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<br />202
<br />
<br />~
<br />
<br />COGGINS ET AL.
<br />
<br />every unmarked fish captured received a tag (the same
<br />ma., is used both to decrement 0 and to increment M).
<br />Note further that multiple recaptures of individual fish
<br />within each year are counted as only one recapture
<br />event (this model works on annual steps).
<br />To make specific predictions of 0 and M for
<br />comparison with the data, we must estimate boundary
<br />conditions Ma.l for all a, Oa,1 for all a in year 1, and
<br />new recruits U2., for t > .1. The initial numbers of
<br />marked fish are obviously Ma,l = 0 for all,a. There are
<br />two choices for dealing with the U boundary
<br />conditions: (1) "forward propagation," in which the
<br />initial abundance at age and recruitment (Oland 02 )
<br />a. ,I
<br />are treated as unknowns to be estimated; and (2)
<br />"backward propagation," in which the oldest age in
<br />each year (OA) and all ages in the terminal y~ (Oa,T)
<br />are treated as the unknowns and the other U are
<br />a,1
<br />calculated using the backward recursion or a VP A
<br />e,quation based on solving equation (1) for Oa.1 given
<br />U 0+1,'+1' that is,
<br />
<br />Ua" = (Ua+1,'+1/Sa) + ma,l. (3)
<br />
<br />In using the backward propagation approach in ASMR,
<br />it is safe to treat 0 A,l as 0 for all t if A is sufficiently
<br />beyond the oldest age that fish can attain, and we are
<br />left with estimating the number of unmarked fish in the
<br />terminal year, Oa,T'
<br />Using the predictions of the numbers at risk of
<br />capture from equations (1-3), we then calculate the
<br />expected numbers of unmarked and marked fish
<br />captured by age and year as follows:
<br />
<br />ma I = Ua.,jJa,
<br />, .,
<br />
<br />Ta,z = Ma"Pa,/l
<br />
<br />where P a,' is the estimated age- and time-s~ecific
<br />capture probability. We assume, conditional on S and
<br />P I' that the observations m and r rep~sent
<br />a, a,1 a,l
<br />independent samples from Poisson distributions with
<br />means given by equations (4) and (5). This is the same
<br />as assuming independent binomial sampling of in-
<br />dividuals in each 0- and M-age subpopulation with
<br />sample capture probability P a,I' Ignoring terms in-
<br />volving only the data, the Poisson assumption leads to
<br />the log-likelihood function
<br />A T
<br />log"L(m,rI9) = 2: 2:[-ma,1 + ma"loge(ma,,)]
<br />a=l ,=1
<br />A T
<br />+ 2: 2:[-Ta" + ra"loge(Ta,,)] (6)
<br />a=l 1=2
<br />
<br />where 9 is the parameter vector to be estimated.
<br />Following Lorenzen (2000), we defined an age-
<br />dependent survival function based on a von Bertalanffy
<br />
<br />growth function (von Bertalanffy 1938). This formula-
<br />tion allows mortality to decrease with age to a minimum
<br />defined by MadUl<' the instantaneous mortality rate
<br />suffered by fish that have reached asymptotic length.
<br />We obtained an independent estimate of the von
<br />Bertalanffy k from growth data presented in the
<br />USFWS humpback chub recovery goals (USFWS
<br />2002; see below). The resulting "Lorenzen model"
<br />(Lorenzen 2000) allows age-specific survival to be
<br />estimated with one unknown parameter, Madull, as
<br />
<br />S = [e"(a+1) _l]i4"'."/k
<br />a eka _ 1 (7)
<br />
<br />The two remammg model specification issues are
<br />estimation of the unmarked fish in the terminal year to
<br />initialize the back propagation (i.e., 0 '1') and estima-
<br />a,
<br />tion of the age- and time-specific capture probabilities
<br />P . We define three alternative formulations of the
<br />a,l
<br />ASMR model to incorporate various options.
<br />
<br />Specific Models
<br />
<br />ASMR 1
<br />
<br />In ASMR 1, we calculate the age- and time-specific
<br />capture probability as Pa.1 = p,va,,' where Va,l is the age-
<br />and time-specific vulnerability to capture gear and P, is
<br />the conditional maximum likelihood estimate of annual
<br />capture probability, which is calculated as
<br />A A
<br />p, = 2)ma,z + ra,z)/ 'L)a,,(Ua,z + Ma,z). (8)
<br />a=l a=l
<br />
<br />(4)
<br />
<br />Following standard fisheries virtual population analysis
<br />techniques (Hilborn and Walters 1992), we estimate the
<br />abundance of unmarked fish in the terminal year to
<br />initialize the back propagation as Oa,T =:: m a,Tlp a,1 This
<br />leaves the parameter vector 9 = (Va,I' Madull, and PT) to
<br />be estimated using nonlinear search routines to
<br />maximize equation (6). To further restrict the problem,
<br />it is possible to assume that Va,1 = 1 for all ages older
<br />than a specified age (i.e., all fish older than a specified
<br />age are equally vulnerable to capture). It is also
<br />possible to estimate identical Va,1 schedules for blocks
<br />of years that have similar sampling intensities (see
<br />Coggins et al. 2006).
<br />
<br />ASMR 2
<br />
<br />ASMR 2 differs from ASMR 1 only in the
<br />initialization of the terminal abundances and in the
<br />calculation of the terminal capture probability. Instead
<br />of estimating an overall terminal capture probability
<br />(fiT)' we directly estimate terminal abundances (Oa,r)'
<br />ASMR 2 maximizes equation (6) by varying 9 = (Val'
<br />Madu1t, and Oa;)' '
<br />
<br />(5)
<br />
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