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## Estimating the magnitude of environmental stochasticity in survivorship data

- Author(s): Kendall, Bruce E.
- et al.

## Published Web Location

http://www.esajournals.org/doi/abs/10.1890/1051-0761%281998%29008%5B0184%3AETMOES%5D2.0.CO%3B2## Abstract

Small populations are often at risk of extinction through processes that are effectively stochastic. Prediction of this extinction risk requires that the observed temporal variation in demographic rates be accurately partitioned between demographic stochasticity (variation among individuals) and environmental stochasticity (variation among years, correlated across individuals). However, studies of population viability analysis that include both forms of stochasticity parameterize the magnitude of environmental stochasticity incorrectly (they overestimate it). I describe and evaluate tests (1) to determine whether all the year-to-year variation in observed survivorship can be explained by demographic stochasticity alone, and (2) if not, to estimate the magnitude of environmental stochasticity in survival. The first issue can be resolved with a G test. I used simulated data to show that this test has an appropriate type I error rate, unless the individual survival probability is either very low or very high. Small amounts of environmental stochasticity often are not detected by the G test (type II error), but the hypothesis of demographic stochasticity alone is consistently rejected when environmental stochasticity is large. In contrast, estimating the magnitude of environmental stochasticity requires explicit hypotheses about the nature of the underlying variation, but I provide a flexible framework in which many such hypotheses can be examined. In particular, I show, using simulated data, that if the temporal variability in individual survival probabilities is distributed according to a beta distribution, then the maximum likelihood estimate of the variance of the survival probability is biased, but in a consistent and correctable way. The estimate obtained with my method is usually superior to an estimate that assumes that all of the variation in the observed survivorship is due to environmental stochasticity. I show how to include deterministic sources of variability, such as density dependence, and how to apply different assumptions about the underlying environmental stochasticity. I illustrate these tests with data from a population of Acorn Woodpeckers (Melanerpes formicivorus). With these data, I can determine that there is strong environmental stochasticity in juvenile survival, whereas variation in adult survival can be explained either by density dependence or by weak environmental stochasticity.

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