80 



Fishery Bulletin 98(1 



0.2 



to harvesting. The analysis of the LesHe 

 matrix offers information not available 

 in the analysis of SSB/R, however. First, 

 the consequences of ovei"fishing or "un- 

 derfishing" are clearly evident from the 

 graph of A, against fishing mortality rate. 

 For example, for an age at entry of 3 

 years, F^^ is 0.465. If the fishing mortal- 

 ity rate was limited to 0.20 (for exam- 

 ple), the population would be expected 

 to increase at a rate of about 8.8% per 

 year (Fig. 1). Likewise if fishing mortal- 

 ity was increased to 1.0, the population 

 would be expected to decline at a rate of 

 about 10.3% per year (Fig. 1). 



When %MSP is plotted against lambda 

 resulting from various levels of fishing 

 mortality and ages at entry into the fish- 

 ery, it is apparent that equal %MSP val- 

 ues are obtained for the same lambda 

 only at two points along each of the 

 curves. The first point is for the unfished 

 population when lambda is at a maxi- 

 mum and there is 100% MSP. The second point where 

 all 7f MSP values are equal is when lambda is equal to 

 1.00 (Fig. 2). These results demonstrate the assertion 

 that fishing mortality rates that result in equal %MSP 

 values do not necessarily result in the same population 

 dynamics (i.e. the same rate of increase or decrease). 

 The reason for this disparity is that in a growing or 

 declining population, the timing of reproduction dur- 

 ing the lifetime is important, as well as the total life- 

 time egg production. For example, when a population is 

 growing, earlier realization of lifetime spawning poten- 

 tial contributes more to population growth than later 

 reproduction. This relationship is evident when the for- 

 mula for lifetime spawning stock biomass (on which 

 %MSP is based) is compared to the formula for repro- 

 ductive value, upon which the rate of population change 

 depends. Observe that lifetime spawning stock biomass 

 per newborn individual is (Gabriel et al., 1989) 



Age at entry 



5 Maintenance 



4 level 



Current 

 3 



0.0 



—\ — 

 0.5 



— I — 

 1.0 



1.5 



— I — 

 2.0 



2.5 



Instantaneous fishing mortality 



Figure T 



Rate of population increase (lambda) in relation to instantaneous fishing 

 mortality (F) for a range of age at entry into the fishery. 



SSB/Nq = 



S(0) W( 1 )PM( 1 ) -t- S( )S( 1 ) W( 2 )PM ( 2 ) -i- 

 S(0)S(1)S(2)W(3)PM(3) + ... . 



This formula is equivalent to that for the "net repro- 

 ductive rate" (Caswell, 1989) which is the expected 

 number of offspring produced by a newborn over its 

 lifetime. With the above notation, the reproductive 

 value (Caswell, 1989) of an age-1 individual can be ex- 

 pressed as 



Reproductive 

 value = 



S(0)S(1)W(2)PM(2)A-' + 

 S( |S( 1 )S( 2 )W( 3 )PM( 3 )A-2 + 

 S(0)S(l)S(2)S(3)W(4)PM(4)A-3 + 



Table 4 



Sustainable fishing mortality (F^,) and %MSP as a function 

 of age at entry (tj for Georges Bank haddock. 



t. F„ '?MSP 



27.20 

 27.20 

 27.20 

 27.20 

 27.20 



These formulae are similar except for the addition of 

 the term A ', where / is the age index. Classical demo- 

 graphic theory shows that the growth rate of a popu- 

 lation is dependent on the reproductive value rather 

 than on the net reproductive rate (Caswell, 1989). 

 These two quantities are clearly related, however. 



Currently, Georges Bank haddock become vulner- 

 able to the fishery at age 2 but are not fully recruited 

 until age 4 (Table 2 1. With this partial recruitment 

 vector, F, is 0.519. The graph of A, against F (Fig. 1) 

 indicates that if F is held at its 1994 value of about 

 0.29, the population would be expected to increase at 

 a rate of 6.35% per year If F is reduced to 0, the stock 

 would be expected to increase at a rate of 18.0% per 

 year. The expected rate of increase when F is below 

 F^i is particularly pertinent to cases where stock re- 

 building is desired because this analysis allows the de- 



