FISHERY BULLETIN: VOL. 74, NO. 3 



observation that the specific growth rate dW/ Wdt 

 of animals and their parts tends to decay expo- 

 nentially with increasing age. They have shown 

 that this relation offers a practical means of 

 analyzing the growth of parts of embryonic and 

 postnatal animals (Laird 1965a), the growth of 

 tumors (Laird 1964, 1965b), whole embryos of a 

 number of avian and mammalian species (Laird 

 1966a), and early stages of postnatal growth of a 

 variety of mammalian and avian organisms (Laird 

 1966b). Further, Laird (1966b, 1967) has shown 

 that postnatal growth of a variety of mammalian 

 and avian organisms can be fitted by compounding 

 this model with a linear growth process beginning 

 at birth and extending on beyond the asymptotic 

 limit of the underlying Gompertz growth process. 

 Overall growth is assumed to be genetically de- 

 termined by programming only the initial specific 

 growth rate and the rate of exponential decay, 

 these governing growth processes then act on a 

 genetically determined original mass to produce 

 the observed course of growth to a final limiting 

 size characteristic of the species and individual. 



Mathematically, these assumptions are de- 

 scribed by the two equations: 



dWjt) 

 dt 



y{t)Wit) 



and^ 



dm = -ay{t) 



dt 



which have the solution 



W{t) = Woe 



^{ 



-at. 



(2) 



where W^ is weight at f = 0, Aq is the specific 

 growth rate at ^ = 0, a is the rate of exponential 

 decay and the specific growth rate at time t, A, = 

 Aoe-'". 



Laird et al. (1965) indicated that an additional 

 growth component not included in the Gompertz 

 equation may be due to the accumulation of 

 products that are not self-reproducing or to 

 renewal systems that are not in exact phys- 

 iological equilibrium and suggested the com- 

 pound growth curve: 



W= W^ + (i 



/ 



t w, 



. M 



dt 



(3) 



where Wq is the mass due to the Gompertz growth 

 process, /3 is the rate of linear growth, and M is the 

 asymptotic limit of the growth process. She also 

 suggests that this linear process starts in the early 

 embryonic period, if not at conception. For the age 

 interval covered in this paper, however, the linear 

 growth component {W - W^) was not found to be 

 important. 



Several characteristics of the curve are worthy 

 of mention: 



The asymptotic limit M is Wq Exp (^ ,)/«)• 

 The point of inflection {t,, W,) = 



[^- 



iAo/a),WoExp{^ 

 a 



»] 



3. The zero point on the time scale may 

 be shifted to any point t^ without changing the 

 form of the equation with new parameters W^ = 

 W(l), A^ = Aoe'"^ where a remains un- 

 altered. 



The fundamental concept of the Laird- 

 Gompertz model is one of change in weight or 

 mass with time, being due primarily to the self- 

 multiplication of cells and genetically determined 

 limitations on the growth parameters. The use of 

 length as the measured variable is thus a matter of 

 convenience due to the fact that weight measure- 

 ments are much more time consuming, especially 

 in early larval growth, but also in juvenile and 

 adult fishes. As indicated in Equation (la), if a true 

 allometric relationship existed, the choice would be 

 unimportant. However, all experimental evidence 

 indicates that both length and weight can be 

 described by a Gompertz-type curve. Hence, it can 

 be shown that 1) the growth rate for both changes 

 continually with time and 2) the form of the 

 length-weight relationship will change continually 

 except for two special instances. Laird et al. (1968) 

 has shown that this occurs only when the rates of 

 exponential decay are the same and either the two 

 measured variables begin growth at difi'erent 

 times at the same initial rate or at difi'erent rates 

 at the same time. In all other cases the allometric 

 plot will be nonlinear. For 



■'In the usual Gompertz representation the rate of exponential 

 growth is assumed to decline logarithmically as W approaches 



the asymptote M = WqC » , i.e., 



dW ^ 



amn(M/W). 



Kl(^ 



and 



L = Loe 



W = WoB^^^^ 



rPt) 



^-at\ 



610 



