FLETCHER: TIME-DEPENDENT SOLUTIONS AND EFFICIENT PARAMETERS 



.1- 



Y{t) = FB, 



1 -(l-(Bo/B*)'"7 exp ((7m/Boo-F)(l-n)/)y^^''"^ 



(19) 



which is valid for all values of n save n ^ 1. Owing 

 to the range of definition on exponent 1/1 -n, I 

 have not found a closed form for the general time 

 integral of Equation (19) (although existence is 

 fairly easy to show for n positive and either less or 

 greater than unity). But the usefulness of the 

 analysis does not suffer too greatly for that omis- 

 sion, since one may accommodate Equation ( 19) to 

 a numerical equation solver for finite measures of 

 yield 8Y on associated intervals 8t. 



When F changes abruptly (as we have assumed 

 throughout), yield rate Y changes abruptly, but 

 the ensuing trends of adjustment are governed, in 

 the Pella-Tomlinson system, by the following rela- 

 tionships: 



<n < 1: 



0<i^<x; stock size B(t)-*B^ (Figure 6), which 

 implies that Y -^FB^. 

 n > 1: 



F < ym/By,: stock size B(t)->B^ (Figure 7), 

 which implies that Y-^FB^.. 

 F^ymlBy.; stock s\zeB(t) -►O (Figure 8), which 

 implies that Y-^O. 

 n > (both ranges): 



F = (\-lln)ym!B~f.\ stock size Bit) ->p, which 

 implies that Y -^■m (and we may identify 

 maximum latent productivity m with 

 maximum yield rate in any of the time- 

 dependent formulations of the analysis). 



The quantity ym/By-, which plays such a promi- 

 nent role in the analysis, can be identified as the 

 "intrinsic growth rate" of the stock whenever ex- 

 ponent n > 1, in direct analogy to quantity k of the 

 Graham system (and, in fact, with n = 2, then y = 

 4 and Am/By. = k). But as a consequence of the 

 indeterminate power form of the Pella-Tomlinson 

 system and the switching of coefficient signs in the 

 governing equations, the intrinsic growth rate 

 turns out to be density-dependent when n takes on 

 values between zero and unity. That is, by Equa- 

 tion ( 12), the intrinsic rate (if we may call it so) has 

 the form 



_ JUL 5"-i 



Boo 



when n falls on the interval < n < 1 (in which 

 case, 7 < 0). 



DISCUSSION 



Any nonlinear stock-production system may be 

 restructured along the lines of the critical-point 

 analysis described in the foregoing sections; such a 

 treatment will generate parametric variables 

 most likely to be those essential to management 

 analysis. A synopsis of the parameters that appear 

 in the restructured Graham and Pella-Tomlinson 

 systems is given by Table 1. 



Table l. — Parameters of the restructured Graham and Pella- 

 Tomlinson systems as they apply to management components. 



For optimization procedures on the Graham sys- 

 tem, the essential parameters are {F, m, By-} 

 augmented by the auxiliary parameters B^ and 

 B;;:. For the Pella-Tomlinson system we may 

 choose the combination {F, m.p.B,^} or the combi- 

 nation {F, m, n,Bx}, either of which constitutes an 

 essential set of mutually independent parameters. 

 In the first set, p and By. determine n; in the 

 second, n and S^ determine p. The relationships 

 in either case are governed by Equation (14). 



Although the parametric influence of n is 

 wholly prescribed by the ratio p/Byr, exponent n 

 also determines the curvature of all graphs of the 

 Pella-Tomlinson system. Therefore, when the par- 

 ticularization of the system depends primarily on 

 general curve fitting, the likelihood always exists 

 that ill-determination of parameters will follow, 



387 



