1. Demographic PVA’s Based on vital rates

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

3. 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

4. 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.

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

6. The problem of negative correlations A hypothetical individual is currently in size class 3 and has mean probability s 3 =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 g 4,3 =0.10 a 33 =s 3 (1-g 43 )=(0.95)(1-0.10) and a 43 =s 3 g 43 =(0.95)(0.10)

7. The Desert Tortoise

8. Size classes and definitions of matrix elements for the desert tortoise assuming a prebreeding census Class01234567 Yearling 0f5f5 f6f6 f7f7 Juvenile 1 1 s2s2 s 2 (1-g 2 ) Juvenile 2 2 s2g2s2g2 s 2 (1-g 2 ) Immature 3 s2g2s2g2 s 3 (1-g 3 ) Immature 4 s3g3s3g3 s 4 (1-g 4 ) Subadult 5 s4g4s4g4 s 5 (1-g 5 ) Adult 1 6 s5g5s5g5 s 6 (1-g 6 ) Adult 2 7 s6g6s6g6 s7s7

9. Estimated vital rates GrowthSurvival Class19701980e1980lMeanVar19701980e1980lMeanVar 2.500.50.330.083.631.65.76.044 3.50.180.1770.280.036.911.98.96.002 4.470.06700.180.065.98.59.81.79.039 5.230.2600.160.020.98.921.96.0018 6.0630.0320 0.001.991.68.89.034 7.781.8.86.015

10. Pearson Correlations Growth g2g2 g3g3 g4g4 g5g5 g6g6 g2g2 1 g3g3.4691 g4g4.382.9951 g5g5 -.597.429.5141 g6g6 -0.014.877.919.8111

11. Pearson Correlations Survival s2s2 s3s3 s4s4 s5s5 s6s6 s7s7 s2s2 1 s3s3.7261 s4s4 -.911-.9451 s5s5 -.946-.465.7291 s6s6.487-.247-.083-.7431 s7s7.997.778-.941-.918.4171

12. Pearson Correlations Survival-Growth s2s2 s3s3 s4s4 s5s5 s6s6 s7s7 g2g2 1-.704.898.956-.514-.994 g3g3.496-.957.810.189.516-.563 g4g4 -.41-.925.75.094.597-.481 g5g5.571-.149-.182-.806.995.505 g6g6 -0.017-.7.428-.307.865-.096

13. 0.5,0.001 0.5,0.01 0.5,0.2 The beta distribution Key distributions for vital rates

14. The beta distribution

15. Lognormal

16. Stretched Beta

17. Matrix selection Element selection Vital rate selection