Porch et al A catch-free assessment model with application to Epinephelus itajora 



91 



where 4> can vary among three eras of exploitation; 

 a "prehistoric" period, during which little data are 

 available; a "modern" period, when presumably there 

 are some data on abundance or mortality rates; and a 

 "future" period, when fishing mortality rates are con- 

 trolled (input). The absence of data during the "prehis- 

 toric era" generally precludes the estimation of annual 

 deviations in recruitment (f) or fishing mortality rate 

 (b) during that period. 



The average weight or fecundity of the plus group is 

 expressed as a function of the average age of the plus- 

 group. Initially, it is assumed that the age composition 

 of the plus-group is in equilibrium consistent with Equa- 

 tion 1, in which case the average age of the plus-group 

 at the beginning of the first year is approximately 



-M,, 



a^i = A + 



1-e 



-M^ 



(7) 



Subsequently, the age of the plus-group is updated as 



AN 



A-l.y 



-'^v''.4-l-'W.4-l 



/ — .1 V AT - i" J' X~ ^1 A 



'.4.y+l 



N 



(8) 



A,y+\ 



Reference points 



Equations 1-4 describe the relative dynamics of a popu- 

 lation apart from its absolute abundance. As such they 

 are suitable for developing management plans where 

 the fishing mortality rate is controlled directly (e.g., by 

 reducing effort) and the biomass reference points are 

 expressed on a relative scale. When the virgin spawn- 

 ing biomass itself is used as the reference point, the 

 estimated value of s^ is a direct measure of the status 

 of the stock. For example, if the management goal is 

 to maintain spawning biomass at or above 50% of the 

 virgin level, then estimates of s below 0.5 may trigger 

 some action to reduce fishing pressure. 



A related reference point is the equilibrium spawning 

 potential ratio (Goodyear, 1993), defined as the expected 

 lifetime fecundity per recruit at a given F (i/>^) divided 

 by the expected lifetime fecundity in the absence of 

 fishing (ly'fl): 



Wo 



a=0 



(9) 



-J^Fi^+M, 



As shown in Appendix 2, the corresponding equilibrium 

 level of relative spawning biomass (denoted by a tilde) 

 may be computed as 



1-1- 



a-1 



Ricker 



Beverton and Holt 



(10) 



Note that s is independent of the vulnerability vector 

 I'. Accordingly, MSST definitions based on s will have 

 the desirable property of being insensitive to changes 

 in fishery behavior. 



Other management plans employ reference points 

 such as ^,„„,. or 

 recruit statistic 



Fqj, which are based on the yield per 



^V-ly- 



1-e 



-(R' +M„) 



-I /;•,+*', 



(11) 



Fv+M„ 



where w^ is some measure related to the average weight 

 of the catch. Inasmuch as there are no terms involving 

 the absolute abundance of the stock, the calculation of 

 such statistics poses no special problems for the relative 

 framework presented in the present study. Prescriptions 

 based on the maximum sustainable yield (MSY) are 

 slightly more complicated because equilibrium yield is 

 the product of equilibrium recruitment R and equilib- 

 rium yield per recruit: 



A 1 .p-'/^'n+^a' -Y.F",*I^, 



Y = RpJ^w,,Fv„ ^ .^ e -0 . (12) 



Fv+M„ 



