Appendix D. Sensitivity to increasing variability vs. longevity in randomly constructed life histories.
The sets of projection matrices we estimated for our study species incorporate demographic complexities that differ among the study species, including the overall level of year-to-year variation in demographic rates, whether stages are based on age or size, whether the environment is IID vs. Markovian, the degree of within-year correlation among different vital rates, and differences among vital rates in their current levels of variability. Therefore, to see if greater longevity would still lead to lower sensitivity to increasingly variable survival or reproduction if we eliminated these differences, we looked for a relationship between the two variables in randomly constructed sets of projection matrices.
The matrices we constructed were all size-based, assumed an IID environment, lacked within-year correlations among vital rates, and had equal levels of variability in all vital rates.
We constructed them following the basic protocol outlined by Morris and Doak (2004). In brief, we constructed 500 sets of size-based matrices in which surviving individuals could either remain in the same size class or grow to the next size class each year, but we varied survival probabilities and we varied the number of stages from 1 to 10 to generate a range of life expectancies. Mean annual survival probabilities and fecundities increased and mean annual growth probabilities decreased with increasing size. We treated annual survival and growth probabilities as beta random variables, each with a variance equal to 10% of its maximum value given the mean, and we assumed annual fecundities were lognormally distributed, each with a variance equal to 10% of the mean. We generated five annual values for each vital rate (close to the median for the demographic studies in our data base), assuming no within- or among-year correlations among rates (i.e., we assumed independent vital rates and an IID environment), and we used them to construct matrices and to compute stochastic elasticities to the vital rate means and SDs. To avoid unrealistic sets of matrices, we retained only those sets that produced a stochastic population growth rate () between 0.5 and 1.5. Finally, we calculated the relative effect of variability as described in the main text, and plotted it against life expectancy (conditional on reaching the second size class in cases with two or more classes).
As for our study species, the relative effect of increasing variability in survival declines with increasing life expectancy in the randomly constructed matrices (Fig. D1A). However, the rank correlation between these two variables is weaker than we observed in our focal species (cf. Fig. 2A in the text). The relative effect of increasing variability in reproduction also declines with life expectancy, and the relationship is also weaker than in the empirical data (compare Fig. 2B in the text to Fig. D1B). However, some simulated data sets (but not ones with a single life history stage) show both short life expectancy and low sensitivity to increasingly variable reproduction. As in the empirical data, increasing variability in reproduction has a weaker effect overall than does increasing variability in survival (compare y-axis scales in Fig. D1A,B). The effect of increasing variability in all vital rates is again negatively correlated with life expectancy (Fig. D1C).
These results obtained for sets of randomly constructed but biologically realistic projection matrices argue that longevity may in general lower sensitivity of population growth to increasing vital rate variability, even when the vital rate means and SDs have not been shaped by natural selection. However, more work remains to be done to examine how real-world demographic features, such as correlations in vital rates within and among years and the buffering of highly influential vital rates against environmental stochasticity (Pfister 1998), influence the sensitivity of population growth to increasing variability.
|FIG. D1. The relationship between life expectancy and the relative effect of increasing variability in (A) survival, (B) reproduction, and (C) all vital rates for randomly constructed sets of vital rates. Points on the y axis represent simulated data sets with only a singe life history stage; all other points represent cases with two or more stages. One point with extremely high life expectancy is omitted to better display other points. is Spearman’s rank correlation.|
Morris, W. F., and D. F. Doak. 2004. Buffering of life histories against environmental stochasticity: Accounting for a spurious correlation between the variabilities of vital rates and their contributions to fitness. American Naturalist 163:579590.
Pfister, C. A. 1998. Patterns of variance in stage-structured populations: evolutionary predictions and ecological implications. Proceedings of the National Academy of Sciences of the United States of America 95:213218.