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. Vital rate Classifying variable Age or size Continuous Stage Discrete
Appropriate Statistical tools for testing associations between vital rates and potential classifying variables
Vital rate
Classifying variable
Survival or reproduction binary
Reproduction
Discrete but not binary
Reproduction or growth
Continuous or so
Age or size
Continuous
Logistic regression
Generalized linear models
Linear, polynomial or non-linear regression
Stage
Discrete
Log-linear models
ANOVAs

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 < y= x x 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. A typical projection matrix

43. A matrix classified by ageF2 F3
P21
P32
A =

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

45. Birth pulse, pre breedingfi
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 √sj √so Average fertility 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

51. Limitations of Matrix selection
The assumption that the precise combinations of values that we observed the limited duration of a demographic study will always occur is unlikely to be correct.

52. A more realistic approach
Use the means, variances, and correlations between vital rates, and then simulate a broader range of possible values

53. The problem of negative correlations
A hypothetical individual is currently in size class 3 and has mean probability s3=0.95 of surviving for one year.
If it survives it will either stay the same size, or grow to be in size class 4 with mean probability g4,3=0.10
a33=s3(1-g43)=(0.95)(1-0.10) and
a43=s3g43=(0.95)(0.10)

54. The Desert Tortoise

55. Size classes and definitions of matrix elements for the desert tortoise assuming a prebreeding census
Class 1 2 3 4 5 6 7
Yearling f5 f6 f7
Juvenile 1 s2
s2(1-g2)
Juvenile 2
s2g2
Immature
s3(1-g3)
s3g3
s4(1-g4)
Subadult
s4g4
s5(1-g5)
Adult 1
s5g5
s6(1-g6)
Adult 2
s6g6 s7

56. Estimated vital rates Growth Survival Class 1970 1980e 1980l Mean Var2 .5
0.5
0.33
0.083
.63 1
.65
.76
.044 3
0.18
0.177
0.28
0.036
.91
.98
.96
.002 4
.47
0.067
0.065
.59
.81
.79
.039 5
.23
0.26
0.16
0.020
.92
.0018 6
.063
0.032
0.001
.99
.68
.89
.034 7
.78 .8
.86
.015

57. Pearson Correlations Growth1
.469
.382
.995
-.597
.429
.514
-0.014
.877
.919
.811

58. Pearson Correlations Survival1
.726
-.911
-.945
-.946
-.465
.729
.487
-.247
-.083
-.743
.997
.778
-.941
-.918
.417

59. Pearson Correlations Survival-Growth1
-.704
.898
.956
-.514
-.994 g3
.496
-.957
.810
.189
.516
-.563 g4
-.41
-.925
.75
.094
.597
-.481 g5
.571
-.149
-.182
-.806
.995
.505 g6
-0.017
-.7
.428
-.307
.865
-.096

60. Key distributions for vital rates
0.5,0.001
0.5,0.2
0.5,0.01
The beta distribution

61. Key distributions for vital rates
The beta distribution

62. Lognormal

63. Stretched Beta

64. Matrix selection
Element selection
Vital rate selection