Homework 7 for FISH 458, revised 26 May 2015
Bayesian population model of short-tailed albatrosses
Your name here1
1
School of Aquatic and Fishery Sciences, Box 355020, University of Washington, Seattle, WA, 981955020, your email address here
Abstract: Summarize your main findings here in less than 200 words, remember to include specifics
(estimated rate of increase, minimum abundance, etc.) [10 points].
Introduction
Short-tailed albatrosses (Phoebastria
albatrus) are one of the most endangered seabirds
in the world (Zador et al. 2008). In the late 1800s
they numbered several million adults, nesting in
at least 14 remote island sites. However, massive
commercial feather hunting from 1887 to 1933 (at
least 5 million were killed by 1902) led to their
being declared extinct in 1949
(http://seabirdbycatch.washington.edu/shorttailed-albatross). In 1950, however, Japanese
weather observers found a few breeding pairs on
Torishima Island.
Since then, a remarkable time series of annual
census counts by Dr. Hasegawa in 1954–1964 and
1979–2013 have revealed an almost perfect
exponential increase in their numbers (Fig. 1),
although their numbers remain far below 1% of
their pre-hunting levels.
Although the population is steadily increasing,
currently more than 80% of the population still
nests on Torishima Island, a remote small island to
the south-east of Japan. Unfortunately, this island is
an active volcano, leading to the possibility that at
any time nearly the entire population could be
wiped out by an eruption (Finkelstein et al. 2010).
This danger is real: in 1902 an eruption on
Torishima Island killed all 125 albatross hunters on
the island.
Like all albatrosses, short-tailed albatrosses are
long-lived (>50 yr) with a high survival rate
(>90%), and a long time from hatching to first
breeding (5 yr). Females lay a single egg in most
years although 1 in 4 females skips breeding each
year. Breeding pairs are faithful to each other and
both forage for food for the single chick that is
1
Figure 1 Census counts of the number of eggs
on Torishima Island over time, collected by Dr.
Hiroshi Hasegawa of Toho University. Egg
counts are a reliable index of the number of
breeding pairs in this population.
Figure 2 Torishima Island, the main breeding
site for short-tailed albatrosses. Source:
http://www.volcano.si.edu/volcano.cfm?vn=2
84090
Homework 7 for FISH 458, revised 26 May 2015
produced. During the non-breeding season they range throughout the North Pacific, but during
breeding they remain close to the two known breeding sites. (All biological information obtained
from http://seabirdbycatch.washington.edu/short-tailed-albatross.)
Accounting for both sexes, subadults aged 0-5 yr, females that skip breeding (25% of the
population), and albatrosses that nest on other islands (20% of the total), allows us to estimate
total population abundance by multiplying the number of eggs on Torishima Island by a factor of
7.1.
The main management concern in the US is that some fisheries occasionally encounter shorttailed albatrosses, and when they are entangled in trawling or long-lining fishing gear, they
drown. These bycatch deaths are therefore strictly managed by the US Fish and Wildlife Service
(USFWS), which currently mandates that bycatch levels may not exceed an average of 4.4 shorttailed albatrosses per year. Actual numbers of bycatch are, to date, much lower than this number:
three in the Alaska longline fisheries since 1998, and one off the U.S. West Coast since 2002.
The USFWS has outlined a recovery plan for short-tailed albatross with specific
requirements for population growth rate (≥ 6%/yr), number of islands with breeding colonies,
and total population size. In this paper we will focus on using a Bayesian exponential model to
estimate population trends and to calculate the impacts on growth rates of a bycatch level of 5
short-tailed albatrosses in 2014.
Methods
To estimate rates of change we fit an exponential model to the numbers of breeding pairs of
birds (which equals the number of eggs counted) starting in the year of minimum abundance,
1954:
(1)
N1954  N0
Nt 1  (1  r )Nt
We will assume that the abundance estimates are lognormally distributed with a very small
minimum CV of 0.02 plus additional variance CVadd, that will be estimated, and will be constant
across years. The total CV2 assumed for each abundance estimate is the sum of the minimum
assumed variance of 0.022 and the additional variance CV2add. The net result is the following
negative log-likelihood, summed over all years with data:


2

lnNt  lnNˆt 
2
2

(2)  lnL   ln 0.02  CVadd 
2

2  0.022  CVadd
t 



The estimable parameters are r, N0, and CVadd.
For part 1 (worth 40 points), read in the data from the data file “STAL data.csv” and store
this in a variable alb.data. Then create a function called getNLL() with parameters r, N0, CVadd,
and alb.data. This function will implement the exponential model and return –lnL for the model
fit to the data. [Check: if r=0.07, N0=10, and CVadd=0.05 then NLL = 45.516]
For part 2 (worth 20 points), create functions rNLprior(), N0NLprior() and CVaddNLprior()
which return the negative-log-priors for the respective model parameters. The priors are assumed
to be broad uniform distributions for all parameters, as follows:
r ~ U(-0.1, 0.2)
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Homework 7 for FISH 458, revised 26 May 2015
N0 ~ U(0, 50)
CVadd ~ U(0,0.3)
[Check: the sum of the three negative-log-priors should be 1.504]
For part 3 (worth 50 points) create a function runMCMC that implements an MCMC
algorithm to find the posteriors for your model parameters. This function should have parameters
ndraws, filename, rinit, N0init, and CVaddinit. In your function, read the data from the data file,
and calculate X = the initial sum of the NLL and the negative-log-priors for all parameters. Loop
through ndraws, and for each draw, sample a candidate draw from the previous draw (draw i) as
follows:
r* = ri + U(-0.01, 0.01)
N0* = N0i + U(-2, 2)
CVadd* = CVaddi + U(-0.05,0.05)
Calculate X* = sum of the NLL and negative-log-priors for r*, N0*, and CVadd*.
Then calculate the ratio R in likelihood space: R  exp( X i  X*) . If a random uniform number
Y~U(0,1) is smaller than R, then accept the draw and set ri+1 = r*, N0i+1=N0*, and
CVaddi+1=CVadd*. Otherwise reject the draw and set ri+1 = ri, N0i+1=N0i, and CVaddi+1=CVaddi.
Save the accepted draws in a matrix called posterior.
For part 4 (worth 40 points), first test runMCMC with ndraws = 100; and when satisfied it is
working, run it with ndraws=100000, filename="STAL data.csv", rinit=0.03, N0init=10,
CVaddinit=0.05. This will take some time (minutes to an hour).
From the resulting posterior, delete the first 20% of the runs (the burn-in period), then sample
every 40th draw using seq() to thin the chain to a manageable number of draws. The resulting
thinned chain of 2000 draws will still be somewhat autocorrelated but sufficient for our needs.
For part 5 (worth 30 points), from the thinned chain, calculate (1) the distribution of the
number of nesting pairs (eggs) on Torishimo Island in 2014: N2014,i  N0,i (1  ri )20141954 , (2) the
distribution of total numbers of birds in the whole population assuming that for every egg
produced there are 7.1 birds: N2014,i  7.1 and (3) the amount that the entire population will
increase by between 2014 and 2015: N2014,i  7.1  ri .
Extra credit part 6 (worth 10 points) if 5 short-tailed albatrosses were caught in US fisheries
in 2014, how much would the population growth rate (from 2014 to 2015) decline?
Results
For Parts 1-3, provide your R code.
For Part 4, from the thinned chain, provide the median and 95% quantiles (function quantile) for
r, N0, and CVadd (Table 1). Create separate histograms showing the posterior probability for
each of these three parameters (Figures 3-5).
For Part 5, from the thinned chain, provide the median and 95% quantile for the following: (1)
number of nesting pairs on Torishima in 2014, (2) total number of birds in the whole population,
and (3) the amount of birds by which the population will increase from 2014 to 2015.
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Homework 7 for FISH 458, revised 26 May 2015
For Part 6, provide an estimate of the population growth rate (from 2014 to 2015) with zero extra
deaths, and with 5 extra deaths.
Discussion
In this section interpret your results. Clearly state and explain your key results, and the
implications of your results for current management of short-tailed albatrosses and of the North
Pacific trawl fisheries. [10 points].
Acknowledgements
Leave this section blank.
References
Finkelstein, M.E., Wolf, S., Goldman, M., Doak, D.F., Sievert, P.R., Balogh, G., and Hasegawa,
H. 2010. The anatomy of a (potential) disaster: Volcanoes, behavior, and population viability
of the short-tailed albatross (Phoebastria albatrus). Biol. Conserv. 143: 321-331.
Zador, S.G., Punt, A.E., and Parrish, J.K. 2008. Population impacts of endangered short-tailed
albatross bycatch in the Alaskan trawl fishery. Biol. Conserv. 141: 872-882.
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