Thompson: Variations on a simple dynamic pool model 



725 



To accommodate this change, the Cushing stock- 

 recruitment relationship (Equation 4) has to be re- 

 written to specify that only the mature biomass BjF') 

 contributes to recruitment: 



where 



6(F',a / )=pB„,(F)". 



(25) 



Equation 2 can be rewritten to express total fishable 

 biomass as 



(26) 



\ M )\ (l + F) 2 ) 

 where K" f = ll[M(a t - a,,)], after Equation 3. 



Polynomial solution 



Age at recruitment exceeds age at maturity In the 



case where a, exceeds a„,, recruitment to the fishery 

 and to the mature stock are related as follows: 



biF ,a.) = b(F ,a„, )( — — <—) e 

 \w(a„,)J 



biF\a„,)(jAe ifr'irj, 



(27) 



where K"„, = l/[M(a„, - a )], after Equation 3. 

 Total mature biomass can then be expressed as 



B, 



= fb(F,a„,)\ | a l[a _ a]e -M^ Jda + B{F 

 \a,„-a J "- 



\ M m ))X + M(a m -a )) 



_e*VO fc«o_ + 1 Yl 



\a„,-a M(a m -a l] ))\ 



IF') (28) 



b(F',a„,){ 1 + K") - b(F',a r ) ( 1 + K") UlJ? , 

 - " - —L '—+ BAF ). 



M 



Equations 25-28 constitute a set of four equations 

 in four unknowns [b(F\a t ), b(F\a,„), B' f {F'), and BjF')]. 

 Solving simultaneously gives 



^•»-(-&)(tSf)[©^f) + ^' (29) 



olK;)jS\t ltK ' 



/A, w l+A. x _ 



(30) 



Multiplying Equation 29 through by MF\ differenti- 

 ating with respect to F\ and setting the resulting ex- 

 pression equal to zero gives the following polynomial 

 solution for F' MSY (which collapses to Equation 6 when 



K" f =K" m y. 



(< 1- q)MK" f )( 1-K'; )+l]-l)F' A/sy 3 + 



{(\-q){a(K" f )(3-K" f )+2K" t +3]-3K" f -2)F , Ms ^+ 



(31) 

 a^)[a(K})(3+K;)+K; i +AK"+3]-2Ky--3K';-l)F' m ,-+ 



( 1^7 \MK" f >( i+k; )+k; 1 +2K';+i] = o. 



The solution for F' ma:i in this model is the same as in 

 the base model (Equation 7). 



Equation 31 contains F' MSY = 1 as a special case, 

 obtained when the following relationship holds: 



9 = 1 



IK} + 2){K", + 1) 

 Aa[K" f ) + (K" f + 2f 



(32) 



Just as the base model required an inverse relation- 

 ship between q and K" in order for F' MSY to equal 1.0 

 (Equation 8), this model requires an inverse relation- 

 ship between q and K" f . Likewise, the upper limit to 

 the ratio B(F UST )/B(0) is the same as in the base model 

 (1/e). 



Age at maturity exceeds age at recruitment Another 

 possible modification is to allow a,„ to exceed a r This 

 requires rewriting Equation 27 as follows: 



biF',a,) = b(F 



'-(§) 



o + F1/1 



fe-a 



(33) 



The previous expressions for recruitment (Equation 

 25) and equilibrium fishable biomass (Equation 26) 

 can be applied without modification. However, because 

 the entire mature stock is subject to both fishing and 

 natural mortality, the previous expression for equilib- 

 rium mature biomass (Equation 28) is simplified to 



BJF') 



_/b(F\a„,r\A+K" ni + F\ 

 ' \ M ) \ (1+FT ) ' 



(34) 



