Geoscience Reference
In-Depth Information
a
b
e
t
X
t
+1
=
+ (
+ 1)
X
t
+
(6.3)
b
a
If both
= 0 and
= 0, the model is known as a random walk:
e
t
X
t
+1
=
X
t
+
(6.4)
Thus to look for density dependence we could plot either
R
or
X
t
+1
against
X
t
. Under the null hypothesis of density independence we could test whether
b
b
+ 1 = 1.
Morris (1959) was the first to use plots of
X
t
+1
against
X
t
to search for den-
sity dependence in population systems, in his case spruce budworm popula-
tions. Smith (1961) used this technique to show density dependence in popu-
lations of the flower thrips, which Andrewartha and Birch (1954) had used as
an example of an insect governed by density independent processes. Equiva-
lently one can plot
R
against
X
t
, which I illustrated with my own data on gypsy
moth density (Elkinton et al. 1996) collected from eight populations over a
10-year period (figure 6.1).
The striking downward trend evident in figure 6.1 seems to indicate a
strong density dependence: Populations decline (
R
< 0) when densities are
high and increase (
R
> 0) when densities are low. Standard regression proce-
dures indicate a significant negative slope (solid line in figure 6.1). However,
there is a statistical problem. The axes are not independent and this produces
a negative bias in regression estimates of slope (Watt 1964; St. Amant 1970;
Eberhardt 1970; Reddingius 1971). Data generated from a random walk
process (no density dependence) will show strong negative slope. To illustrate
this I fit a random walk model to the data given in figure 6.1 and selected 100
time series of length
n
= 10 (years) based on values of
= 0 or
e
t
chosen at random from
a normal distribution with a variance that matches that of the data. The result-
ing average slope (dotted line in figure 6.1) is strongly negative, implying that
population growth declines with density, even though there is no density
dependence and hence no stability in this model of the population system.
This example illustrates that usual methods of statistical inference based on
regression of
X
t
+1
or
R
on
X
t
are fundamentally flawed as a way of detecting
density dependence in a population time series.
Various investigators have suggested solutions to the problem just illus-
trated. Varley and Gradwell (1968) advocated plotting
X
t
+1
against
X
t
, inter-
changing them as independent and dependent variables. Only if both regres-
sions were significantly different from the slope of the null model (
b
+ 1 = 1)
