A More Detailed Analysis

The previous analysis is based on some extreme simplifying assumptions. There is good evidence that survivorship and fecundity differ among three classes: juveniles, subadults (1-2 year olds), and adults. Using an approach similar to the one above,⁸ let

$$\begin{eqnarray*} f_1 &=& \hbox{fecundity of juveniles} \\ f_2 &=& \hbox{fecund... ...les} \\ s_a &=& \hbox{survivorship of subadults and adults} \\ \end{eqnarray*}$$

Then $\lambda$ is found as the solution to the following equation

$$1 = f_1\left({s_j \over \lambda}\right) + f_2\left({s_js_a ... ...lambda^3}\right) \left(1 \over 1 - s_a/\lambda\right) \quad .$$

Data from fourteen demographic study areas that had at least four years and as many as nine years of capture-recapture data found substantial differences in $\lambda$ among site: range 0.83-1.02, mean 0.93. Noon and Blakesley [5] point out that more recent analyses provide an estimate of $\lambda=0.976 \pm 0.014$ for eight study areas on federal lands and an estimate of $\lambda=0.942$ for five study areas on non-federal lands.

We now have five additional years of data and a more detailed model and we discover that the spotted-owl is even worse off than we initally imagined. Why?

Parameter	Value	Sensitivity	Elasticity
$f_1$	0.068	0.020	0.00147
$f_2$	0.205	0.018	0.00400
$f_a$	0.339	0.196	0.07200
$s_j$	0.258	0.278	0.07773
$s_a$	0.844	1.008	0.92193
$\lambda$	0.923

As argued above, population dynamics are predominantly determined by adult survivorship.

These estimates are biased by the underestimate of juvenile survivorship due to individuals that migrate outside the study area and survive. When this is accounted for $\lambda = 0.95 \pm 0.02$.

Juvenile survivorship is $0.38 \pm 0.06$ and emigration is $0.32 \pm 0.05$. For $\lambda > 1$ we would require juvenile survivorship $>$ 0.57 and emigration $>$ 0.51. In addition to this, we have other strong reasons for thinking the decline is very real.

• Average adult female survivorship declined from approximately 0.88 in 1985 to 0.80 in 1994.

• $\lambda$ calculated using average adult survivorship of 0.84.⁹ If declining trend were included, $\lambda$ would be even smaller.

• Although adult survival has the highest elasticity coefficient, there may be little that can be done to improve it, since it's already relatively high. Focusing on improving juvenile establishment is likely to have greater returns.

14 demographic study areas with a minimum of four years capture-recapture data. Approximately 50 biologists spent two weeks collating and analyzing the data to reach these conclusions. $\lambda$ barely distinguishable from 1, i.e., barely demonstrable that population is declining. In fact, the results in [3] suggest that its even worse than that (Figure 2). The results we've presented so far pretend (a) that we know the values of each demographic parameter without error and (b) that the values of those parameters do not vary among individuals or populations. Both of those assumptions are wrong. As a result, the projections underestimate both the process error, i.e., the probabilistics uncertainty of outcomes because of the intrinsic stochasticity of the underlying processes (demographic and environmental stochasticity), and the parameter error, i.e., the uncertainty about the correct values of the demographic parameters. The result is that the results overstate the degree to which we can be confident about the projections.

Figure 2: Comparison of projection uncertainty with (H-Bayes) and without (NH) parameter uncertainty (from [3]).

So we are left with a dilemma. It is clear from first principles that populations of northern spotted owls will continue to decline if there habitat continues to become less common. It is also clear from first principles that at some point the amount of habitat remaining will be insufficient to support a self-sustaining wild population. But after all these years of effort, it is still very difficult to provide clear, quantitative advice that would allow land managers to determine whether the amount of remaining old-growth is sufficient or whether forests must be managed to provide even more in the future. Moreover, ``[t]he causal factors of spotted owl declines from 1990 to 2003 are poorly known,'' and invasion of the barred owl into the region appears to pose a new threat to persistence of the northern spotted owl [5].

Major research question: Is there a way to reach management conclusions more rapidly?