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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
600
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100
0
Std. Dev = 130.13 Mean = 73.6N = 1058.00
Stretched Beta
rep 1994
600
500
400
300
200
100
0
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