26
27
Sensitivity is the effect on λ of an absolute change
in the vital rates (
a
ij
, the arcs in the life cycle graph
[
Figure 8] and the cells in the matrix, A [
Figure 9]).
Sensitivity analysis provides several kinds of useful
information (see Caswell 2001, pp. 206–225). First,
sensitivities show how important a given vital rate is
to λ, which Caswell (2001, pp. 280–298) has shown to
be a useful integrative measure of overall fitness. One
can use sensitivities to assess the relative importance
of survival (P
ij
) and fertility (F
ij
) transitions. Second,
sensitivities can be used to evaluate the effects of
inaccurate estimation of vital rates from field studies.
Inaccuracy will usually be due to a paucity of data, but
it could also result from use of inappropriate estimation
techniques or other errors of analysis. In order to
improve the accuracy of the models, researchers should
concentrate additional effort on transitions with large
sensitivities. Third, sensitivities can quantify the effects
of environmental perturbations, wherever those can be
linked to effects on stage-specific survival or fertility
rates. Fourth, managers can concentrate on the most
important transitions. For example, they can assess
which stages or vital rates are most critical to increasing
the population growth of endangered species or the
“weak links” in the life cycle of a pest. Figure 10
shows the “possible sensitivities only” matrices for this
analysis (one can calculate sensitivities for non-existent
transitions, but these are usually either meaningless or
biologically impossible – for example, the biologically
impossible sensitivity of λ to the transition from Stage 2
“adult” back to being a Stage 1 first-year bird).
The summed sensitivity of λ to changes in
survival (65.2 percent of total sensitivity accounted
for by survival transitions) was greater than the
summed sensitivity to fertility changes (34.8 percent
of total). The single transition to which Stage l was
most sensitive was first-year survival (47.4 percent of
total). The second most important transition was first-
year reproduction (21.8 percent of total). The major
conclusion from the sensitivity analysis is that survival
rates and both kinds of first-year vital rates are most
important to population viability.
Elasticity analysis
Elasticities are useful in resolving a problem
of scale that can affect conclusions drawn from the
sensitivities. Interpreting sensitivities can be somewhat
misleading because survival rates and reproductive
rates are measured on different scales. For instance,
an absolute change of 0.5 in survival may be a large
alteration (e.g., a change from a survival rate of 90
percent to 40 percent). On the other hand, an absolute
change of 0.5 in fertility may be a very small proportional
alteration (e.g., a change from a clutch of 3,000 eggs to
2,999.5 eggs). Elasticities are the sensitivities of λ to
proportional changes in the vital rates (a
ij
) and thus
partly avoid the problem of differences in units of
measurement (for example, we might reasonably equate
changes in survival rates or fertilities of 1 percent).
The elasticities have the useful property of summing
to 1.0. The difference between sensitivity and elasticity
conclusions results from the weighting of the elasticities
by the value of the original arc coefficients (the a
ij
cells
of the projection matrix). Management conclusions will
depend on whether changes in vital rates are likely to
be absolute (guided by sensitivities) or proportional
(guided by elasticities). By using elasticities, one can
further assess key life history transitions and stages as
well as the relative importance of reproduction (F
ij
) and
survival (
P
ij
) for a given species. It is important to note
1
2
3
1
0.393
1.27
0.796
2
0.28
3
0.67
0.42
1
2
3
1
P
21
m
1
P
32
m
a
P
a
m
a
2
P
21
3
P
32
P
a
Figure 9a. Symbolic values for the projection matrix of vital rates, A (with cells a
ij
) corresponding to the long-billed
curlew life cycle graph of
Figure 8. Meanings of the component terms and their numeric values are given in
Table
1.
Figure 9b. Numeric values for the projection matrix of vital rates, A (with cells
a
ij
) corresponding to the long-billed
curlew
life cycle graph of Figure 8.
26
27
that elasticity as well as sensitivity analysis assumes that
the magnitude of changes (perturbations) to the vital
rates is small. Large changes require a reformulated
matrix and reanalysis.
Figure 11 shows elasticities for the long-billed
curlew. λ was most elastic to changes in first-year
survival (e
21
= 29.7 percent of total elasticity). Next
most elastic were first- and second-year reproduction
(e
11
= 19.1 percent; e
12
= 17.3 percent of total elasticity).
Survival of older birds was relatively unimportant (e
12
= 17.3 percent of total elasticity). The sensitivities
and elasticities for long-billed curlew were generally
consistent in emphasizing first-year transitions. Thus,
first-year transitions, particularly survival rates, are the
data elements that warrant careful monitoring in order
to refine the matrix demographic analysis.
Other demographic parameters
The stable stage distribution (SSD; Table 2)
describes the proportion of each stage or age-class
in a population at demographic equilibrium. Under
a deterministic model, any unchanging matrix will
converge on a population structure that follows the
stable age distribution, regardless of whether the
population is declining, stationary, or increasing. Under
most conditions, populations not at equilibrium will
converge to the SSD within 20 to 100 census intervals.
For long-billed curlew at the time of the post-breeding
annual census (just after the end of the breeding season),
fledglings represent 62.6 percent of the population,
yearlings (second-year birds) represent 17.4 percent of
the population, and older birds represent 20 percent of
the population. Reproductive values (Table 3) can be
thought of as describing the value of a stage as a seed for
population growth relative to that of the first (newborn
or, in this case, fledgling) stage (Caswell 2001). The
reproductive value is calculated as a weighted sum of
the present and future reproductive output of a stage
discounted by the probability of surviving (Williams
1966). The reproductive value of the first stage is, by
definition, 1.0. A second-year female individual (Stage
2) is “worth” 2.2 fledglings, and older females are worth
1.4 fledglings. The second-year females are the core of
the population under this model. The cohort generation
time for this species was 2.1 years (SD = 1.1 years).
Stochastic model
We conducted a stochastic matrix analysis for
long-billed curlew. We incorporated stochasticity
in several ways (Table 4), by varying different
combinations of vital rates, and by varying the amount
of stochastic fluctuation. We varied the amount of
fluctuation by changing the standard deviation of the
truncated random normal distribution from which the
stochastic vital rates were selected. To model high levels
of stochastic fluctuation, we used a standard deviation
of one quarter of the “mean” (with this “mean” set at the
value of the original matrix entry [vital rate], a
ij
under
the deterministic analysis). Under Case 1, we subjected
all the fertility arcs (F
11
, F
12
, and F
13
) to high levels
of stochastic fluctuations (SD one quarter of mean).
Under Case 2, we varied all the survival arcs (P
21
,
P
32
and P
33
) with high levels of stochasticity (SD one
quarter of mean). Under Case 3, we varied the first-year
transitions (P
21
and F
11
) with high levels of stochastic
fluctuation. In Case 4, we varied those same first-year
transitions, but with only half the stochastic fluctuations
(SD one eighth of mean). Each run consisted of 2,000
census intervals (years) beginning with a population
1
2
3
1
0.489
0.136
0.157
2
1.065
3
0.186
0.214
1
2
3
1
0.191
0.173
0.125
2
0.297
3
0.125
0.09
Figure 10. Possible sensitivities only matrix, S
p
(blank cells correspond to zeros in the original matrix, A). The λ of
long-billed curlew is most sensitive to changes in first-year survival (Cell
s
21
= 1.065).and first-year fertility (Cell s
11
= 0.489).
Figure 11. Elasticity matrix, E (remainder of matrix consists of zeros). The elasticities have the property of summing
to 1.0. The λ of long-billed curlew is most elastic to changes in first-year survival (e
21
= 0.297), followed by first- and
second-year fertility (
e
11
= 0.191, e
12
= 0.173).