biological grounds, and since it was practical for 

 analog computation, I employed it. 



It may be noted that growth rates of fisii typically 

 decline throughout life and can be represented most 

 simply by a positive instantaneous rate which de- 

 creases exponentially with time, as in the Gompertz 

 curve. The relations can be expressed in the follow- 

 ing formulas (where G represents the initial exponen- 

 tial growth and g governs the exponential rate of 

 decline) : 



(2a) 



, = Wre''e-°'-''"-''^ 



iv, = u\e 



G-ae-«"-'T) 



(2b) 



This formula can be combined with formula (1) 

 to express total weight of survivors at any time. It 

 is of interest, also, that it has an upper asymptote 

 u'„ similar to the "LJ' of the von Bertalanffy 

 equation. Thus: 



as I > cc, e-""-''' >0 



and e-o'-'"-''' 



1.00 



ca from ecjuation 



so that w, > wve," as t 



(2a) above. The limiting value of this expression is 

 w^. If extended from w, = to !(',^m'„, the Gompertz 

 curve has a point of inflection lacking in the von 

 Bertalanffy curve. This inflection is found in age- 

 weight curves of many fishes. The total weight of 

 survivors from a single year class may be expressed ; 



P, = w',^V, 



(3) 



Substituting in (3) for w, and .V, their equivalents 



in (1) and (2b): 



Because I dealt with weight rather than numlier 

 of recruits, I set 



R^ = RlVr 



and obtained as my working equation: 



For convenience, clarity, and ready comparability 

 with other work, I have dealt with all relationships 

 so far in algebraic form. Although the computer 

 requires differential e<iuations, the differentiation 

 can be performed on the final equation, (4) above. 



34 



THE COMPUTER AND PLOTTER 



Since analog machines probably are not familiar 

 to most fishery biologists, a brief descrijition seems 

 in order. Modern analog computers (fig. 1) are 

 electronic and perform operations on voltages. The 

 voltage is made numerically equal (analogous) to 

 \ariablcs in the proljlem (e.g., 1 volt = age of fish of 

 2 years), and component building blocks on the 

 computer i)erforni mathematical operations. Vari- 

 ous blocks perform: (1) algebraic summation, (2) 

 multiplication or division by a constant, (3) multi- 

 plication and division of two variables, (4) integra- 

 tion, and (5) generation of nonlinear functions. This 

 last building block makes it possible to produce 

 ftnictions if a graph of the function is availat>lc even 

 though the equations describing the graph are 

 unknown. 



The analog computer is primarily a device for 

 solving differential ecjuations with time as the inde- 

 pendent variable. It, therefore, becomes evident 

 that if a biological process can be expressed as a dif- 

 ferential equation, the etjuation can be mechanized 

 by interconnecting analog computer components 

 corresponding to the mathematical operations. 



I'liaiiE 1. — .\iiulog cdinputer and plotter. 



U.S. FISII ASD WILDLIFE SERVICE 



