Projections from museum collections

Sometimes the only data we have are data from museum collections, and many of those data are now online through databases like the Global Biodiversity Information Facility (GBIF: http://www.gbif.org). Skarpaas and Stabbetorp [2] suggest a way to use these data for a population viability analysis. Since collectors typically collect only one individual from a site, each specimen reflects presence or absence of a species at a site rather than the number of individuals present. Thus, Skarpaas and Stabbetorp treat the number of sites at which a species is present as their measure of viability. A species goes extinct when the number of sites it occupies goes to zero. Their approach also recognizes that there are two sources of variability in the number of collected specimens over time:

1. Population variability: Real variability in the number of sites at which a species is present over time.

2. Observation error: Specimens may not be availalbe for some sites at which the species is present.

They model population variability with a very simple model of exponential growth:

$$\begin{eqnarray*} \ln(N_{t+1}) &=& \ln(N_t) + \mu + \epsilon_t \\ N_{t+1} &=& e^{\mu\epsilon_t}N_t \\ &\approx& (1 + \mu\epsilon_t)N_t \quad . \end{eqnarray*}$$

$e^\mu$ is the average grawth rate of the species. $\epsilon_t$ is a normal random variable with mean 0 and variance $\sigma^2$. $e^{\mu\epsilon_t}$ is the growth rate in a particular year. If it is negative, the number of sites occupied declines. If it is positive, the number of sites occupied increases.

The model for observation error is also simple. If $P_i$ is the probability that a species is observed at a site, they let

$$P_t = \frac{kN_t}{s} \quad ,$$

where $k$ accounts for the possibility that species are present in a site but not collected, and $s$ is the number of sites included in the sample. Given an estimate of sampling effort over time, $c_t$, and an independent estimate of $k$, the other parameters in the model can be estimated, and the probability of extinction over time can be calculated.

Skarpaas and Stabbetorp present simulation results suggesting that their method tends to overestimate quasi-extinction probabilities, but the overestimation is not severe unless (a) the number of sites included in the sample is small (100 or fewer) or (b) the detection probability is low (0.75 or less). The approach seems promising, though I do have to wonder how often species of concern will have distributions for which it is reasonable to imagine that there are 100 or more sites where it could have been seen. The good news is that the simulations suggest that we are unlikely to underestimate the risk a species faces if we use their approach.