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independent model; we also analyzed a Beverton-Holt type 

 model (where per-capita productivity reaches an asymptote 

 instead of declining to zero as population size increases to 

 infinity) and found that it gave similar predictions to the 

 Ricker model (results not shown). The insensitivity of ex- 

 tinction risk to the form of density dependence is perhaps 

 not surprising because density dependence is considered to 

 have little influence on the extinction process if populations 

 are well below carrying capacity (Emlen, 1995), although 

 it has the potential to create both compensatory popula- 

 tion growth that can increase population persistence and 

 oscillatory or chaotic dynamics that can reduce population 

 persistence (Ginzburg etal., 1990; Mills et al., 1996; Be- 

 lovsky et al., 1999). The probability of recovery, however, 

 was strongly dependent on whether density dependence 

 was included: regardless of striped bass stocking level, the 

 recovery probability predicted by the density-independent 

 model was about threefold higher than that predicted by 

 the density-dependent model. Although the density de- 

 pendence parameter was not well-identified by the data, 

 winter-run chinook salmon are currently restricted to a 

 limited portion of the Sacramento River and it is certainly 

 possible that there is not enough habitat to support a 

 spawning run of 20000 adults. Further study of the Sac- 

 ramento River's carrying capacity for winter-run chinook 

 salmon is warranted. 



The dynamics of food web and predator-prey models can 

 also be sensitive to the form of the predator's functional 

 response to prey abundance (Overholtz et al., 1991; Ber- 

 ryman, 1992). The models presented here assumed that 

 the predation-related per-capita mortality of winter-run 

 chinook salmon is a linear function of striped bass abun- 

 dance only. It is possible, however, that this mortality rate 

 depends on winter-run chinook salmon abundance as well, 

 through the feeding response of individual striped bass to 

 winter-run chinook salmon abundance. In deterministic 

 models, the form of the functional response (as well as 

 predator abundance and prey productivity) determines the 

 equilibrium prey population size. In particular, whether a 

 prey population can persist may depend on whether the 

 predator's functional response is sigmoidal or increases 

 monotonically to an asymptote with increasing prey abun- 

 dance (Sinclair etal., 1998). In cases where the prey is 

 the major food source of the predator, it can be possible to 

 detect a nonlinear functional response from the time series 

 themselves (Jost and Arditi, 2000), especially if the system 

 is perturbed (Carpenter etal., 1994). Winter-run chinook 

 salmon are not the main prey of striped bass, and any pos- 

 sible depensatory effect of predation may be reduced by 

 alternate prey, including juvenile chinook salmon of other 

 races. Juvenile fall chinook salmon, in particular, are abun- 

 dant, and often co-occur with winter-run chinook salmon 

 (Healey, 1991). If the abundance of fall chinook salmon is 

 uncorrelated with, and high in relation to, winter-run chi- 

 nook salmon, then the striped bass predation rate may be 

 related to fall chinook salmon abundance and unrelated to 

 winter-run chinook salmon abundance. In the absence of 

 relevant data, further consideration of nonlinear feeding 

 responses and effects of alternate prey (e.g. Spencer and 

 Collie, 1995), is beyond the scope of this paper. 



Another aspect of model uncertainty is the assumption 

 that the future will be like the present. The future will 

 probably include increased conservation efforts, changing 

 ocean productivity, and perhaps further habitat degrada- 

 tion. Although the level of absolute risk would change sub- 

 stantially if these processes were included in the simula- 

 tions, the relative risks posed by the different striped bass 

 stocking schemes would change much less. The main goal of 

 this work was to compare these relative risks; a secondary 

 goal was to predict what would happen if things continued 

 in the future as they are now. We therefore feel confident in 

 stating that a large striped bass stocking program would be 

 risky and that further winter-run chinook salmon restora- 

 tion actions are needed. 



Data uncertainties 



Imprecise estimates of predator and prey abundance 

 limit the precision of parameter estimates and can bias 

 parameter estimates if not accounted for (Seber and Wild, 

 1989; Carpenter et al., 1994). For the bulk of the winter- 

 run chinook salmon series, observation error is probably 

 quite low because all fish were counted directly; the CV 

 for the striped bass population estimates is thought to be 

 about 25% (Stevens, 1977). We ignored measurement error 

 in both the striped bass and winter-run chinook salmon 

 population abundance data. Further work is required to 

 assess how much of an influence these errors might have 

 on parameter estimates for the model presented here. 



Informative priors 



A major advantage of the Bayesian approach is the ability 

 to include informative prior probability distributions for 

 model parameters. Informative priors can greatly improve 

 the precision of posterior parameter estimates and model 

 predictions. In the example presented here, the estimate 

 of the striped bass predation rate could be improved, and 

 uncertainty in stocking impacts reduced, by incorporating 

 direct information on the rate of striped bass predation on 

 winter-run chinook salmon into an informative prior on a. 

 Such information would include estimates of the number of 

 salmon that striped bass eat per day (obtainable from food 

 habits and metabolic studies), and the number of juvenile 

 salmon that are vulnerable to striped bass predation. Some 

 information on these quantities is available for the Sacra- 

 mento system (Stevens, 1966; Thomas, 1967). 



A Bayesian meta-analysis of the available food habits 

 data was performed to estimate the number of salmon 

 that striped bass eat per day, and the number of juvenile 

 salmon passing through the Sacramento River system was 

 estimated from ocean catches, spawning escapements, and 

 considerations of smolt-to-adult survival rates. Unfortu- 

 nately, including the informative prior did not substan- 

 tially improve the precision of the posterior distribution of 

 a, nor did it significantly alter the central tendency. Given 

 the number of necessary assumptions, the complexity of 

 the meta-analysis, and the minimal impact of including 

 the informative prior on the posterior distribution of a, we 

 opted to retain a noninformative prior on a. Should better 



