He et al : A prior for steepness In stock-assessment relationships, based on an evolutionary persistence principle 



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h value 



Figure 3 



Relative probability distributions of the steepness (/;) prior for 

 four different natural mortality (M) values with parameter 

 setting a = 1.0, A^,, = O.lAf^, and T = 500 years. 



characterizing critical abundance. This parameter is 

 more or less a theoretical concept that defines the low- 

 est population abundance deemed to be viable, and its 

 value interacts with the time horizon, T. Although we 

 used a relatively large value of TV., the associated time 

 horizon was short (7=500 yr) in the evolutionary time 

 scale. Because the stock-recruitment dynamics near 

 the origin are nearly linear, a relatively large value of 

 N^. is approximately equivalent to using a lower value 

 of N^, over a longer time horizon. Also, Mangel and 

 Clark (1988) showed that once the time horizon is suf- 

 ficiently large, the distributions have the same shape 

 and become stationary. Test runs on our models with 

 large values of T (T=3000 yr) showed almost the same 

 results. In a complex marine environment like that of 

 the U.S. west coast, it would be very difficult to de- 

 termine "true" values of A^^ and T, which could be, for 

 example, at a level where the Allee effect is likely to 

 take place and over a time scale of major environmental 

 change, respectively. However, the results indicated that 

 the model is not very sensitive to the N^, values (Fig. 

 2). For a range of A^,/Af„ values from 0.05 to 0.125, the 

 prior probability curves were very similar. In contrast, 

 recruitment variability (a) had much greater effects on 

 the distributions of the prior probability (Fig. 1). As re- 

 cruitment variability increases, minimum /; values need 

 to be significantly higher in order to ensure population 

 viability. Stock assessment reports on other west coast 

 groundfish species have indicated a very high recruit- 

 ment variability in the past few decades, such as for 

 bocaccio, (a=1.0, MacCall'), and darkblotched rockfish, 

 (a=0.85, Rogers-^). However, some recent stock assess- 

 ment models for the west coast groundfish species have 

 used a lower value of o. such as 0.6 (Punt, 2003). In 

 our example, if lower values of a are used, the derived 

 h prior curves will be very flat for h>0.23 (Fig. 2), in- 



dicating that the population would be sustainable with 

 very low h values. 



Other methods for deriving h priors include using 

 expert opinion, borrowing values from other taxo- 

 nomically or ecologically related species, and using 

 regional meta-analysis (Myers, 1998, 2001; Chen and 

 Holtby, 2002; Dorn, 2002; Millar, 2002; Myers et 

 al., 2002). For example, Myers et al. (2002) used an 

 empirical Bayesian approach to derive prior distri- 

 butions for steepness for nine species. As compared 

 to our method, their method, based on a combined 

 method of taxonomic and ecological criteria, still re- 

 quires the collection of biological data from related 

 species, as well as expert opinions on the life history of 

 each species, which are somewhat "subjective" values. 

 Dorn (2002) used Bayesian meta-analysis to derive 

 stock-recruitment relationships for a group of west 

 coast groundfish species. He used the results from the 

 previous stock assessments for widow rockfish and 

 other west-coast groundfish species and indicated that 

 the prior distribution for steepness for widow rockfish 

 could have a median value around 0.72 and a prob- 

 ability of 0.0033 for /jsO.225. This result could indi- 

 cate that our method is more conservative because our 

 method indicates much lower median value for widow 

 rockfish-like species. Minte-Vera et al. (2005) also sug- 

 gested that the priors derived by Dorn (2002) might 

 not be appropriate for stock assessments that include 

 the same data that were used in the meta-analysis to 

 derive priors. 



We believe that the method used in our study provides 

 a scientific way of estimating the prior for steepness and 

 avoids the pitfall of imposing a preconception of what 

 the true distribution is thought to be. For example, it 

 seems reasonable that very high values of h will not be 

 likely because of tradeoffs between individual survival 



