The Cerulean Warbler is exhibiting severe declines which, if they
persist, will lead to the extirpation of this species over much of its
range. In the animation below, we show a potential timeframe for this
extirpation, projecting trends in the species abundance as determined in
our earlier study of the species in the Prairie Hardwood Transition
(Thogmartin et al. 2004). The peak predicted abundance in the Prairie
Hardwood Transition occurs in southeastern Michigan, at Allegan State
Game Area.

![](media/image1.jpeg){width="6.492361111111111in"
height="5.144444444444445in"}

[This animated
projection](http://www.umesc.usgs.gov/terrestrial/migratory_birds/bird_conservation/cerulean_warbler.swf)
is a deterministic assessment and does not recognize the uncertainty
associated with the estimates of trend.

The U.S. Fish and Wildlife Service is considering listing the Cerulean
Warbler under the Endangered Species Act. To help the U.S. Fish and
Wildlife Service determine whether this listing is appropriate, we
applied the best available science in estimating probabilities of
extinction for this species. We employed a diffusion approximation
approach to solving this problem, which accounts for uncertainty in the
estimation of the population trend. Available methods for parameterizing
a diffusion model include maximum likelihood methods (Dennis et al.
1991), these same methods but using a running sum transformation of the
data (Holmes 2001), maximum likelihood methods employing a Kalman filter
(Harvey 1989, Lindley 2003), slope methods (Holmes 2001, Holmes and
Fagan 2002), and asymptotically unbiased estimators for random matrix
products (Heyde and Cohen 1985). Holmes (2004) provides a thorough
treatment of these methods.

METHODS

The various diffusion models require a time series of population
estimates to predict probability and time to extinction. However, for
the Cerulean Warbler, absolute estimates of population size do not
exist. Indices of relative abundance do however exist, as derived from
modeling of population trends with Breeding Bird Survey data (Link and
Sauer 2002, Sauer et al. 2003). Additionally, K. Rosenberg and P.
Blancher, as part of the North American Landbird Conservation Plan,
derived a continental estimate of Cerulean Warbler population size (Rich
et al. 2004). These population size estimates are ostensibly relevant to
the mid-point of the 1990s. The Bird Conservation Regions we examined
included all regions for which there were at least five Breeding Bird
Survey routes informing trend estimates. These included BCR29 (Piedmont,
*n* = 7 survey routes), BCR28 (Appalachian Mountains, *n* = 143), BCR25
(West Gulf Coastal Plain / Ouachitas, *n* =5), BCR24 (Central Hardwoods,
*n* = 35), BCR23 (Prairie Hardwood Transition, *n* = 15), BCR22 (Eastern
Tallgrass Prairie, *n* = 9), and BCR13 (Lower Great Lakes / St. Lawrence
Plain, *n* = 19) (Figure 1). We also used the survey-wide estimate of
trend resulting from 238 survey routes.

The Rosenberg and Blancher population size estimate was associated with
the annual index of relative abundance for 1995. These estimates were
extrapolated to the years 1966--1994 and 1996--2005 by multiplying the
ratio of the relative abundance estimates (RA~t~/RA~1995~) by the
population size estimate for 1995 (Table 1).

We calculated the probability of a 90% decline using μ~run~, σ^2^~μ~,
and σ^2^~slp~ as the data. μ~run~ is the rate at which median log
population size (log(N~t~/N~0~), changes through time as estimated by
maximum likelihood methods described by Dennis et al. (1991), but using
a running sum transformation of the data as identified by Holmes (2001).
σ^2^~slp~ is the slope of the variance in the log population size versus
τ, the time span 1966--2005. σ^2^~μ~ is the point estimate for the
variability or error in the diffusion process.

The risk metric, *p*(ψ = φ \| data), is calculated by integrating the
probability of the data given a particular set of parameter values times
the probability of that parameter set. Standard reference priors for
normal variances were uniform. The data provide much information on
μ~run~, resulting in posterior distributions which are considerably
different than the priors. The same cannot be said for σ^2^, but Holmes
(2004) provided insight into appropriate priors for the variance terms.
Holmes (2004) employed both uniform and gamma priors, but the results
are largely the same for these data, so we concentrated our inference on
those resulting from the uniform priors.

σ^2^~np~, *r*^2^, λ (%) are additional parameters derived in this
process. σ^2^~np~ is the nonprocess variability, which is equal to the
nonprocess error in the age-structured population process plus
nonprocess error from sampling error in the censuses of the population.
*r*^2^ is for the linearity of variance slope. λ (%) is the population
rate of and percent change in the population size.

RESULTS

Eight models and their diagnostics were calculated to derive point
estimates for the regions and the entire survey (Table 2). Point
estimates for the regional and survey-wide diffusion models suggested
that Cerulean Warblers were declining across the survey by 3.1% per
annum (Table 3). There was heterogeneity in this decline across the
various Bird Conservation Regions, with the decline ranging from 1% per
annum in BCR29 and 16.9% per annum in BCR25. This latter decline
constitutes a population collapse, though we are reminded that it is
derived from data collected at only five survey sites.

The majority of the posterior probability distribution of the
survey-wide λ is in the region λ \< 1 (Figure 2), indicating that the
data give high support to long-term declining dynamics. The probability
that the population as a whole and at the level of the individual Bird
Conservation Regions ever experiences a 90% decline within the next
century is outlined in Figure 3. There is approximately an 80%
probability of a 90% decline within 80 years for the species (the
survey-wide estimate). The probability of a 90% decline occurs within
the next couple decades for the population in BCRs 22 and 25, whereas
BCRs 13, 24, and 28 do not begin to exhibit a 90% decline for 5 decades.
In the core of the species range, BCR28, there was approximately a 50%
probability of a 90% decline within 80 years. All of the Bird
Conservation Regions except for BCR29 exhibit a substantial probability
of a 90% decline within the next century; all but BCR28 and 29 exhibited
\>90% probability of a 90% decline within the century.

LITERATURE CITED

Dennis, B., P. L. Munholland, and J. M. Scott. 1991. Estimation of
growth and extinction parameters for endangered species. Ecological
Monographs 61:115--143.

Harvey, A. C. 1989. Forecasting, structural time series models and the
Kalman filter. Cambridge University Press, Cambridge, UK.

Heyde, C. C., and J. E. Cohen. 1985. Confidence intervals for
demographic projections based on products of random matrices.
Theoretical Population Biology 27:120--153.

Holmes, E. E. 2001. Estimating risks in declining populations with poor
data. Proceedings of the National Academy of Sciences (USA)
98:5072--5077.

Holmes, E. E. 2004. Beyond theory to application and evaluation:
diffusion approximations for population viability analysis. Ecological
Applications 14:1272--1292.

Holmes, E. E., and W. F. Fagan. 2002. Validating population viability
analysis for corrupted data sets. Ecology 83:2379--2386.

Lindley, S. T. 2003. Estimation of population growth and extinction
parameters from noisy data. Ecological Applications 13:806--813.

Link, W. A., and J. R. Sauer. 2002. A hierarchical model of population
change with application to Cerulean Warblers. Ecology 83:2832--2840.

Rich, T. D., C. J. Beardmore, H. Berlanga, P. J. Blancher, M. S. W.
Bradstreet, G. S. Butcher, D. W. Demarest, E. H. Dunn, W. C. Hunter, E.
E. Iñigo-Elias, J. A. Kennedy, A. M. Martell, A. O. Panjabi, D. N.
Pashley, K. V. Rosenberg, C. M. Rustay, J. S. Wendt, T. C. Will. 2004.
Partners in Flight North American Landbird Conservation Plan. Cornell
Lab of Ornithology. Ithaca, NY. Partners in Flight website. URL:
\<http://www.partnersinflight.org/cont_plan/\> (VERSION: March 2005).

Sauer, J. R., J. E. Fallon, and R. Johnson. 2003. Use of North American
Breeding Bird Survey data to estimate population change for bird
conservation regions. Journal of Wildlife Management 67:372--389.

Thogmartin, W. E., J. R. Sauer, and M. G. Knutson. 2004. A hierarchical
spatial model of avian abundance with application to Cerulean Warblers.
Ecological Applications 14:1766--1779.

Table 1. Reconstructed time series of regional abundance for the
Cerulean Warbler in the United States.

  ------- -------- ----------- ---------- ---------- --------- --------- --------- -------------
  YEAR    BCR29    BCR28       BCR25      BCR24      BCR23     BCR22     BCR13     Survey-wide

  1966    2879.0   985830.2    154114.9   174414.7   37550.6   37860.6   49395.9   1365935.6

  1967    3327.9   879475.3    147598.8   171807.2   37110.5   27281.1   48662.3   1223530.7

  1968    3098.7   997195.3    137555.6   166006.3   36227.1   25813.8   47311.3   1325883.0

  1969    2822.8   986090.8    72095.6    159501.9   32179.6   40497.4   46408.4   1283967.3

  1970    2644.2   1064193.6   68419.1    152099.8   31577.4   21540.4   45888.4   1331935.5

  1971    2469.5   956771.8    68090.3    144636.3   29615.9   19486.8   44525.3   1213042.9

  1972    2659.8   924887.1    59242.8    141520.1   27330.5   18217.1   43559.7   1172557.9

  1973    2465.2   876192.7    39724.3    132024.6   26487.8   18003.2   41782.9   1103628.5

  1974    2354.5   787737.2    32670.2    128507.2   24350.9   16843.7   38711.6   1004883.6

  1975    2401.9   738949.8    27986.4    124758.7   23994.2   14831.0   37011.4   944715.5

  1976    2316.3   863952.7    23066.4    120960.6   23626.6   13566.3   36201.7   1058190.6

  1977    2487.5   716010.8    19013.3    117478.4   23509.4   12680.9   34912.6   908093.4

  1978    2232.9   619877.4    18122.6    110401.3   20054.2   11633.4   33121.7   799484.4

  1979    2355.4   675108.8    13510.5    106411.0   19084.4   17823.8   33244.1   851653.3

  1980    2243.2   591633.9    11286.6    106045.0   18335.7   11191.8   33112.5   764264.2

  1981    2097.6   607967.9    9451.3     100003.2   18112.1   9173.7    31181.3   767281.6

  1982    2142.2   514101.2    7912.0     98709.8    18767.2   11337.9   31414.9   678256.0

  1983    2274.0   627813.7    6635.7     93913.2    16634.4   10657.3   28059.6   775552.5

  1984    2775.1   546110.3    5499.8     90093.9    14599.7   8556.8    27642.2   690340.1

  1985    2092.1   569129.6    4639.0     88321.3    14538.9   7784.7    26836.0   705632.9

  1986    2123.5   530863.2    3852.9     85798.2    13850.6   9252.0    26646.5   666951.6

  1987    2342.2   510878.5    3840.9     80922.0    13242.4   6672.5    27216.8   639460.5

  1988    1936.9   520344.1    2673.7     81160.0    13487.8   14132.2   26941.7   654750.1

  1989    2248.2   496513.4    2190.1     76895.2    12861.1   9056.7    25982.2   621192.1

  1990    2233.5   506646.0    2177.5     77657.9    12022.5   13909.6   24076.8   633009.4

  1991    1975.7   474933.2    1542.3     70944.1    11612.2   5973.3    25379.8   590704.6

  1992    1957.7   467387.9    1285.3     69713.0    10142.6   5632.8    24910.4   577046.7

  1993    1975.7   432442.4    1075.8     71090.1    10292.4   5283.6    27822.2   549666.5

  1994    2215.0   408693.9    893.7      67941.7    9613.4    4806.5    26078.1   524316.8

  1995    1990     445174      900        64542      9854      4664      23619     560000

  1996    1940.0   489921.9    621.7      62982.7    9930.8    4467.3    22989.7   590341.3

  1997    2156.7   429632.4    515.3      62657.4    9369.0    3678.8    25680.8   540079.0

  1998    1932.3   369321.6    431.6      59142.6    8203.8    5407.4    22189.0   466786.7

  1999    1889.0   317250.2    359.6      57930.0    8078.5    3311.5    21956.6   419557.9

  2000    1835.7   318387.1    302.2      56243.3    7784.5    3114.8    21310.2   412438.0

  2001    1838.5   313830.0    247.5      54460.4    7878.1    3082.9    20920.8   406499.6

  2002    2167.3   318073.7    207.7      52709.9    7253.6    3123.4    21728.5   409745.3

  2003    2194.4   357751.7    171.6      53636.4    6971.5    2849.4    21948.3   451212.6

  2004    2150.6   328832.7    142.3      49439.4    7503.7    2670.0    22533.0   417685.1

  2005    1915.4   296880.4    117.5      50754.1    7385.6    2663.6    24043.8   390954.4
  ------- -------- ----------- ---------- ---------- --------- --------- --------- -------------

Table 2. Basic diagnostics for estimating parameters and risk metrics
from regional time series of Cerulean Warbler abundance using the
Dennis-Holmes method. The diagnostics include tests of normality
(Kolmogorov Smirnov), trend (Least-squares) and autocorrelation (first
order).

  ----------------- ----------------- ----------------- -----------------
  Region (*n*)      GOF *P*-value     TREND *P*-value   AUTO *P*-value

  BCR29             0.498             0.822             0.069

  BCR28             0.736             0.873             0.079

  BCR25             \<0.001           0.950             0.040

  BCR24             0.882             0.259             \<0.001

  BCR23             0.308             0.285             0.140

  BCR22             \<0.001           0.818             0.008

  BCR13             0.081             0.123             0.522

  Survey-wide       0.506             0.820             0.089
  ----------------- ----------------- ----------------- -----------------

Table 3. Point estimates for the diffusion approximation of regional
predictions of Cerulean Warbler population change. μ~run~ is the rate at
which median log population size (log(N~t~/N~0~), changes through time
as estimated by maximum likelihood methods described by Dennis et al.
(1991), but using a running sum transformation of the data as identified
by Holmes (2001). σ^2^~slp~ is the slope of the variance in the log
population size versus τ, the time span 1966--2005. σ^2^~μ~ is the point
estimate for the variability or error in the diffusion process. σ^2^~np~
is the nonprocess variability, which is equal to the nonprocess error in
the age-structured population process plus nonprocess error from
sampling error in the censuses of the population. *r*^2^ is for the
linearity of variance slope. λ (%) is the population rate of and percent
change in the population size.

  ------------- ---------- ----------- --------- ---------- --------- ------------
  Region        μ~run~     σ^2^~slp~   σ^2^~μ~   σ^2^~np~   *r*^2^    λ (%)

  BCR29         -0.01011   0.00107     0.00003   0.00445    0.996     0.990 (-1.0)

  BCR28         -0.03012   0.00390     0.00005   0.00390    0.965     0.970 (-3.0)

  BCR25         -0.18569   0.00037     0.00001   0.00732    0.866     0.831
                                                                      (-16.9)

  BCR24         -0.03276   0.00010     0.00001   0.00030    0.980     0.968 (-3.3)

  BCR23         -0.04422   0.00044     0.00001   0.00112    0.967     0.957 (-4.4)

  BCR22         -0.06815   0.01374     0.00040   0.03773    0.995     0.934 (-6.6)

  BCR13         -0.02094   0.00087     0.00002   0.00089    0.999     0.979 (-2.1)

  Survey-wide   -0.03155   0.00113     0.00003   0.00022    0.978     0.969 (-3.1)
  ------------- ---------- ----------- --------- ---------- --------- ------------

![](media/image2.jpeg){width="6.490972222222222in"
height="5.011805555555555in"}

Figure 1. Bird Conservation Regions for which the probability of
extinction was predicted for Cerulean Warblers.

![](media/image3.wmf){width="5.9319444444444445in"
height="7.677777777777778in"}

Figure 2. Posterior probability distribution of λ, the population rate
of change, as estimated for the Cerulean Warbler using a diffusion
approximation for Breeding Bird Survey-derived estimates of population
abundance for the period 1996--2005.

Figure 3. Expected total probability of 90% decline within the next
century. The prior uniform expectation is provided as a dashed line; the
posterior expectation given the data is the solid line.

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