Brian Dennis

Estimation of Growth and Extinction Parameters for Endangered Species

Survival or extinction of an endangered species is inherently stochastic. We develop statistical methods for estimating quantities related to growth rates and extinction probabilities from time series data on the abundance of a single population. The statistical methods are based on a stochastic model of exponential growth arising from the biological theory of age- or stage-structured populations. The model incorporates the so-called en- vironmental type of stochastic fluctuations and yields a lognormal probability distribution of population abundance. Calculation of maximum likelihood estimates of the two un- known parameters in this model reduces to performing a simple linear regression. We describe techniques for rigorously testing and evaluating whether the model fits a given data set. Various growth- and extinction-related quantities are functions of the two param- eters, including the continuous rate of increase, the finite rate of increase, the geometric finite rate of increase, the probability of reaching a lower threshold population size, the mean, median, and most likely time of attaining the threshold, and the projected population size. Maximum likelihood estimates and minimum variance unbiased estimates of these quantities are described in detail. We provide example analyses of data on the Whooping Crane (Grus americana), grizzly bear (Ursus arctos horribilis) in Yellowstone, Kirtlands Warbler (Dendroica kirtlandii), California Condor (Gymnogyps californianus), Puerto Rican Parrot (Amazona vittata), Palila (Loxioides balleui), and Laysan Finch (Telespyza cantans). The model results indicate a favorable outlook for the Whooping Crane, but long-term unfavorable prospects for the Yellowstone grizzly bear population and for Kirtlands Warbler. Results for the California Condor, in a retrospective analysis, indicate a virtual emergency existed in 1980. The analyses suggest that the Puerto Rican Parrot faces little risk of extinction from ordinary environmental fluctuations, provided intensive management efforts continue. However, the model does not account for the possibility of freak catastrophic events (hurricanes, fires, etc.), which are likely the most severe source of risk to the Puerto Rican Parrot, as shown by the recent decimation of this population by Hurricane Hugo. Model parameter estimates for the Palila and the Laysan Finch have wide uncertainty due to the extreme fluctuations in the population sizes of these species. In general, the model fits the example data sets well. We conclude that the model, and the associated statistical methods, can be useful for investigating various scientific and management questions concerning species preservation.

DENSITY DEPENDENCE IN TIME SERIES OBSERVATIONS OF NATURAL POPULATIONS: ESTIMATION AND TESTING'

We report on a new statistical test for detecting density dependence in uni- variate time series observations of population abundances. The test is a likelihood ratio test based on a discrete time stochastic logistic model. The null hypothesis is that the population is undergoing stochastic exponential growth, stochastic exponential decline, or random walk. The distribution of the test statistic under both the null and alternate hy- potheses is obtained through parametric bootstrapping. We document the power of the test with extensive simulations and show how some previous tests in the literature for density dependence suffer from either excessive Type I or excessive Type II error. The new test appears robust against sampling or measurement error in the observations. In fact, under certain types of error the power of the new test is actually increased. Example analyses of elk (Cervus elaphus) and grizzly bear (Ursus arctos horribilis) data sets are provided. The model implies that density-dependent populations do not have a point equilibrium, but rather reach a stochastic equilibrium (stationary distribution of population abundance). The model and associated statistical methods have potentially important applications in conservation biology.

Estimating community stability and ecological interactions from time-series data

Natural ecological communities are continuously buffeted by a varying environment, often making it difficult to measure the stability of communities using concepts requiring the existence of an equilibrium point. Instead of an equilibrium point, the equilibrial state of communities subject to environmental stochasticity is a stationary distribution, which is characterized by means, variances, and other statistical moments. Here, we derive three properties of stochastic multispecies communities that measure different characteristics associated with community stability. These properties can be estimated from multispecies time-series data using first-order multivariate autoregressive (MAR(1)) models. We demonstrate how to estimate the parameters of MAR(1) models and obtain confidence intervals for both parameters and the measures of stability. We also address the problem of estimation when there is observation (measurement) error. To illustrate these methods, we compare the stability of the planktonic commun...

ESTIMATING DENSITY DEPENDENCE, PROCESS NOISE, AND OBSERVATION ERROR

We describe a discrete-time, stochastic population model with density depend ence, environmental-type process noise, and lognormal observation or sampling error. The model, a stochastic version of the Gompertz model, can be transformed into a linear Gaussian state-space model (Kaiman filter) for convenient fitting to time series data. The model has a multivariate normal likelihood function and is simple enough for a variety of uses ranging from theoretical study of parameter estimation issues to routine data analyses in population monitoring. A special case of the model is the discrete-time, stochastic exponential growth model (density independence) with environmental-type process error and lognormal observation error. We describe two methods for estimating parameters in the Gompertz state-space model, and we compare the statistical qualities of the methods with computer simulations. The methods are maximum likelihood based on observations and restricted maximum likelihood based on first differences. Both offer adequate statistical properties. Because the likelihood function is identical to a repeated-measures analysis of variance model with a random time effect, parameter estimates can be calculated using PROC MIXED of SAS. We use the model to analyze a data set from the Breeding Bird Survey. The fitted model suggests that over 70% of the noise in the populations growth rate is due to observation error. The model describes the autocovariance properties of the data especially well. While observation error and process noise variance parameters can both be estimated from one time series, multimodal likelihood functions can and do occur. For data arising from the model, the statistically consistent parameter estimates do not necessarily correspond to the global maximum in the likelihood function. Maximization, simulation, and bootstrapping programs must accommodate the phenomenon of multimodal likelihood functions to produce statistically valid results.

The evidence for Allee effects

Allee effects are an important dynamic phenomenon believed to be manifested in several population processes, notably extinction and invasion. Though widely cited in these contexts, the evidence for their strength and prevalence has not been critically evaluated. We review results from 91 studies on Allee effects in natural animal populations. We focus on empirical signatures that are used or might be used to detect Allee effects, the types of data in which Allee effects are evident, the empirical support for the occurrence of critical densities in natural populations, and differences among taxa both in the presence of Allee effects and primary causal mechanisms. We find that conclusive examples are known from Mollusca, Arthropoda, and Chordata, including three classes of vertebrates, and are most commonly documented to result from mate limitation in invertebrates and from predator–prey interactions in vertebrates. More than half of studies failed to distinguish component and demographic Allee effects in data, although the distinction is crucial to most of the population-level dynamic implications associated with Allee effects (e.g., the existence of an unstable critical density associated with strong Allee effects). Thus, although we find conclusive evidence for Allee effects due to a variety of mechanisms in natural populations of 59 animal species, we also find that existing data addressing the strength and commonness of Allee effects across species and populations is limited; evidence for a critical density for most populations is lacking. We suggest that current studies, mainly observational in nature, should be supplemented by population-scale experiments and approaches connecting component and demographic effects.

NONLINEAR DEMOGRAPHIC DYNAMICS: MATHEMATICAL MODELS, STATISTICAL METHODS, AND BIOLOGICAL EXPERIMENTS'

Our approach to testing nonlinear population theory is to connect rigorously mathematical models with data by means of statistical methods for nonlinear time series. We begin by deriving a biologically based demographic model. The mathematical analysis identifies boundaries in parameter space where stable equilibria bifurcate to periodic 2-cy- cles and aperiodic motion on invariant loops. The statistical analysis, based on a stochastic version of the demographic model, provides procedures for parameter estimation, hypothesis testing, and model evaluation. Experiments using the flour beetle Tribolium yield the time series data. A three-dimensional map of larval, pupal, and adult numbers forecasts four possible population behaviors: extinction, equilibria, periodicities, and aperiodic motion including chaos. This study documents the nonlinear prediction of periodic 2-cycles in laboratory cultures of Tribolium and represents a new interdisciplinary approach to un- derstanding nonlinear ecological dynamics.

Sixty years of environmental change in the world's largest freshwater lake – Lake Baikal, Siberia

High-resolution data collected over the past 60 years by a single family of Siberian scientists on Lake Baikal reveal significant warming of surface waters and long-term changes in the basal food web of the worlds largest, most ancient lake. Attaining depths over 1.6 km, Lake Baikal is the deepest and most voluminous of the worlds great lakes. Increases in average water temperature (1.21 °C since 1946), chlorophyll a (300% since 1979), and an influential group of zooplankton grazers (335% increase in cladocerans since 1946) may have important implications for nutrient cycling and food web dynamics. Results from multivariate autoregressive (MAR) modeling suggest that cladocerans increased strongly in response to temperature but not to algal biomass, and cladocerans depressed some algal resources without observable fertilization effects. Changes in Lake Baikal are particularly significant as an integrated signal of long-term regional warming, because this lake is expected to be among those most resistant to climate change due to its tremendous volume. These findings highlight the importance of accessible, long-term monitoring data for understanding ecosystem response to large-scale stressors such as climate change.

Discussion: Should Ecologists Become Bayesians?

Bayesian statistics involve substantial changes in the methods and philos- ophy of science. Before adopting Bayesian approaches, ecologists should consider carefully whether or not scientific understanding will be enhanced. Frequentist statistical methods, while imperfect, have made an unquestioned contribution to scientific progress and are a workhorse of day-to-day research. Bayesian statistics, by contrast, have a largely untested track record. The papers in this special section on Bayesian statistics exemplify the diffi- culties inherent in making convincing scientific arguments with Bayesian reasoning.

Joint effects of density dependence and rainfall on abundance of San Joaquin kit fox

We analyzed time-series abundances of San Joaquin kit fox estimated during 1983-95 on the Naval Petroleum Reserves in California (NPRC). For the analysis, we modified a model of density-dependent, stochastic population growth to include the lagged effects of a weather covariate (vegetation growing season rainfall). Without the covariate in the model, a statistical test failed to detect significant density dependence in fluctuating kit fox abundances. However, when the covariate was added, strong density dependence was detected. According to an information-theoretic model-selection index, the model with both density dependence and rainfall is far superior to the models that result from deleting one or more of these factors. The 2-year time lag in the response of kit fox abundances to changes in rainfall is consistent with biological expectations of how rainfall affects habitat carrying capacity for kit fox. An additional covariate, a coyote abundance index, failed to improve the model. A population viability analysis (PVA) performed with the combined density dependence-rainfall model suggests that the San Joaquin kit fox on NPRC could face a risk of up to 52% of falling to low levels within 20 years.

The gamma distribution and weighted multimodal gamma distributions as models of population abundance

Abstract The gamma probability distribution is a general model of a population fluctuating around a steady state. We show this using stochastic differential equations (SDEs), constructed by adding white noise to the specific growth rate in deterministic models of population abundance. The gamma is an approximate stationary solution for almost any SDE having an underlying deterministic equilibrium. If the deterministic model possesses multiple stable and unstable equilibria, the approximate stationary solution to the stochastic case is a weighted gamma distribution. Modes of the stationary distribution roughly correspond to the equilibria of the deterministic model. Stochastic forces have effects similar to harvesting. These findings provide: (1) a theoretical basis for certain descriptive uses of the gamma in statistical ecology, (2) a concise graphical summary of the interactions between density dependent and density independent population regulation, (3) a statistical framework for fitting catastrophe-theoretic models to ecological data sets.