Stochastic Projection Matrix Modeling for Demographic Studies

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

32. 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

33. 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

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

48. Demographic PVA’s Based on vital rates

49. Basic types of vital rates • Fertility rates • Survival rates • State transition, or growth rates

50. The estimation of Vital rates • Accurate estimation of variance and correlation in the demographic rates • We need to know: • The mean value for each vital rate • The variability in each rate • The covariance or correlation between each pair of rates