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Demographic PVAs
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Structured populations
Populations in which individuals differ in their contributions to population growth
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Population projection matrix model
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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
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States Different variables can describe the “state” of an individual
Size Age Stage
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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
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Disadvantages These models contain more parameters than do simpler models, and hence require both more data and different kinds of data
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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
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Vital rates Survival rate State transition rate (growth rate) Fertility rate The elements in a projection matrix represent different combinations of these vital rates
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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
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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
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Ideally The mark should be permanent but should not alter any of the organism’s vital rates
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Determine the state of each individual
Measuring size (weight, height, girth, number of leaves, etc) Determining age
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Sampling Individuals included in the demographic study should be representative of the population as a whole Stratified sampling
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Census at regular intervals
Because seasonality is ubiquitous, for most species a reasonable choice is to census, and hence project, over one-year intervals
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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
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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
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Special procedures Experiments Seed Banks Juvenile dispersal
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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
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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
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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
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P (survival) P(survival) (i,t+1)=exp (ßo +ß1*area (i,t) ) /(1+ exp (ßo +ß1*area (i,t)))
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Growth Area (i,t+1) =Area (i,t)*(1+(exp(ßo +ß1*ln(Area (i,t) ))))
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P (flowering) P (flowering) (i,t+1) =
exp (ßo +ß1*area (i,t) ) /(1+ exp (ßo +ß1*area (i,t)))
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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
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Number of flowers and fruits
CUBIC r2 =.701, n= 642 P < y= x x x3
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Classifying individuals
Hypericum cumulicola
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Age 2-3 different years
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Stage different years same cohort
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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
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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
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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
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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
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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
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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
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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
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State transition rates
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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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Building the projection matrix
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A typical projection matrix
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A matrix classified by age
F2 F3 P21 P32 A =
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A matrix classified by stage
P11 F2 + P12 A = P21 P22 P32 P33
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Birth pulse, pre breeding
fi fi*so so Census t Census t +1
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Birth pulse, post breeding
sj*fi sj Census t Census t +1
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Birth flow √sj*fi *√so √sj √so Average fertility Actual fertility
Census t Census t +1
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Demographic PVA’s Based on vital rates
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Basic types of vital rates
Fertility rates Survival rates State transition, or growth rates
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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
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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.
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A more realistic approach
Use the means, variances, and correlations between vital rates, and then simulate a broader range of possible values
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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)
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The Desert Tortoise
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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
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Estimated vital rates Growth Survival Class 1970 1980e 1980l Mean Var
2 .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
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Pearson Correlations Growth
1 .469 .382 .995 -.597 .429 .514 -0.014 .877 .919 .811
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Pearson Correlations Survival
1 .726 -.911 -.945 -.946 -.465 .729 .487 -.247 -.083 -.743 .997 .778 -.941 -.918 .417
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Pearson Correlations Survival-Growth
1 -.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
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Key distributions for vital rates
0.5,0.001 0.5,0.2 0.5,0.01 The beta distribution
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Key distributions for vital rates
The beta distribution
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Lognormal
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Stretched Beta
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Matrix selection Element selection Vital rate selection
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