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Fishery Bulletin 101(2) 



W, = W,,3 + W,,,. 



(1) 



The number of age a spawning fish in year t depends on the 

 number of spawning fish a years before, the productivity 

 (g) of these fish, and the propensity to spawn at age a (n^) 

 given survival to spawning; 



W =W s K 



(2) 



For winter-run chinook salmon, a e 13,4) and we set K., = 



l-n, 



0.89 (Botsford and Brittnacher, 1998), assuming 



that the maturation rate of age-3 fish and the annual 

 mortality rate of age-4 fish is constant across years. We 

 modeled \og(gf) as the sum of several effects, 



log(^,) = /J + zi/, - aS,^j - PW, + £,, e, - Normal(0,cj2), (3) 



including a mean population growth rate in the absence 

 of striped bass and density dependence (jU); a possible 

 change (A) in the mean population growth rate resulting 

 from conservation measures initiated in 1989 (Williams 

 and Wilhams, 1991) (/,=0 for ?<1989; I,=l for <>1989); an 

 effect due to variations in the abundance of striped bass 

 (oSj^j, where S,^j is the abundance of adult striped bass 

 in year /+1 and a is the per-bass predation rate); a density 

 dependence effect (/3W,); and a normally distributed process 

 error (e,) having mean = and variance = a^. We ignored 

 the measurement error in (W,,?>1985) for simplicity; the 

 main effect of including measurement error would be to 

 increase the uncertainty in A. Together, Equations 2 and 3 

 imply that W, ^ is a lognormal random variable, and that 

 Wi (see Eq. 1) is distributed as the sum of two lognormal 

 random variables. 



Density dependence in this formulation is equivalent 

 to the Ricker model of stock-recruitment (Ricker, 1954): 

 as stock size increases to infinity, per-capita productivity 

 declines exponentially to zero. Because population viabil- 

 ity analysis (PVA) model predictions can be sensitive to 

 density dependence, we also considered Equation 3 with 

 P set to zero. 



Equation 3 states that predation by an individual 

 striped bass is a linear function of winter-run chinook 

 salmon abundance, ignoring the possibility of satiation or 

 a minimum prey abundance to initiate feeding. Although 

 the actual functional response of striped bass to winter-run 

 chinook salmon is probably more complex, it is unlikely 

 that satiation is a major issue for a rare prey species such 

 as winter-run chinook salmon. We use S,^, rather than S, 

 because striped bass population size is estimated in the 

 spring, and the juvenile winter-run chinook salmon born 

 in year t are vulnerable to striped bass predation as they 

 develop and migrate to sea the following winter and spring 

 (year ^-i-l). 



Parameter estimation 



In this section, we couple the time series data and the 

 winter-run chinook salmon population dynamics model 

 with a prior distribution of the model's parameters to yield a 

 Hayes posterior distribution for those parameters. For con- 



venience, we denote the vector of model parameters as 6 = 

 (/J, A. a. p. a), and the data vectors as W = (Wjgg,, W,ggg, ...), 

 S = ''S'igg7,Si9gg, ...), and / = (/i9g7, /i9gg, ...). We denote a 

 probability density as p(-) and a conditional probability 

 density asp(- 1 •). The unnormalized Bayes posterior distri- 

 bution of the model parameters is given by 



p(d\'w,i,S)ocp(e)p(w\i,s,e), 



(4) 



where piQ) is the prior distribution of 0, and p( W \ I,S, 9) is 

 the model probability density function of W conditional on 

 /, S, and 0, given by 



p(W\ 1,5,6) = YlP{W,\W,_,,W,_,J,_,J,_,,S,_„S,_„e). (5) 



From the previous subsection, p(W, | •) on the right hand 

 side of Equation 5 is the probability density for a sum of 

 two lognormal random variables. We evaluated p(lV, |-) 

 using the analytic expression provided in Johnson et al. 

 ( 1994, Eq. 14.20), solving the integral contained therein by 

 adaptive quadrature. 



The prior density p( 9) is the joint probability of the com- 

 ponents of 6: p( 6) - X\p( 6, ). Because we have little informa- 

 tion about Q that is independent of the data used in our 

 analysis, we desired a prior that would have little influence 

 on the posterior. There are many ways such a noninforma- 

 tive prior could be specified. In the results presented here, 

 we set p(\X, A, a, P) ^ 1 over the range of the parameters 

 (a and j3 are restricted to positive values) and p(a) « (T^, 

 following the recommendations of Lee (1989) and Gelman 

 et al. (1995) based on the work of Jeffreys (1961). We also 

 examined the effects of using other reference priors, such 

 as normal and exponential distributions with very large 

 variances, and found there to be little difference in the 

 results (not shown). 



We did not attempt to derive a closed-form analytical ex- 

 pression for the posterior distribution of 0. Instead, we used 

 the Metropolis-Hastings algorithm (Metropolis et al., 1953; 

 Hastings, 1970; Gilks et al., 1996), a Markov chain Monte 

 Carlo method. The Metropolis-Hastings algorithm produces 

 a Markov chain with a stationary distribution equivalent to 

 the posterior of 6. Estimates of parameter means, medians, 

 and credible intervals were obtained from samples of the 

 stationary Markov chain. We used a multivariate normal 

 distribution centered on the current value of Q for the algo- 

 rithm's proposal distribution. The variance-covariance ma- 

 trix of the proposal distribution was adjusted by trial and er- 

 ror until the resulting Markov chain was well-mixed and the 

 probability of accepting candidates fell in the range of (0.15, 

 0.50 ) ( Gilks et al, , 1996 ). Note that the proposal distribution 

 form does not presume anything about the distribution of 

 the unknown parameters, and as long as certain criteria are 

 met (see Gilks et al. 1 1996] ), only the convergence speed and 

 mixing are affected, not the stationary distribution of the 

 chain. To assess convergence, we initiated chains from many 

 widely varying starting places and observed convergence 

 to the same distribution. We found that 50,000 iterations 

 following an initialization of 10,000 iterations provided 

 stable parameter estimates. For an additional convergence 



