A Metapopulation Approach

The analysis we've done so far presumes that we can treat this geographically widespread species as a single, homogeneous population. This clearly doesn't make a lot of sense. Instead, it may make more sense to treat it as a metapopulation.¹⁰Lande [4] showed how the metapopulation approach, which we'll see applied in a different way to the bay checkerspot butterfly, can be applied to territorial animals by treating the individual territory as the unit of extinction and colonization.

Parameter	Definition
$\epsilon$	probability that a juvenile female inherits the territory of its mother
$m$	number of territories a juvenile can disperse through before dying from predation, starvation, etc.
$h$	proportion of the region that is habitable
$p$	proportion of habitable sites that are occupied

The probability of not finding a suitable unoccupied territory in $m$ trials is $(1-\epsilon)(ph + 1 - h)^m$. Thus, for a population in demographic equilibrium

$$[1 - (1-\epsilon)(ph + 1 - h)^m]R'_0 = 1 \quad ,$$

where

$$R'_0 = \sum_{x=0}^\infty l'_xf_x$$

is the mean lifetime production of offspring per female, given that she finds a territory, and $l'_x$ is the probability of survival to age $x$, given that she finds a territory. For our model $R'_0 = l'_\alpha b/(1-s)$.

We can solve the above equation for the equilibrium number of patches occupied.

$$\hat p = \left\{ \begin{array}{rl} 1 - \frac{1 - k}{h} & \hbox{for }h > 1-k \\ 0 & \hbox{for }h \le 1-k \end{array}\right.$$

where

$$k = [(1-1/R'_0)/(1-\epsilon)]^{1/m} \quad .$$

$k$ is the equilibrium occupancy of patches when the entire region is habitable ($h=1$). The population can persist only if $h > 1-k$.

National forests in the Douglas-fir region of Washington and Oregon contained about 38% forest greater than 200 years old in 1987. Thus, $h \approx 0.38$. Recent surveys have suggested that about 44% of the appropriate sites are currently occupied, i.e., $p \approx 0.44$. Assuming that the population is demographically stable, as suggested by the Leslie matrix model, then we can solve for $k$ as

$$\begin{eqnarray*} k &=& 1 - h(1-p) \\ &\approx& 0.79 \end{eqnarray*}$$

Future Forest Service plans suggest leaving 7% to 16% of forest in stands older than 200 years. $1-k = 0.21 > 0.07-0.16$. Therefore, this course of action seems likely to doom spotted owls to extinction. Note: Even if habitat occupancy estimates are highly inaccurate, this conclusion does not change much. Suppose 60% of all suitable habitat is occupied. Then $k \approx 0.85, 1-k \approx 0.15$

These estimates for $k$ may be optimistic because older owls are likely to pack more densely into existing habitat as harvest continues. Thus, recent estimates of habitat occupancy may be overestimates. If equilibrium occupancy rates are 50% lower than those currently seen (not unreasonable given the long life-span of these birds), then $k = 0.70$.

Moreover, these calculations are overly optimistic about the persistence possibilities. They are, after all, based on the assumption that the population is demographically stable, which it is not.