Detecting Stability and Causes of Change
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detected. A typical conclusion is that evidence of density dependence is rare
(Dempster 1983; Stiling 1987). Part of the reason for this is that most life table
studies are of short duration and the problems of statistical power indicated
earlier limit our ability to detect density dependence when it exists. In addition, data from most studies contain a considerable degree of measurement
error. Such error can also obscure density-dependent relationships (Hassell
1985). Finally, the action of many density-dependent mortalities may lag
behind those of their hosts. The methods described earlier will not detect their
action. I discuss this in the next section.
Detection of Delayed Density Dependence
Time lags in density-dependent responses are common in population systems.
For example, it is typical for a predator or parasitoid to respond numerically to
changes in density of its host, but this response typically lags behind that of its
host by at least one generation. The result is that peak predator density and
hence peak mortality of the host occurs after the host has declined dramatically
from peak density. Plots of mortality against density may reveal no positive
relationship between the two, even if it is clear that the predator is regulating
its host. Such responses are known as delayed density dependence. Different
techniques have been developed to detect it.
The first of these techniques were graphical in nature (Hassell and Huffaker 1969; Varley et al. 1973). If one plots mortality against density and connects successive years, a counterclockwise spiral is evident (figure 6.3). If the
data consist of census data rather than mortality, that is, successive generations
of density counts, then connection of successive years on a graph of R plotted
against N t or X t yields a clockwise spiral (figure 6.3).
A major advance in detection of delayed density dependence was developed by Turchin (1990). He applied time series analyses (Box and Jenkins
1976) that have had wide application in econometrics and the physical sciences. The methods involve fitting a model similar to equation 6.2 but with
terms representing the effects of density in generations before the last one:
X t +1 = α + β X t + γ X t –1 . . . + ε t
(6.5)
Partial autocorrelation analysis tells you whether there is significant delayed
density dependence.
To illustrate this method, I give two examples. The first one is undoubtedly
199
detected. A typical conclusion is that evidence of density dependence is rare
(Dempster 1983; Stiling 1987). Part of the reason for this is that most life table
studies are of short duration and the problems of statistical power indicated
earlier limit our ability to detect density dependence when it exists. In addition, data from most studies contain a considerable degree of measurement
error. Such error can also obscure density-dependent relationships (Hassell
1985). Finally, the action of many density-dependent mortalities may lag
behind those of their hosts. The methods described earlier will not detect their
action. I discuss this in the next section.
Detection of Delayed Density Dependence
Time lags in density-dependent responses are common in population systems.
For example, it is typical for a predator or parasitoid to respond numerically to
changes in density of its host, but this response typically lags behind that of its
host by at least one generation. The result is that peak predator density and
hence peak mortality of the host occurs after the host has declined dramatically
from peak density. Plots of mortality against density may reveal no positive
relationship between the two, even if it is clear that the predator is regulating
its host. Such responses are known as delayed density dependence. Different
techniques have been developed to detect it.
The first of these techniques were graphical in nature (Hassell and Huffaker 1969; Varley et al. 1973). If one plots mortality against density and connects successive years, a counterclockwise spiral is evident (figure 6.3). If the
data consist of census data rather than mortality, that is, successive generations
of density counts, then connection of successive years on a graph of R plotted
against N t or X t yields a clockwise spiral (figure 6.3).
A major advance in detection of delayed density dependence was developed by Turchin (1990). He applied time series analyses (Box and Jenkins
1976) that have had wide application in econometrics and the physical sciences. The methods involve fitting a model similar to equation 6.2 but with
terms representing the effects of density in generations before the last one:
X t +1 = α + β X t + γ X t –1 . . . + ε t
(6.5)
Partial autocorrelation analysis tells you whether there is significant delayed
density dependence.
To illustrate this method, I give two examples. The first one is undoubtedly