Geoscience Reference
In-Depth Information
between
X
t
and
X
t
-2
, albeit a weaker one (figure 6.4). The
PACF
removes these
effects due to shorter lags and measures the correlation that remains. For most
population systems, delayed density dependence is manifest in a significantly
negative
PACF
at lag 2 (figure 6.4), where significance (
p
< 0.05) is indicated by
a values that cross the dashed, horizontal cutoff lines. In figure 6.4 I give an
example from Turchin (1990) of the pine spinner moth,
Dendrolimus pini,
in
Germany, where fluctuations are erratic and there is no evidence of cyclicity in
the
ACF
and yet significant delayed density dependence in the
PACF
(figure 6.4).
Turchin (1990) used these procedures to analyze data from 14 species of forest
lepidoptera to show that nearly all of them had significant lag 2 or higher
effects. In contrast, Hanski and Woiwod (1991) analyzed 5,715 annual time
series of moths and aphids captured in survey traps. They found a high inci-
dence (67-91 percent) of density dependence but less than the 5 percent inci-
dence of delayed density dependence they would have expected by chance
alone. Holyoak (1994b) suggests that part of the unexpectedly low incidence
of delayed density dependence may be caused by the multiple generations that
elapsed between the annual samples for many of these species in contrast to the
forest lepidoptera analyzed by Turchin (1990).
Several of the same limitations of tests for direct density dependence also
apply to tests for delayed density dependence. The techniques have little or no
statistical power for the short time series that are typical of most ecological data
(Holyoak 1994a). Furthermore, autocorrelations in the error terms can lead to
spurious conclusions of delayed density dependence (Williams and Liebhold
1995). Berryman and Turchin (1997) argue that Williams and Liebhold's con-
clusion is overly pessimistic and based on simulations using unrealistic param-
eter values. Williams and Liebhold (1997) reply that the parameter values were
typical of populations analyzed earlier by Turchin (1990).
Detection of Causes of Population Change
j
KEY FACTOR ANALYSIS
Many population ecologists are more interested in determining the causes of
changes in density than in the causes of stability. For example, ecologists who
work with animals that exhibit outbreak dynamics may want to uncover the
cause of these outbreaks. Key factor analysis was developed to identify such
causes, or at least the life stage on which they act (Morris 1959; Varley and
Gradwell 1960). Like the density dependence techniques discussed earlier, key
