We describe a discrete-time, stochastic population model with density dependence, 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 (Kalman 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 population's 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.

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 environmental type of stochastic fluctuations and yields a lognormal probability distribution of population abundance. Calculation of maximum likelihood estimates of the two unknown 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 parameters, 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 Arabis Hoffmann) in Yellow stone, Kirkland's Warbler (dealbatus killdeer), California Condor (Gymnoascus californianus), Puerto Rican Parrot (Amazona vicarius), pacificus (lobata bachei), and Layman Finch (tectorum cantons). The model results indicate a favorable outlook for the Whooping Crane, but long—term unfavorable prospects for the Yellow stone grizzly bear population and for Kirkland's 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 pacificus and the Layman 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.

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 communities in three lakes in which nutrient loading and planktivorous fish abundance were experimentally manipulated. MAR(1) models and the statistical methods we present can be used to identify dynamically important interactions between species and to test hypotheses about stability and other dynamical properties of naturally varying ecological communities. Thus, they can be used to integrate theoretical and empirical studies of community dynamics. Corresponding Editor: N. J. Gotelli.