Abstract. -This paper is devoted 



to a theoretical examination of two 

 rules of thumb commonly used in 

 fishery management: (I) the fishing 

 mortality rate associated with max- 

 imum sustainable yield (Fm^y ) equals 

 the natural mortality rate, and (II) 

 the equilibrium stock biomass at 

 maximum sustainable yield equals 

 one-half the pristine stock biomass. 

 Taken together, these rules of thumb 

 are shown to be inconsistent with 

 any simple dynamic pool model in 

 which three conditions hold; (1) the 

 first derivative of the stock-recruit- 

 ment relationship is uniformly non- 

 negative, (2) the second derivative of 

 the stock-recruitment relationship is 

 uniformly nonpositive, and (3) the 

 first derivative of the weight-at-age 

 relationship is uniformly positive. An 

 example of such a model is presented 

 and the equilibrium solution derived 

 analytically. In this model, Fmsy can 

 be either greater than or less than 

 the natural mortality rate, while the 

 equilibrium stock biomass at max- 

 imum sustainable yield is consistent- 

 ly less than one-half the pristine 

 stock biomass. To illustrate the util- 

 ity of the theoretical framework de- 

 veloped, the model is applied to the 

 eastern Bering Sea stock of rock sole 

 Pleuronectes bilineatus. 



Management advice from 

 a simple dynamic pool model 



Grant G. Thompson 



Resource Ecology and Fisheries Management Division 



Alaska Fisheries Science Center, National Marine Fisheries Service, NOAA 



7600 Sand Point Way NE, Seattle, Washington 981 15-0070 



Two rules of thumb 



Despite its acknowledged shortcom- 

 ings (e.g., Larkin 1977), management 

 for maximum sustainable yield (MSY) 

 remains a common strategy among 

 fisheries professionals. Under a con- 

 stant harvest rate policy, this strate- 

 gy is implemented by exploiting the 

 stock at the fishing mortality rate 

 corresponding to MSY (Fmsy)- Alter- 

 natively, this strategy could be imple- 

 mented by exploiting the stock so as 

 to maintain its biomass at the level 

 corresponding to MSY, B(Fmsy)- To 

 estimate Fmsy and B(Fmsy). fishery 

 scientists and managers employ a 

 variety of approaches, ranging from 

 highly sophisticated simulation 

 models to simple "rules of thumb." 

 Frequently used examples of the 

 latter can be found in the form of two 

 hypotheses employed by Alverson 

 and Pereyra (1969) in their analysis 

 of the potential yield of certain fish 

 stocks. These hypotheses (hereafter 

 referred to as Rules I and II) are 



and 



Fmsy 

 M 



B(Fmsy) 

 B(0) 



0.5, 



(I) 



(11) 



Manuscript accepted 4 June 1992. 

 Fishery Bulletin, U.S. 90:552-560 (1992). 



where F is the instantaneous rate of 

 fishing mortality per year, Fmsy is 

 the value of F that produces MSY in 

 equilibrium, M is the instantaneous 

 rate of natural mortality per year, 

 B(F) is the equilibrium stock biomass 

 corresponding to a fishing mortality 



rate of F, B(Fmsy) is the equilibrium 

 stock biomass when F = Fmsy. and 

 B(0) is the pristine stock biomass 

 (i.e., equilibrium stock biomass when 

 F = 0). 



Alverson and Pereyra (1969) pre- 

 sented a sketchy derivation of Rules 

 I and II, leaving open the question of 

 which models might be capable of 

 leading to the hypothesized relation- 

 ships. A number of authors have sub- 

 sequently examined specific models 

 in this context and shown them to be 

 inconsistent with Rules I and II. 

 Gulland (1971) and Beddington and 

 Cooke (1983) cast doubt on the 

 robustness of Rules I and II in terms 

 of their appHcability to the "simple" 

 model of Beverton and Holt (1957), 

 but did not generalize their conclu- 

 sions beyond that particular model. 

 Likewise, Francis (1974) demon- 

 strated inconsistencies between 

 Rules I and II and a set of assump- 

 tions derived from the Schaefer 

 (1954) model, although his argument 

 was weakened somewhat by com- 

 puting MSY in terms of numbers, not 

 biomass. Deriso (1982) showed that 

 the discrete fishing mortality rate 

 generated by his delay-difference 

 model at MSY was consistently 

 higher than the discrete natural mor- 

 tality rate when recruitment was 

 constant, while under several other 

 stock-recruitment assumptions the 

 relationship was reversed. Shepherd 

 (1982) also demonstrated that Rules 

 I and II did not adequately describe 

 the behavior of a particular surplus 

 production model. 



Since none of these authors ad- 

 dressed the possibility that other 



552 



