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<br />LITTLE COLORADO RIVER HUMPBACK CHUB
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<br />245
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<br />Appendix: Likelihood Function for Tag Recaptures Including Movement
<br />
<br />For each tagged fish i, we have a recapture history of the form
<br />Y, = {O, k, 0, 0, . . ., k, 0, . . .}, where for each possible
<br />sampling date an observation of Yt = 0 denotes no recapture
<br />and an observation of Yt = k denotes at least one recapture in
<br />location k (in this case either the Little Colorado River or the
<br />Little Colorado River inflow reach). We calculate the
<br />likelihood P(Y,) of each history using the recursive method
<br />reviewed in DeValpine and Hastings (2002), where P(Y,) is
<br />represented by the equation
<br />
<br />PlY,) = P(Y,-I)P(YtIYt-1).
<br />
<br />To use this representation, we note that the probability
<br />P(y,!Y,_,) can be written as
<br />
<br />P(Yt!Y,-J) = LP(St!Yt-I)P(Ytls" Y,-J), (A.2)
<br />
<br />where s, represents possible fish states (dead, alive in location
<br />I, alive in location 2, etc.) and P(y,ls" Yt-I) is the probability
<br />of the observation Y given that the fish is in state s (i.e., the
<br />capture probability for fish in state s at time t if; > 0, I
<br />t t
<br />minus this capture probability if Y, = 0). Representing
<br />P(y,IY,_,) in terms of states St expresses the problem of
<br />calculating it as two simpler problems, namely, calculating
<br />location state probabilities P(s,IYt-J) over time and the capture
<br />probabilities.
<br />
<br />The location state probabilities are updated over time using
<br />Bayes' theorem. We first calculate the "posterior" probabil-
<br />ities as
<br />
<br />P(s,IY,) = p(Y,ls,)P(stIYt_I)/P(Yt), (A.3)
<br />
<br />where the total probability of the Y, data is given by P(y,) =
<br />L,P(y,!s,)P(s,IY,I) and p(y,ls,)P(s,!Y, I) is simply I or 0 (0 if
<br />the fish is either dead or not recaptured in location s" I if the
<br />fish is recaptured in location s,). Calculation of P(s,IY,) is
<br />nontrivial only for the case Y, = O. We then calculate P(s'+lIY,)
<br />(which is P(S,!Yt_l) for the next time step) as follows: .
<br />
<br />P(s'+IIY,) = SLP(s,ly,)M(s,s'), (A.4)
<br />
<br />(A. I)
<br />
<br />where S is a (possibly age- and time-varying) survival rate to
<br />be estimated from the data and M(s, Sf) is a mO'Vement
<br />probability matrix representing the probability of a live fish's
<br />moving from location state s at time t to state s/ at time t + I
<br />(the elements are set to 0 for s = dead; M(s, s) is the
<br />probability of a fish's staying in location s).
<br />
<br />In calculating the likelihood function, the capture probabilities
<br />P(k,ls" Y 1-1) are set to their conditional maximum likelihood
<br />estimates given by
<br />
<br />P(k,ls" Y,--J) = nk,J L Pj(S,IY'-I), (A.5)
<br />
<br />where n is the number of fish captured at location k at time t,
<br />and the ~um over fish i of the probabilities of being alive and
<br />at location k represents the expected number of fish at risk of
<br />capture in location k at time t.
<br />
<br />Seasonal and age dependencies are incorporated in the
<br />estimation by (1) assigning each fish i an apparent age at
<br />first capture based on length, then varying S for subsequent
<br />recapture times using apparent age and the Lorenzen survival
<br />function (Lorenzen 2000); (2) calculating the capture
<br />probabilities in equation (A.5) separately by apparent fish
<br />ages up to age 10; and (3) using a different movement rate
<br />matrix M (s, S'), m = 1, . . ., 8, for at least the first eight
<br />calendar ~1Onths of each year (we believe that there is
<br />relatively little movement in late summer and fall; Valdez and
<br />RyeI1995). These assumptions require estimating at least 2 X
<br />8 + 1 leading parameters (assuming capture probabilities to be
<br />given by equation A.5), 16 parameters for movements to and
<br />from the LCR and 1 parameter for the asymptotic mortality
<br />rate, Madul" In some estimation trials we have multiplied the
<br />M(s, Sf) movement rates by a logistic age factor representing
<br />lower probabilities of movement out of the LCR for younger
<br />fish, thereby adding two parameters.
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