The bowhead whale

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The bowhead whale
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Reference
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The problem

• Bowhead whales summer in Arctic ocean and winter
  in the Bering sea.
• Discovered by Yankee whalers in 1848 and rapidly
  depleted.
• Whaling effectively stopped in 1915 due to poor
  catch rates and collapse of markets.
• Officially protected since 1946 by IWC.
• Aboriginal whaling remains

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Current problem

• Estimate current removals in relation to
  replacement yield
• Replacement yield is the harvest that would keep
  the population at its current level

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Data available

• Historical catches
• Recent counts at Pt. Barrow
• Biological information on age distribution,
  pregnancy rates etc

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Count data

• From 1978-1984 visual counts of individuals
  passing Pt. Barrow
• From 1984 onwards this has been supplemented by
  hydrophone recordings
• From these data they estimate a probability
  distribution of the 1988 population size
• And a rate of increase from 1978 to 1988

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Model births and survival
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Assumptions

• assumed to be an immature survival rate,
• a mature survival rate
• and a maximum age, at which time all individuals
  die.
• Fecundity is assumed to be density dependent, but
  age independent

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Fecundity
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Parameters
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Model procedure SIR

• For each parameter draw a value from the prior
  distribution
• Project forward using known catches to generate
  estimate of 1988 population size and growth rate
  from 78 to 88
• Calculate the likelihood (goodness of fit) for
  the projection
• store the parameters and the goodness of fit
• Repeat the draw from the prior 100,000 or more
  times
• Now sample from the 100,000 draws about 5,000
  based on the goodness of fit
• Plot posterior distributions of quantities of
  interest

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Model outputs

• Probabilistic distributions of historical
  trajectories
• Probability distribution of current stock size
  (or stock size in any year)
• Probability distribution of current replacement
  yield

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Praise and Criticisms

• Received an award for best applied paper of year
  in JASA
• Subject to criticism with IWC scientific
  committee over formulation of priors,
  particularly prior on initial population size

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Key Points

• No demographic stochasticity
• Integrated data with projections
• Almost exactly the opposite of the Doak approach
• No alternative policies considered

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How uncertainty considered