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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

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Demographic PVAs

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  1. Demographic PVAs

  2. Structured populations • Populations in which individuals differ in their contributions to population growth

  3. Population projection matrix model

  4. 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

  5. States • Different variables can describe the “state” of an individual • Size • Age • Stage

  6. 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

  7. Disadvantages • These models contain more parameters than do simpler models, and hence require both more data and different kinds of data

  8. 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

  9. Vital rates • Survival rate • State transition rate (growth rate) • Fertility rate The elements in a projection matrix represent different combinations of these vital rates

  10. 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

  11. 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

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

  13. Determine the state of each individual • Measuring size (weight, height, girth, number of leaves, etc) • Determining age

  14. Sampling • Individuals included in the demographic study should be representative of the population as a whole • Stratified sampling

  15. Census at regular intervals • Because seasonality is ubiquitous, for most species a reasonable choice is to census, and hence project, over one-year intervals

  16. 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

  17. 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

  18. Special procedures • Experiments • Seed Banks • Juvenile dispersal

  19. 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

  20. 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

  21. Appropriate Statistical tools for testing associations between vital rates and potential classifying variables

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

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

  24. P (flowering) P (flowering) (i,t+1) = exp (ßo +ß1*area (i,t) ) /(1+ exp (ßo +ß1*area (i,t)))

  25. 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

  26. Number of flowers and fruits CUBIC r2 =.701, n= 642 P < .0001 y= 2.8500 -1.5481 x + .0577 x2 + .0010 x3

  27. Classifying individuals Hypericum cumulicola

  28. Age 2-3 different years

  29. Stage different years same cohort

  30. Stage different cohorts and years

  31. 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

  32. 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

  33. Early wedding?!! Do not use too few classes More formal procedures to make these decisions exist: Vandermeer 1978, Moloney 1986

  34. 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

  35. 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

  36. 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

  37. 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

  38. 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

  39. State transition rates

  40. 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

  41. Building the projection matrix

  42. a13 a11 a12 a21 a22 a23 a31 a32 a33 A typical projection matrix A =

  43. F3 0 F2 P21 0 0 0 P32 0 A matrix classified by age A =

  44. A matrix classified by stage F3 P11 F2 + P12 A = P21 P22 0 0 P32 P33

  45. Birth pulse, pre breeding fi fi*so so Census t Census t +1

  46. Birth pulse, post breeding sj*fi sj Census t Census t +1

  47. Birth flow √sj*fi *√so Average fertility √sj √so Actual fertility Census t Census t +1

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