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Lloyd Goldwasser, Scott Ferson, and Lev Ginzburg
difference between the actual and the expected tosses is negligible. Their divergence is linear, however, so that the eventual excess of heads over tails lies well
outside the predicted range (Fig. 11.3). Although the analogy between coin tossing and population dynamics is only heuristic, it suggests that measurement error
can play a significant role in limiting our ability to project population trajectories
into the future. In general, for long periods of time, measurement error may be a
dominant cause of our uncertainty about the future, whereas in the short run
natural variability may dominate.
In assessing the effects of measurement error on our extinction risk analysis of
owl population dynamics, we need to use the standard errors of the demographic
rates (Table 11.1) to construct simulations reflecting the potential variation under
the uncertainty in the estimates at the outset (as opposed to temporal variability
that enters the picture during the simulation). There are several ways we might
proceed. One way would be to let each mean vital rate be selected at random from
a distribution (whose dispersion is characterized by its standard error) at the start
of each trajectory and letting it be fixed at this random value for the 100 years of
the simulation. Annual variation in the vital rate would be simulated as before
with reference to the standard deviation for each vital rate. The resulting trajectories would therefore express uncertainty both from measurement error and from
temporal variability. However, the output summaries would also confound both
kinds of uncertainty.
Another way to assess measurement error’s effect on our risk analysis would be
to explore how it causes divergence under a fixed sequence of environmental
variation. For instance, each trajectory would use the same sequence of environmental fluctuations, so that any spreading out of this trajectory over time shows
the growing effects of the initial inaccuracy on the uncertainty of the projection.
When the lower limit of the 95% confidence interval of the population size
reaches 15 breeding pairs (or whatever is the smallest population size that could
be self-sustaining), then the uncertainty of the demographic rates has rendered the
model unable to distinguish between persistence and extinction of the population.
Projections beyond this time would clearly be of limited use. Accordingly, we
would take the maximum time scale for which meaningful estimates of population
viability are possible to be the number of years before the 95% confidence interval
drops below the extinction threshold, regardless of the time scales used by any
arbitrary endangerment criteria. Such a time scale would be the longest one for
which it is appropriate to compare different scenarios, and subsequent comparisons of the viability of this population should be based on it.
A third, more straightforward, way to explore the effects of measurement error
is simply to bump the vital rates simultaneously up or down by magnitudes
proportional to their respective measurement errors and compute the resulting
variation in quasi-extinction risk. This technique is not generally possible in other
applications of risk assessment because factors are generally related to the endpoint variable in complex ways. In this case, however, it yields an interpretable
result because all the elements of the vital rate matrix contribute positively to
population growth and contribute negatively to extinction risk.
Lloyd Goldwasser, Scott Ferson, and Lev Ginzburg
difference between the actual and the expected tosses is negligible. Their divergence is linear, however, so that the eventual excess of heads over tails lies well
outside the predicted range (Fig. 11.3). Although the analogy between coin tossing and population dynamics is only heuristic, it suggests that measurement error
can play a significant role in limiting our ability to project population trajectories
into the future. In general, for long periods of time, measurement error may be a
dominant cause of our uncertainty about the future, whereas in the short run
natural variability may dominate.
In assessing the effects of measurement error on our extinction risk analysis of
owl population dynamics, we need to use the standard errors of the demographic
rates (Table 11.1) to construct simulations reflecting the potential variation under
the uncertainty in the estimates at the outset (as opposed to temporal variability
that enters the picture during the simulation). There are several ways we might
proceed. One way would be to let each mean vital rate be selected at random from
a distribution (whose dispersion is characterized by its standard error) at the start
of each trajectory and letting it be fixed at this random value for the 100 years of
the simulation. Annual variation in the vital rate would be simulated as before
with reference to the standard deviation for each vital rate. The resulting trajectories would therefore express uncertainty both from measurement error and from
temporal variability. However, the output summaries would also confound both
kinds of uncertainty.
Another way to assess measurement error’s effect on our risk analysis would be
to explore how it causes divergence under a fixed sequence of environmental
variation. For instance, each trajectory would use the same sequence of environmental fluctuations, so that any spreading out of this trajectory over time shows
the growing effects of the initial inaccuracy on the uncertainty of the projection.
When the lower limit of the 95% confidence interval of the population size
reaches 15 breeding pairs (or whatever is the smallest population size that could
be self-sustaining), then the uncertainty of the demographic rates has rendered the
model unable to distinguish between persistence and extinction of the population.
Projections beyond this time would clearly be of limited use. Accordingly, we
would take the maximum time scale for which meaningful estimates of population
viability are possible to be the number of years before the 95% confidence interval
drops below the extinction threshold, regardless of the time scales used by any
arbitrary endangerment criteria. Such a time scale would be the longest one for
which it is appropriate to compare different scenarios, and subsequent comparisons of the viability of this population should be based on it.
A third, more straightforward, way to explore the effects of measurement error
is simply to bump the vital rates simultaneously up or down by magnitudes
proportional to their respective measurement errors and compute the resulting
variation in quasi-extinction risk. This technique is not generally possible in other
applications of risk assessment because factors are generally related to the endpoint variable in complex ways. In this case, however, it yields an interpretable
result because all the elements of the vital rate matrix contribute positively to
population growth and contribute negatively to extinction risk.