What do we do with limited data?

What I've just shown you is wonderful stuff, but we can only use it if we have data on survival probabilities and fecundities for every age or stage in our population. When you consider that the elements of the matrix are likely to vary from season to season¹³ and that to make reasonable forecasts about the fate of the population we'd need good data on both the mean and the variance of every element in the matrix, it seems as if we're doomed. Only a few of the most important endangered species will ever have enough data collected to do a complete analysis. Does that mean we should give up on population viability analysis? Obviously, I don't think so, or I wouldn't be lecturing about it, but how should we proceed?

One possibility is to construct an explicit simulation model that incorporates as much of the biology of the species you're concerned with as possible, to select a broad range of plausible values for the parameters in that model with values based on best guesses (derived from existing data, comparison with related species, or back-of-the envelope calculations), and to do an extensive simulation study incorporating all of the plausible parameter estimates. You would probably then try to find a combination of management alternatives that made the worst-case scenario the most unlikely, but there are other alternatives.

Another alternative, a variant of which we'll see applied to the northern spotted owl, is to do some relatively simple calculations based on a minimal amount of data [4]. Dennis et al. [2] propose a relatively simple model of population dynamics based on the observation that age-structured populations are well-approximated by a stochastic, discrete time model with exponential growth. They illustrate how to calculate extinction properties of a population from such a model knowing only the mean rate of population growth and its variance.¹⁴ Holmes [4] presents relatively simple method for estimating $\bar R$ and $\sigma^2_R$ from a time series of population censuses. Her simulations show that the method she proposes provides reasonable estimates of extinction paramters when compared with projections based on fully-specified age-structured models. Morris and Doak [3, pp. 434] suggest two rules in conducting a PVA that I urge you to remember:¹⁵

1. Let the available data tell you how complex your PVA should be. We've seen that a complete demographic analysis, even for a plant or animal with a fairly simple life cycle requires measuring a lot of stage transitions, ideally for at least 4-5 years. If you don't have that much data, don't try to make a precise, quantitative model. Explore simpler alternatives, try some plausibility analyses, and don't overinterpret your results. They will be a qualitative guide to action more than a quantitative prediction.

2. Make sure you understand what the model is doing. If you're collaborating with someone who says ``Don't worry about the model. I'll handle that,'' make sure you understand what that person is up to. You don't need to know how eigenvalues were calculated, but you must understand how your data are being fed into the model, what assumptions are embedded in the model, and how to interpret the output. And you need to do a ``gut check'' of the output. If the results don't seem plausible, don't dismiss them out of hand, but examine the inputs and the structure of the model very carefully before accepting a projection that doesn't match your instincts.