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ation is included in the estimates, consider an example population of 10 ani-
mals with a constant survival rate of 0.55. Thus, the actual temporal variation
is zero, yet a sequence of estimates of survival from this population suggests
considerable variation. That is, the estimates of survival would have a variance
of 0.55(1 - 0.55)/10 = 0.02475 if all 10 animals had a survival probability of
0.55. Furthermore, the only observed values of survival would be 0, 0.1, . . . ,
1.0. However, if the size of the population were increased to 100, you would
find that the variance of the sequence of estimates becomes 0.002475, a con-
siderable decrease. Thus randomly sampling the estimates from a population
of size 10 results in considerably more variation than from a population of
100. As a result, the demographic variation from the sampled population is
incorporated into the persistence model if the bootstrap approach is used.
A similar example can be used to demonstrate that sampling variation is
also inherent in bootstrapping from a sample of observed estimates. Suppose a
sample of 10 radiocollared animals is used to estimate survival for a population
of 100,000 animals (i.e., the finite sample correction term can be ignored).
The sampling variation of the estimates would be S (1 - S )/10, where S is the
true survival rate for the population assuming all animals had the same survival
rate. Now if a sample of 100 radiocollared animals is taken, the sampling vari-
ation reduces to S (1 - S )/100. Thus randomly sampling estimates with a boot-
strap procedure incorporates the sampling variation of the estimates into the
persistence model. As a result of the increased variation, persistence values will
be underestimated.
Therefore, I suggest not using the bootstrap approach demonstrated by
Stacey and Taper (1992) if unbiased estimates of persistence are required. Per-
sistence estimates developed with this procedure will generally be too low; that
is, you will conclude that the population is more likely to go extinct than it
really will. However, methods such as shrinkage estimation of variances (K. P.
Burnham, personal communication 1997) may prove useful in removing sam-
pling variance from the estimates and make the bootstrap procedure more
applicable to estimating population persistence.
Basic Population Model and Density Dependence
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Leslie matrix models (Leslie 1945, 1948; Usher 1966; Lefkovitch 1965;
Caswell 1989; Manly 1990) are commonly used as the modeling framework
for population viability models. Density dependence must be incorporated
into the model; that is, basic parameters must be a function of population size.
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