433 



Comparison of two approaches for 



estimating natural mortality based on longevity* 



David A. Hewitt 



John M. Hoenig 



Virginia Institute of Marine Science 



The College of William and Mary 



P.O. Box 1346 



Gloucester Point, Virginia 23062 



E-mail address (for D A Hewitt) dhewittiq'vimsedu 



Vetter (1988) noted that her review 

 of the estimation of the instanta- 

 neous natural mortality rate (M) 

 was initiated by a discussion among 

 colleagues that identified M as the 

 single most important but least 

 well-estimated parameter in fishery 

 models. Although much has been 

 accomplished in the intervening 

 years, M remains one of the most 

 difficult parameters to estimate in 

 fishery stock assessments. A number 

 of novel approaches using tagging 

 and telemetry data provide promise 

 for making reliable direct estimates 

 of M for a given stock (Hearn et al., 

 1998; Frusher and Hoenig, 2001; 

 Hightower et al., 2001; Latour et al., 

 2003; Pollock et al., 2004). However, 

 such methods are often impracticable 

 and fishery scientists must approxi- 

 mate M by using estimates made 

 for other stocks of the same or simi- 

 lar species or by predicting M from 

 features of the species' life history 

 (Beverton and Holt, 1959; Beverton, 

 1963; Alverson and Carney, 1975; 

 Pauly, 1980; Hoenig, 1983; Peterson 

 and Wroblewski, 1984; Roff, 1984; 

 Gunderson and Dygert, 1988; Chen 

 and Watanabe, 1989; Charnov, 1993; 

 Jensen, 1996; Lorenzen, 1996). 



We are concerned with two ap- 

 proaches for predicting M based 

 solely on the longevity of the mem- 

 bers of a stock — an approach that 

 can be used when data are not 

 available to make direct estimates 

 of the parameter. One is a linear re- 

 gression model (Hoenig, 1983) and 

 the other is a simple rule-of-thumb 

 approach. Hoenig (1983) found that 



M was inversely correlated with lon- 

 gevity across a wide variety of taxa 

 and recommended use of the follow- 

 ing predictive equation relating the 

 maximum age observed in the stock 



U max ) to M: 



ln(M) = 1.44-0.982xln(? max ). 



(1) 



The rule-of-thumb approach consists 

 of determining the value of M such 

 that 100(P)% of the animals in the 

 stock survive to the age t max ; thus, 



M- 



-ln(P> 



(2) 



The challenge in this approach is 

 determining an appropriate value for 

 the proportion P. 



The rule-of-thumb approach has 

 the potential to be used widely be- 

 cause it is presented in Quinn and 

 Deriso (1999) and stock assessment 

 manuals of the Food and Agriculture 

 Organization of the United Nations 

 (FAO; Sparre and Venema, 1998; 

 Cadima, 2003). The approach has re- 

 cently been used extensively, in the 

 specific form M~3/t max , in work relat- 

 ed to stock assessments for blue crab 

 (Callinectes sapiclus). In this note, 

 we 1) show that the regression model 

 and the rule-of-thumb approach can 

 be compared directly; 2) illustrate 

 the difference in the estimates of M 

 generated by the two approaches; 3) 

 discuss the origins and current use 

 of the rule-of-thumb approach; and 4) 

 recommend that the regression model 

 be used instead of the rule-of-thumb 

 approach. 



Methods 



With the rule-of-thumb approach, the 

 fraction of a population that survives 

 to a given age is used to estimate 

 M. This approach is equivalent to a 

 quantile estimator (Bury, 1975). Sup- 

 pose the fraction surviving to age / is 

 described by the negative exponential 

 function 





~-zt 



(3) 



where Z is the total instantaneous 

 mortality rate. The quantile estima- 

 tor is of the form 



-Zr P 



(4) 



where r p is the age at which 100(P)% 

 of the population remains. In the case 

 where P = 0.05, the estimator, based 

 on data from a sample of the popula- 

 tion, is 



0.05 = 



(5) 



where 595 of the animals in the sample 

 are older than age t 005 . 



To estimate M, an empirical ap- 

 proach is usually taken where f 05 

 is replaced with t max : 



0.05: 



-», 



(6) 



where t ma!i is either the oldest age 

 observed in the stock or the oldest 

 age found in the literature for the spe- 

 cies of interest. When age composition 

 data are used from an exploited stock. 

 Equation 6 will provide an estimate 

 of M only if fishing mortality is rea- 

 sonably close to zero iM=*Z) or if there 

 is a refuge where older animals can 

 accumulate. If exploitation affects all 



* Contribution 2637 of the Virginia Insti- 

 tute of Marine Science, The College of 

 William and Mary, Gloucester Point, 

 VA 23062. 



Manuscript submitted 25 March 2004 

 to the Scientific Editor's Office. 



Manuscript approved for publication 

 12 October 2004 by the Scientific Editor. 



Fish. Bull. 103:433-437(2005). 



