FISHERY BULLETIN: VOL. 76. NO. 2, 



B{t) 



5,^-"+Cexp ((ymlBoo-F)il-n)?j 



lia-n) 



(16) 



B* = 



[ym-FB^j 



l/d-M) 



B. 



By setting initial time ^^ arbitrarily at zero, the 

 integration constant C in Equation (16) becomes 



C = B, 



l-n 



B. 



Biomass Equation (16) will apply immediately 

 upon a change in F and remain valid thereafter for 

 the time that F remains constant. Over such time, 

 population biomass will trend up or down in accord 

 with Equation (16) from initial size B^ towards 

 adjustment level B^. Should nonzero root B^. be 

 negative (which is possible only when n > 1), then 

 the adjustment level corresponds to the zero root of 

 Equation (15) and the population tends to extinc- 

 tion by Equation (16). 



The critical relationships between fishing mor- 

 tality, productivity, and time-dependent yield rate 

 in the Pella-Tomlinson system are considerably 



more complex than the relationships between F, 

 P, and Y in the Graham system. Figure 6 illus- 

 trates the behavior of P - Y when n < 1, and 

 Figures 7 and 8 illustrate P -Y when n > 1. The 

 ratio ymlB-j- becomes the critical quantity in the 

 Pella-Tomlinson system [AmlB-j. being its coun- 

 terpart in the Graham system). 



As indicated by Figure 6, the biomass level p 

 where maximum productivity occurs must lie on 

 the range <p <B.y_/e when <n < 1 . And when 

 n takes any such value, the corresponding Pella- 

 Tomlinson stock will exhibit nonzero adjustment 

 levels of biomass for all values of fishing mortality 

 however large; such a stock cannot be fished to 

 extinction. That is, nonzero root B.^ of Equation 

 ( 15) will always have a positive value when <n 

 < 1 , its range of variation being < fi* ^B^ for F 

 unrestricted on ^F < x. But F is partitioned into 



• • • • 



subranges accordingly as Y < P or Y > P. And 

 those values of F, for which Sff,* either increases or 

 decreases to B.,.. , depend on the critical ratio ym/B^c 

 and initial biomass value B^. To insure, for arbi- 



b<0 



5 



Tttnzshoid &co/& 



.T"^ — 



/ fZorujc of p 

 / when 



1 & 



5>0 



e>it) 



F > 



Ym 



1- 



itnplies Y>Py E>o^£>ifc 

 ddfustmant 





n—i 



levcJ 



1- 



ao"-' 



B, "-' 



'«» 



impUcs Y<Py B>0<G>^ 



B 



Figure 6.— a. Typical phase-plane graph of net productivity Equation ( 15), the Pella-Tomlinson system, for values ofn where <n < 

 1. For any such value ofn, root B^ of Equation (15) is always positive irrespective of the magnitude ofF. Should removal rate Y exceed 

 latent productivity P, then B < and the negative branch of Equation (15) applies. Should P exceed removal rate Y. then B > and the 

 positive branch applies. B. Typical solution graphs of stock biomass Br ^J, Equation (16), when < n < 1. Should Y >P, biomass 



declines from initial value B^ towards adjustment level B+ 



384 



But when Y < P, biomass increases from B^ towards B^, . 



