Lindley and Mohr: Impact of population manipulations of Morone saxatilis on Oncorhynchus tshawytscha 



325 



check, we compared the modes of 6 to maximum hkelihood 

 estimates of Q obtained using a quasi-Newton method for 

 minimization of a multivariate function with simple bounds 

 ( IMSL Fortran Numeric Library subroutine BCONF, Visual 

 Numerics, Inc., Houston, TX). 



Extinction and recovery probabilities 



Given our winter-run chinook salmon population dynamics 

 model, alternative striped bass population levels, and the 

 posterior distribution of 6, we used Monte Carlo methods 

 to determine the probability distribution of winter-run 

 chinook salmon abundance in each of the next 100 years. 

 From these distributions, the probability (P) that winter- 

 run chinook salmon abundance is below a quasi-extinction 

 threshold or above a recovery benchmark can be estimated 

 directly. It was assumed, for simplicity, that striped bass 

 abundance over the next 100 years will be held constant 

 (Sj=S) at a level depending on the intensity of striped bass 

 stocking. We considered three levels of striped bass abun- 

 dance of interest to fishery managers, corresponding to no 

 stocking (S=.5 12,000 adults), moderate stocking ( 5=700,000 

 adults), and heavy stocking (5=3,000,000 adults). For com- 

 parative purposes, we also examined the effect of removing 

 all striped bass (5=0 adults). Of particular interest is the 

 increase in extinction probability due to striped bass stock- 

 ing in relation to the no-stocking alternative. We denote 

 this increase in extinction risk as 5. We generated the 

 distribution of extinction and recovery probabilities and 8 

 under the four striped bass population levels as follows: 



1 Initialize the model by setting |W,, ^=-3, -2, -1, 0) 

 equal to the four most recent observations of spawn- 

 ing escapement. 



2 Randomly select a value for 6 according to its poste- 

 rior density using the Metropolis algorithm. 



3 For each striped bass abundance level S, |W,, t =-3, 

 -2, -1,0), and the particular value of 0, simulate 1000 

 100-year trajectories of winter-run chinook salmon 

 spawning escapement according to Equations 1, 2, 

 and 3. 



4 For year t, the fraction of simulations in which spawn- 

 ing escapement was below the quasi-extinction thresh- 

 old or above the recovery benchmark (levels specified 

 below) approximates P,(quasi-extinction | 5,6) and 

 P,(recovery | 5,0), respectively, for /=1,2 100. 



5 For year t=bQ and each level of striped bass abun- 

 dance 5, the increase in extinction probability in rela- 

 tion to that for the no-stocking level is approximated 

 by 5(5, 9) = P,^5(|( quasi-extinction | 5,6) - P,^5g(quasi- 

 extinction | 5=512,000, 0). 



6 Repeat steps 2-5 10,000 times. For each year t and 

 striped bass abundance level 5, the average values 

 of P,(quasi-extinction | 5,6) and P,(recovery | S,Q) over 

 these repetitions approximates their expected values 

 with respect to 6 given 5. For brevity, in the "Results" 

 and "Discussion" sections, we refer to these values as 

 simply the probabilities of quasi-extinction and recov- 

 ery in year / given striped bass abundance 5. 



We focused on quasi-extinction to avoid the problems of 

 modeling depensatory effects, such as demographic stochas- 

 ticity, inbreeding depression, and Allee effects ( Allee, 1931 ). 

 Estimates of absolute extinction risk are very sensitive to 

 how these processes are modeled and parameterized, and 

 relevant data are lacking. Quasi-extinction is less sensitive 

 to these processes and is more likely to occur over short 

 time horizons; therefore it is a more useful management 

 benchmark than absolute extinction (Beissinger and West- 

 phal, 1998). The draft recovery plan for winter-run chinook 

 salmon (NMFSM defines the quasi-extinction level as 100 

 females and the recovery level as 10,000 adult females; 

 therefore we set the quasi-extinction threshold at 200 

 adults and the recovery threshold at 20,000 adults on the 

 working assumption that the sex ratio is approximately 1. 



Results 



Parameter estimates and model fit 



Table 1 lists summary statistics for parameter estimates 

 for both the density-dependent and density-independent 

 models; Figure 4 shows posterior marginal distributions 

 and pairwise bivariate density contour plots for the 

 winter-run chinook salmon density-dependent population 

 dynamics model. The posterior median of ;U was -0.69 per 

 generation and the 0.90 credible interval (CI lower and 

 upper endpoints of the posterior distribution equal to the 

 0.05 and 0.95 percentiles, respectively) for /j was (-1.2, 

 -0.046), which indicates that the decline of winter-run 

 chinook salmon most probably reflects a real trend rather 

 than solely a series of random events. The median of the 

 posterior distribution of the listing effect parameter A 

 was positive, which suggests that the winter-run chinook 

 salmon population growth rate has increased since initia- 

 tion of conservation measures in 1989. The present-day 

 realized growth rate, \og(g), as determined from the joint 

 posterior distribution and current winter-run chinook 

 salmon and striped bass abundance according to Equa- 

 tion 3, has a median of -0.19 (0.90 CI=(-1.08, 0.66)), which 

 indicates that the winter-run chinook salmon population 

 may still be in decline in spite of the conservation measures 

 and the decline in striped bass abundance. 



At current striped bass abundance, the median estimate 

 of a translates into about a 9% chance of an individual ju- 

 venile chinook salmon being consumed by a striped bass. 

 Because log(^,) is highly variable (median of a estimate was 

 1.18), only fairly large values of a are inconsistent with the 

 data. Furthermore, there was positive correlation between 

 the estimates of a and ^ (correlation coefficient=0.77), 

 meaning that fairly high predation rates are consistent 

 with the data if the underlying population growth rate 

 was also high. The negative correlation of J with ji and a 

 indicates that the potential improvement in winter-run chi- 

 nook salmon population growth rate could have been due to 

 either conservation measures or reduced predation. 



At recent population sizes, there is little reduction in 

 winter-run chinook salmon population growth due to den- 

 sity-dependent effects: the median of /3 translates into a 



