Estimating Sperm Whale Abundance with Bayesian Mark-Recapture

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Estimating Sperm Whale Abundance with Bayesian
Mark-Recapture
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Authors

• Joe Liddle
• Jan Straley
• University of Alaska
• Southeast, Sitka Campus

• Milo Adkison
• Terry Quinn
• University of Alaska Fairbanks,
• School of Fisheries and Ocean Sciences

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Collaborators
The Southeast Alaska Sperm Whale Avoidance
Project (SEASWAP) Team F/V EH, Myriad, Swan,
Vallee Lee, Cherokee, Cobra, Norfjord, Kamilar,
Ginny C, Katie-J, Kelly Marie, Ida June Jen
Cedarleaf, UAS Aaron Thode, Scripps Institution
of Oceanography Linda Behnken, Alaska Longline
Fishermens Association Tory OConnell, Alaska
Department of Fish and Game
Three Year Study Funded by North Pacific Research
Board
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Black Cod 4/lb Avg. 8lbs
CPUE reduced -3 p0.023
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Study Area
Shelf Break 12-20 miles offshore
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Prior to sampling What was known?

• Fishermen Biologists routinely see whales
  implying N gt 50
• North Pacific ballpark estimate
• 4000lt N lt 5000
• Eastern Gulf of Alaska study area is several
  hundred square miles
• Maybe N500 whales max?
• Ngt1000 is unrealistic

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Whale Marks

• Photograph flukes
• Fluke patterns are as unique as fingerprints
• Match photographs from year to year
• Difficult to matchlots of eyeball time
• Rain, waves, boat motion can interfere
• Poor quality photos discarded

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Whales Identified by Shape and Nicks on Trailing
Edge of Flukes
Same whale outlined in red
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Captures and Recaptures

• 2003 n114 whales photographed
• 2004 n225 m23 re-sightings
• 2005 n316 m36 re-sightings

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The Chapman Estimator
95 Confidence interval (35.05,157.9)
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ISSUES

• Small sample size.
• Variance of Chapman estimator too large.
• Upper limit is imprecise due to large variance.
• Lower confidence limit is less than known unique
  whales n1n2-m236.
• Prior information is available that is not used
  by Chapman estimator.

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Parametric Bootstrap3000 realizations of m2
hyper-geometric 95 confidence interval,
percentile method
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Beta prior distribution for p

• n1/N
• The prior was chosen so that expected value of p
  lt 0.5

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Beta posterior distribution for p

• Posterior is proportional to priorlikelihood.
• We get a beta posterior distribution with these
  updating rules.
• aam2
• bbn2-m2

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But what about N?
Given that pbeta(a,b) and
I then derived this distribution for N
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Bayesian Credible Interval for N
Percentiles for Nk can be found with a numerical
integral.
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METHOD COMPARISON (2004)
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Sensitivity of mean N to choice of prior
beta(a,b)
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2004-05 COMPARISON
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Conclusions

• Precise estimate of abundance.
• Narrow credible interval.
• Useful distribution for N.
• Utilizes prior information.
• Works with extremely small sample sizes.
• Widely applicable to other studies.
• Not sensitive to choice of prior.
• Between 65 and 128 sperm whales are depredating
  long line gear near Sitka.
• (95 Bayesian credible interval).

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Sperm Whale Male 18 m, Female 11 m, Deep divers
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Comparison with other multi-year methods
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SIMULATION

• 3000 Re-samplings of the data
• Fixed N100, n114, n225
• Chapman coverage90.1
• 95 confidence interval
• sampling w/o replacement
• Bayesian coverage95.1
• 95 credible interval
• replacement

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Parametric BootstrapR-code fragment

• parmwhalelt-function(iterations,n1,n2,m2)
• Nstar lt-(n11)(n21)/(m21)-1
• mboot lt-rhyper(iterations, n1, Nstar, n2)
• Nhats lt-(n11)(n21)/(mboot1)-1
• (various output statements)

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