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 Tracks the contribution of individuals in each class at one census to all classes in the following census

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

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

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

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 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 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 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

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

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

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

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 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 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 Experiments Seed Banks Juvenile dispersal

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 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

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

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

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

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

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

Classifying individuals Hypericum cumulicola

Age 2-3 different years

Stage different years same cohort

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 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

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 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 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 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 We must also estimate the probability that a surviving individual undergoes a transition from its original class to each of the other potential classes

State transition 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

Building the projection matrix

A typical projection matrix

A matrix classified by age F2 F3 P21 P32 A =

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

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

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

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

Demographic PVA’s Based on vital rates

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

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

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.

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

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)

The Desert Tortoise

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

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

Pearson Correlations Growth 1 .469 .382 .995 -.597 .429 .514 -0.014 .877 .919 .811

Pearson Correlations Survival 1 .726 -.911 -.945 -.946 -.465 .729 .487 -.247 -.083 -.743 .997 .778 -.941 -.918 .417

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

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

Key distributions for vital rates The beta distribution

Lognormal

Stretched Beta

Matrix selection Element selection Vital rate selection