Extreme events are often considered to be occurrences imposed on a population from outside, in the form of unusual, stochastically fluctuating, environmental conditions. However, unusual events, even extreme events, can also arise from processes operating within the population.
Internal extreme events may arise from heterogeneity (some individuals have extreme properties) or from individual stochasticity (some individuals are extremely lucky). Demographic models can be used to quantify and account for these events. Of particular interest are longevity, healthy longevity, and lifetime reproductive output (LRO).
Results/Conclusions
These quantities are subject to large amounts of stochasticity, and comparisons to date agree that the variance due to stochasticity usually exceeds that due to heterogeneity. Values are also usually positively skewed, sometimes extremely so. In particular, we show that skewness in LRO in human populations at birth ranges from about 0.1 to 1.0, but increases dramatically with age, so that remaining LRO at age 40 has a skewness of about 8. Skewness in LRO calculated for selected bird species ranges from approximately 3 to 10, and skewness for large trees can be as high as 60. For comparison, the exponential distribution has a skewness of 2.
In showing these results, we will present new methods to calculate the variance and skewness of demographic outcomes, including a generalization that promises to permit combining individual stochasticity with environmental stochasticity.
Positive skewness implies that the process is subject to rare, large deviations from the mean, and that empirical observations with small sample sizes are unlikely to reveal the full range of variation, leading to apparent extreme events. Cohen has recently shown that certain kinds of sampling from skewed distributions leads to Taylor’s law relationships between mean and variance.