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 0 1 2 3 4 5 6 7Yearling 0 f5 f6 f7

Juvenile 1 1 s2 s2(1-g2)

Juvenile 2 2 s2g2 s2(1-g2)

Immature 3 s2g2 s3(1-g3)

Immature 4 s3g3 s4(1-g4)

Subadult 5 s4g4 s5(1-g5)

Adult 1 6 s5g5 s6(1-g6)

Adult 2 7 s6g6 s7

Estimated vital rates Growth Survival

Class 1970 1980e 1980l Mean Var 1970 1980e 1980l Mean Var

2 .5 0 0.5 0.33 0.083 .63 1 .65 .76 .044

3 .5 0.18 0.177 0.28 0.036 .91 1 .98 .96 .002

4 .47 0.067 0 0.18 0.065 .98 .59 .81 .79 .039

5 .23 0.26 0 0.16 0.020 .98 .92 1 .96 .0018

6 .063 0.032 0 0.032 0.001 .99 1 .68 .89 .034

7 .78 1 .8 .86 .015

Pearson CorrelationsGrowth

g2 g3 g4 g5 g6

g2 1

g3 .469 1

g4 .382 .995 1

g5 -.597 .429 .514 1

g6 -0.014 .877 .919 .811 1

Pearson CorrelationsSurvival

s2 s3 s4 s5 s6 s7

s2 1

s3 .726 1

s4 -.911 -.945 1

s5 -.946 -.465 .729 1

s6 .487 -.247 -.083 -.743 1

s7 .997 .778 -.941 -.918 .417 1

Pearson CorrelationsSurvival-Growth

s2 s3 s4 s5 s6 s7

g2 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

0.5,0.001

0.5,0.010.5,0.2

The beta distribution

Key distributions for vital rates

Key distributions for vital rates The beta distribution

Lognormal

rep 1994

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0

Std. Dev = 130.13 Mean = 73.6N = 1058.00

Stretched Beta

rep 1994

600

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Std. Dev = 130.13 Mean = 73.6N = 1058.00

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.50

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0.5 1 1.5 2 2.5 3 3.5 4 4.50

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0 1 2 3 4 5 60

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Matrix selection

Element selection

Vital rate selection