434 



Fishery Bulletin 103(2) 



animals in the stock, Equation 6 is unlikely to provide 

 a reliable estimate of M. 



The rule of thumb for approximating M follows di- 

 rectly from Equation 6: 



-ln(0.05) = M xt n 



M = 



2.996 



(7) 



Most importantly, note that the use of 0.05 or any other 

 proportion in the equations is arbitrary because we have 

 no reason to believe that t max pertains to any particular 

 quantile. 



We show in the present study that this arbitrary rule 

 of thumb for approximating M is unnecessary, as an 

 empirical method (Hoenig, 1983) provides an analogous 

 estimate based on a substantial data set. Equation 1 is 

 based on the same model as that in Equation 3 and was 

 developed from a regression of In (AD on ln(< max ) from 

 data on 134 stocks of 79 species of fish, mollusks, and 

 cetaceans. It can be shown to be of the same form as 

 the rule-of-thumb approach as follows: 



InlM) _ 1.44-0.982xln(« max ) 



M = - 



.0.982xln<(l 



4.22 



(8) 



(t ) 



u tnax ' 



4.22 



982 



Results 



We substituted 1.0 for 0.982 in Equation 8 to allow the 

 development of a simple, approximate rule of thumb for 

 direct comparison with 3/t max . As a result, this rule of 

 thumb strictly applies only to the case where t m3X = 1. 

 Estimates from the regression estimator in Equation 

 1 are always greater than estimates from Equation 8 

 for £ max >l, although the difference is usually small 

 (Fig. 1). 



Estimates from the regression estimator are typically 

 40-50% greater than estimates from 3/t max (Fig. 2). 

 For example, if a maximum age of eight years is used 

 for blue crab in Chesapeake Bay (Rugolo et al., 1998), 

 3/t max gives an estimate for M of 0.375/yr and the re- 

 gression estimator gives 0.548/yr. 



Perhaps the most significant result is the finding that 

 rearrangement of the regression model yields an esti- 

 mate of an appropriate value for P in Equation 2. The 

 value of 4.22 in Equation 8 approximately corresponds 

 to -ln( 0.015), indicating that the average longevity for 

 stocks in the data set used by Hoenig (1983) is the age 

 at which about 1.57c of the stock remains alive (versus 

 5% in 3/t max ). 



Discussion 



Development of the rule-of-thumb approach 



The rule-of-thumb approach appears to have arisen inde- 

 pendently in four different places. Cadima (2003) sup- 

 ported the approach by citing the early work of Tanaka 

 (1960). Sparre and Venema (1998) based their presen- 



