^{¤}

^{1}

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JMD conceived and designed the experiments, performed the experiments, analyzed the data, contributed reagents/materials/analysis tools, and wrote the paper.

¤ Current address: National Center for Ecological Analysis and Synthesis, Santa Barbara, California, United States of America

The authors have declared that no competing interests exist.

Understanding population extinctions is a chief goal of ecological theory. While stochastic theories of population growth are commonly used to forecast extinction, models used for prediction have not been adequately tested with experimental data. In a previously published experiment, variation in available food was experimentally manipulated in 281 laboratory populations of

Manipulating the environment of

As changing global climate and anthropogenic modification of the biosphere increasingly threaten global biodiversity [

Two components contribute to random variation in per capita population growth rates: demographic stochasticity and environmental variability [_{t+1} − _{t}, a model for per capita population growth rate with demographic and environmental stochasticity is

where _{e}_{d}

Recent research suggests that demographic stochasticity might have a density-dependent component in the sense that the variance contributed to the population's growth rate by demographic stochasticity, represented by _{d} in _{d}

To test for effects of environmental variation on persistence, 281 laboratory populations of water fleas

After rescaling to isolate density dependence in _{d} (see _{e}-transformed deviations and population size was highly significant (Spearman rank-order correlation: ρ = −0.23,

Deviations from expected population size were rescaled by multiplying the observed deviation by initial population size for the interval to isolate density dependence in _{d} (see

Extinction rates of populations in experimental treatments with low and medium levels of variation were not significantly different, although these were different from the extinction rate of populations subject to a high level of environmental variation (see

Estimates of the extinction rate in populations of

Model-predicted extinction rates were obtained by fitting a simple Ricker model for population growth, _{0}^{−bN}

The accuracy of model predictions can be assessed by comparing model-predicted extinction rates with the 95% CI for the estimated extinction rate of populations in the model-testing half of the experimental dataset (see

A stochastic Ricker model of population growth with density-dependent demographic stochasticity accurately predicted the chance of extinction, within the power of this experiment to reject the null hypothesis of a difference, or was only slightly biased (

In this analysis, the accuracy of model predictions was improved by relaxing the usual assumption that demographic stochasticity is density-independent [_{low} = 342, _{med} = 300, and _{high} = 280). In general, field data will not be so abundant. Thus, an important goal for population biology is to develop methods for obtaining reliable predictions from sparse, low-quality datasets [

Planning for increasing threats to rare species from diverse sources, including climate change, resource extraction, habitat modification, and invasive species will require greater and more precise estimates of extinction risk than ever before. While the reliability of theoretical models for predicting extinction in natural ecosystems remains to be established, the results presented here show that accurate predictions of population extinction in variable environments are indeed possible.

Experimental microcosms (

Because populations in the low-variability treatment were not exposed to any experimentally induced environmental variation, most variation in observed growth rates in these populations should be attributable to demographic stochasticity and is not confounded with environmental stochasticity. Thus, only these data were used for exploratory analysis of density-dependent demographic stochasticity. Estimates of the pairwise multiplicative population growth rate λ̂(_{t}_{t}_{+τ}/_{t}^{λ}(_{t}_{t}_{t}_{0}^{−bNt}_{j}^{2}) were retained. To account for the scaling of demographic variance with population size [_{d}^{2}^{− αN+β} was used. Hyperbolic models of demographic stochasticity were also explored, but these were poorly supported by formal model selection criteria such as AIC, relative to the exponential model.

Because populations were independent, each population represents a Bernoulli trial for which the possible outcomes were extinction or persistence. Thus, the chance of extinction for a population in treatment level _{i}_{i}_{i}_{i}

Initially, it was unclear if there was an effect of experimental treatment. Using regression on pooled observations from the entire experimental dataset, Drake and Lodge [_{0}, _{1},

The theoretical variance in λ is given by _{e}^{2}_{e}^{2}

where _{0} and _{N}

where _{e} is a parameter representing the average level of variation from environmental stochasticity. Models were fit to data in the model-fitting dataset by minimizing the negative log-likelihood function using the Nelder-Mead simplex. Goodness of fit was quantified using AIC = 2

Overall, the inclusion of the parameter _{e} for data from all experimental treatments were not significantly different from 0, even for the treatment with a high level of experimentally induced variation.

Predicted extinction rates were obtained by simulating 100,000 iterations of the Ricker model with density-dependent demographic stochasticity at maximum likelihood estimates of all parameters for populations in the pooled low- and medium-variability treatments and populations in the high-variability treatment separately using Euler's method [

The model for stochastic population growth considered here accounts for two factors commonly ignored when predicting the chance of population extinction: density-dependent changes in expected population size and density-dependent demographic stochasticity. Although the effect of density-dependence in _{d}^{2}

As above, model predicted extinction rates were obtained by simulating 100,000 iterations of the population growth process at maximum likelihood estimates of all parameters.

The author thanks R. Schwartz, P. Baggenstos, and J. Frentress for assistance with this experiment; and J. Hellman, T. Coulson, M. Vellend, and two anonymous referees for comments on the manuscript. This research was supported by a Great Lakes Fishery Commission grant (to principal investigator, D. Lodge), a scholarship from the Illinois-Indiana Sea Grant (to JMD), an Environmental Protection Agency Science to Achieve Results Graduate Research Fellowship (to JMD), and a University of Notre Dame Schmitt Graduate Student Research Fellowship (to JMD). It was also supported by a contribution from the Integrated Systems for Invasive Species project (principal investigator, D. Lodge), funded by the National Science Foundation (DEB 02–13698) and the University of Notre Dame.

Akaike's information criterion

confidence interval