Geoscience Reference
In-Depth Information
of stabilizing the population. Plots of some measure of percentage mortal-
ity versus density of the population reveal density dependence. For example,
Varley and Gradwell (1968) presented data on the winter moth, a defoliator
of oak trees in Great Britain. They collected data on sources of mortality on
successive instars or life stages over a period of 15 generations (years) at one
site. They expressed mortality as
k
-values (
k
= -log
10
(proportion surviving))
and plotted
k
-values for each cause of mortality against the log density of the
individuals present at the beginning of the stage or age category on which the
agent of mortality acted. They used standard linear regression to determine
whether mortality increased or decreased with density. The strong negative
bias described earlier for analysis of density time series was not present here
because the measurements of mortality differed from the measures of density.
However, a number of statistical problems involve violations of the usual
assumptions of linear regression. The regression may be nonlinear, measure-
ment error may affect the estimates of both density and mortality, the variance
of the
k
-values may vary systematically with density, and the error terms may
not be independent because the data are obtained from time series. Solutions
have been proposed for several of these problems (for example, see Hassell et
al. 1987). However, Vickery (1991) analyzed the various extant methods and
found them all either biased or lacking in statistical power. He advocated using
a randomization test identical to that of Pollard et al. (1987), but applied to
data on stage-specific mortality instead of time series of density.
Use of these techniques rests on the assumption that the data collected for
mortality and density are accurate and unbiased. This may not be easy to
achieve, particularly when ages or life stages overlap temporally. Various tech-
niques may be used to convert densities or numbers present in periodic sam-
ples to estimates of numbers entering particular life stages or age categories (see
reviews in Bellows et al. 1992). Similar techniques exist to convert mortalities
or rates of infection obtained from periodic samples to the stage-specific
mortality that best represents the overall impact of the agent of mortality (van
Driesche et al. 1991). Additional techniques may be required when two or
more agents of mortality act contemporaneously (Elkinton et al. 1992). It is
beyond the scope of this review to describe these techniques here.
Density dependence is often reported in studies in which data on mortal-
ity or survival are obtained simultaneously from several different populations
that vary in density. It is important to realize that the processes that give rise to
density dependence in such studies may not be the same as those producing
density dependence in studies wherein mortality and density are shown to vary
over time from one or more populations. For example, Gould et al. (1990)
