FLETCHER: TIME-DEPENDENT SOLUTIONS AND EFFICIENT PARAMETERS 



trary fig, that Y <P in Equation (,15), mortality F 

 must have a value such that 



< F < 



ym 



oo _ 



Bo 



n-1 



Be 



n-l 



in which case B > and the positive branch of 

 Equation (15) applies. Trajectory B(t) then in- 

 creases, in accord with Equation ( 16), from initial 

 value 5p towards a higher adjustment level B... 

 But for any value of F such that 



F > 



ym 



Ban 



B, 



n-l -I 



Be 



n-l 



then y > P and the negative branch of Equation 

 (15) applies; trajectory B(t) decreases from B^ to- 

 wards a lower adjustment level B^. 



Although the sign of B and the consequential 

 course of B(t) is a density-dependent process for 

 given F, we should note here that when 



n 



1 



ym 



Ban 



(17) 



then B(t)->'Y and -► m, irrespective of initial con- 

 ditions. Accordingly, we may identify parameter 

 m with MSY in any of the (reformulated) rate 

 equations of the system. 



As indicated by Figures 7 and 8, the biomass 

 level p where m occurs must lie on the range By^/e 

 <p <B.j. when n > 1. And with n so prescribed, 

 root B.., of Equation (15) may have positive or 

 negative values accordingly as F has a value less 

 or greater than the critical ratio ymlB-x_. Figure 7 

 illustrates the behavior of Equations (15) and ( 16) 

 for the constraints 



n > 1 

 < F < 



ym 



Boo 



< B* < Boo, 



in which case, root Bi.. of Equation (15) becomes 

 the adjustment level such that B(t}-^B:i, by 

 Equation (16). But whether Bf^ trends up or down 

 to B:^ depends on the further partitioning of F with 

 respect to initial biomass value B^. To insure, for 

 arbitrary B„, that Y <P in Equation (15), mortal- 

 ity F must be further constrained to the interval 



< F < 



ym 



b7~ 



oo I— 



D '1-1' 



Bo 



B 



oo -■ 



thus B  and the positive branch of Equation 

 (15) applies as indicated by Figure 7a. Trajectory 

 B(t) then increases, in accord with Equation (16), 

 from initial value B„ towards a higher adjustment 

 level B.:., as indicated by the lower curve of Figure 

 7b. But for any value of F on the interval 



ym 



Ban 



['- 



Bo 



Be 



M-1 



< F < 



ym 



Boo 



then Y > P, B < 0, and the negative branch of 

 Equation (15) applies; trajectory B(t) decreases 

 from By towards a lower adjustment level B^. as 

 indicated by the upper curve of Figure 7b. 



Should mortality F equal or exceed the critical 

 ratio ym/By. in a Pella-Tomlinson system where n 

 exceeds unity, the corresponding stock, over 

 sufficient time, will trend to extinction. Figure 8 

 illustrates the behavior of Equations ( 15) and (16) 

 for the constraints 



n > 1 



ym 



F > 



Br 



B. < 0, 



in which case the zero root of Equation (15) 

 applies, and we have B < and B(t) -►O, irrespec- 

 tive of initial conditions. 



By expanding the asymptotic bound of Equation 

 (16) to a region of radius e'Bn:, and by appealing to 

 arguments similar to those that led to the delay 

 estimate (Equation (8)) of the Graham system, we 

 calculate from Equation ( 16) the transition times 

 for a Pella-Tomlinson stock as being 



Be 



4ag 



(l-n)(7m-FiBoo) 



In 



"l-(l±6)^-" " I 

 _l-(Bo/Bi)i-"J 



(18) 



where e represents the imprecision of stock- 

 abundance estimates, and where B^, andBj signify 

 initial and adjustment levels as they correspond to 

 mortality values F„ and Fj . Again we suppose that 

 F changes abruptly at zero reference time from 

 value F„ to the new value Fj, the plus sign of 

 Equation (18) applying when Fj > F^ and the 

 minus sign when Fj < F^. 



By Equation (4) and the assumption that F var- 

 ies in time by taking on fixed values of finite dura- 

 tion, we can write the transient yield rate for the 

 Pella-Tomlinson system in the consolidated form 



385 



