FLETCHER: TIME-DEPENDENT SOLUTIONS AND EFFICIENT PARAMETERS 



7 = 



n 



n-1 



(13) 



Be 



With the coefficients so cast, the sign reversals at 

 turning point n ^ 1 become automatic. In con- 

 sequence, the consolidated interval of definition 

 for n becomes < n < x (the point n ^ 1 being a 

 removable singularity). With parameter m thus 

 separated from n in Equation (12), the undeter- 

 mined exponent n can be defined solely by the 

 fraction pBy. in the relationship 



Br 



= n"^' 



■n) 



(14) 



Consolidated Equation ( 12) now takes on the role 

 in the Pella-Tomlinson system that Equation ( la) 

 takes on in the Graham system. In fact, when n = 

 2, Equation (12) reducesto Equation (la), in which 

 case y — A and p/B^ = V2. As an interesting aside 

 here, we note that Equation (12), at the turning 

 point n = I, takes on the form 



P = 



e m 



[Boo] [Poo_ 



(e being Napier's constant), while ratio ( 14), in the 

 limit as n-^l, has the value 



In fact. Fox (1970) constructed a stock-production 

 model around this special case, but since the ratio 

 p/Bx has the fixed value 1/e, Fox's model "has as 

 rigid a form as the Graham model" (Ricker 1975: 

 331). 



Quantities m, p, and B^ constitute a complete, 

 minimum set of independent parameters for la- 

 tent productivity in the Pella-Tomlinson system. 

 Collectively they control the behavior of govern- 

 ing Equation (12), but the influence of any one 

 parameter remains independent of the remaining 

 two. Figure 5 illustrates their separate effects on 

 the graph of Equation (12). 



By appealing to the same piecewise constraints 

 that enter the Graham productivity equations, we 

 substitute Equation (12) into the general produc- 

 tivity formula (Equation (5)) and net productivity 

 in the Pella-Tomlinson system becomes 



B = ym 



[boo] 



ym 



B 



Be 



FB. (15) 



And over any time interval, however brief, that 

 mortality F might be presumed to have a fixed 

 value, biomass variable B in Equation ( 15) has the 

 general time-dependent solution 



B^: unexploited stock 

 level [the nonzero 

 root of Equation ( 12)]. 



p: biomass level for 

 maximum productivity 

 [the coordinate of S in Equa- 

 tion (12) where m occurs]. 



m: maximum productivity 

 [the extremum coordinate 

 fmax i"^ Equation (12)]. 



Figure 5. — The graph of Equation ( 12), latent productivity in the Pella-Tomlinson system, as controlled by independent parameters 



m, p. and By.. 



383 



