Estimation of Growth and Extinction Parameters for Endangered Species

Abstract

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.

Modelling population persistence on islands: mammal introductions in the New Zealand archipelago

“…Notably, the observed distribution of persistence times closely followed that predicted by stochastic population growth theory in which extinction results from fluctuations in population size caused by environmental and/or demographic stochasticity (Lande & Orzack 1988;Dennis et al 1991;Engen et al 2005; both the inverse Gaussian and the demographic distributions fitted the data well).…”

Status of Baltic grey seals: Population assessment and extinction risk

“…Risk assessment is a tool used to investigate the vulnerability of populations to different future scenarios (Dennis et al 1991, Burgman et al 1993. The probability that the population declines to a lower threshold is estimated by projecting the population size forward in time, taking the stochastic nature of population growth into account.…”

Stochastic dynamics and logistic population growth

“…The effects of internal fluctuations have been studied in predator-prey models [15,16], epidemic models [17][18][19][20][21][22][23], cell biology [24], and ecological systems [13]. In particular, extinction of a stochastic population [11,25,26], which is a crucial concern for population biology [27] and epidemiology [28,29], has also attracted scrutiny in cell biochemistry [30] and in physics [31,32].…”