse, post breeding



Census t

Census t +1

birth flow1
Birth flow

√sj*fi *√so

Average fertility



Actual fertility

Census t

Census t +1