Is it meaningful to estimate a probability of extinction?

Ecology, Jan, 1999 by Donald Ludwig

INTRODUCTION

Population viability analysis (PVA) has been a popular tool for conservation biology since its development by Shaffer, Gilpin, and Soule (Shaffer 1981, Gilpin and Soule 1986, Shaffer 1987). PVA offers the prospect that sophisticated mathematical tools can be used to provide quantitative estimates of important quantities relating to threatened populations, such as the expected time to extinction or the probability of surviving for a target period such as 100 yr. More recently several authors have raised doubts about the usefulness of estimates derived from PVA. Here I raise some additional doubts based upon statistical difficulties. I consider the effect of possible errors in counts or estimates in population data, and the consequences for the precision of the resulting estimates of the probability of survival. If unaccounted for, such errors may produce a serious bias in estimates of quantities such as the intrinsic growth rate and carrying capacity. In addition, data containing observation errors are less informative than data that are free of error. When this difference is taken into account, confidence intervals are lengthened. In the majority of cases where I have applied the technique to population data, extremely wide confidence intervals result from the corrected analysis. The confidence intervals are so wide that the analysis provides little or no information about the magnitude of extinction probabilities. An additional difficulty is the possibility of natural catastrophes, which may produce much larger probabilities of extinction than otherwise (Mangel and Tier 1994, Ludwig 1996). Since catastrophes are rare by definition, data will seldom be available to estimate the frequency or size distribution of catastrophes. These severe statistical difficulties preclude simple patterns of decision-making that are appropriate for well-identified and predictable systems. Instead, some form of decision-making that takes explicit account of uncertainty appears to be required.

There is substantial disagreement in the literature concerning the usefulness of PVA for conservation purposes. I first describe some general surveys of PVA and its uses, and then I examine three possible defects of the method: (1) the lack of precision of the estimates for the probability of extinction, (2) the sensitivity of such estimates to model assumptions, and (3) the lack of attention to important factors influencing the extinction of populations.

PREVIOUS DISCUSSIONS OF PVA

Surveys

The literature on PVA was reviewed by Boyce (1992). Boyce mentioned effects such as demographic fluctuations, environmental fluctuations, natural catastrophes, genetic effects (e.g., inbreeding), Allee effects (low per capita growth rate at low densities), spatial structure, population structure, multispecies effects, and interactions between the preceding processes. In his concluding remarks Boyce raised grave doubts about the "viability" of PVA in view of these difficulties and the lack of sufficient data to validate models. Boyce advocated an adaptive approach to management (Walters 1986).

Burgman et al. (1993) covered many of the same points as Boyce (1992), and they provided algorithms for many of the most important calculations. However, their emphasis was more upon the modeling than the statistical aspects. They stated that the lack of data to parameterize models does not detract from their usefulness (Burgman et al. 1993:269). Akcakaya and Burgman (1995) amplified this point by citing the advantages of building a model for its own sake. These include the requirement to make all assumptions explicit and the identification of the processes and parameters that strongly influence conclusions. Perhaps this apparent divergence of opinion between Boyce (1992) and Burgman et al. (1993) is a consequence of differing objectives: Burgman et al. emphasize the value of model building as part of the process of performing PVA, while Boyce was concerned with the usefulness of the final product.

Lindenmeyer et al. (1993) surveyed the use of PVA in Australia. They concluded that it was a useful tool for managers to appreciate the gravity of extinction threats for small populations. They stated that PVA can be crucial in the process of policy formulation, implementation, and appraisal, as part of an adaptive management approach. They were optimistic about the prospects for improving parameter estimates on the basis of more comprehensive biological data, but did not provide any detailed calculations to support this view.

Precision of population viability analyses

The issue of precision may be addressed by calculating confidence intervals for estimates. Statistical aspects of the problem were treated in great detail by Dennis et al. (1991). They considered a density-independent model for population growth and used a diffusion approximation to compute the probability of quasi-extinction, i.e., reaching a low threshold size. Under these circumstances parameters can be estimated using a maximum likelihood or linear regression approach, and an explicit formula for the quasi-extinction probability is available. Dennis et al. used a linearized approximation for the variance of the estimated quasi-extinction probabilities. They applied their methods to a variety of practical examples and remarked that the estimated variances for quasi-extinction probabilities were "extremely large." Several of their data sets provide strong evidence for a high probability of extinction. Of the remainder, the only example they present that appears to provide evidence for a low probability of reaching a small threshold level is the Puerto Rican parrot. The latter data set shows a fairly steady increase 1975-1989. But Dennis et al. (1991) pointed out that the data do not show the number of breeding pairs, which is the most important indicator of recovery. The numbers also reflect very intensive recovery efforts (including release of captive-reared birds), and hence it is doubtful whether the usual sort of population dynamics model applies.