PREHATCH AND POSTHATCH GROWTH OF FISHES- 



A GENERAL MODEL 



James R. Zweifel and Reuben Lasker^ 



ABSTRACT 



The developmental stages of fish eggs and the growth of larval fishes of several species can be 

 represented by a Gompertz-type curve based on the observation that in widely different living systems, 

 exponential growth tends to undergo exponential decay with time. Further, experimental studies and 

 field observations have shown that the effect of temperature on the growth process follows the same 

 pattern, i.e., the rate of growth declines exponentially with increasing temperature. Evidence suggests 

 that prehatch growth rates determine ideal or optimum trajectories which are maintained after hatch 

 in the middle temperature range but not at either extreme. Also, posthatch growth exhibits a 

 temperature optimum which is not apparent in the incubation period. These studies have also .shown 

 that for the same spawn both the prehatch and yolk-sac growth curves reach asymptotic limits 

 independent of temperature. Other biological events (e.g., jaw development) occur at the same size for 

 all temperatures. 



The growth of post-yolk-sac larvae follows a curve of the same type and hence the posthatch growth 

 trajectory may be represented by a two-stage curve. For starving larvae, the second stage shows a 

 decline in size but maintains the same form, i.e., the rate of exponential decline decreases exponentially 

 with time. 



Recent success in spawning and rearing marine 

 fish larvae at the Southwest Fisheries Center 

 (SWFC) (Lasker et al. 1970; May 1971; Leong 1971) 

 has made possible a much more fundamental 

 examination of the growth process than has here- 

 tofore been possible. Controlled laboratory exper- 

 iments can now be utilized to investigate both the 

 inherent nature of the growth process as well as 

 the effect of some environmental factors. 



Considerable care is required, however, in con- 

 structing a model- which is meaningful both 

 mathematically and biologically. For example, 

 almost all growth models currently in use can be 

 derived as variations of the differential equation: 



dW 



dt 



= r]W -kW 



or 



dL n 



dt ' 



K'L' 



(1) 



(la) 



(von Bertalanffy 1938; Beverton and Holt 1957; 

 Richards 1959; Chapman 1961; Taylor 1962) where 



'Southwest Fisheries Center La Jolla Laboratory, National 

 Marine Fisheries Service, NOAA, La Jolla, CA 92038. 



^A model is here conceived to be a mathematical representa- 

 tion of change in length or weight with time under measureable 

 environmental conditions. 



W is weight, L is length, and tj, k, m, n, m', and n'are 

 arbitrary constants. These are the equations used 

 most often to describe growth as a function of 

 anabolic and catabolic processes of metabolism. 

 The rate of anabolism, tj, is considered to be 

 proportional to W"' and the rate of catabolism, k, 

 proportional to W". Equation (la) requires, in 

 addition, the allometric relationship W = qL" , 

 where again q and p are arbitrary constants. In 

 practice a dilemma arises from the fact that while 

 such models yield a good empirical fit to the data, 

 the estimates of parameters r\ and k are often 

 negative, thereby negating the assumptions on 

 which the model is based. For n = \ and m = 0, 1, 

 2, respectively. Equation (1) gives rise to the von 

 Bertalanffy growth in length, Gompertz, and 

 logistic growth functions. Although we have not as 

 yet made any extensive comparisons, the fact that 

 for m>l and n = I, t] and k must be negative, 

 suggests that in many instances the Gompertz and 

 logistic rather than the von Bertalanffy functions 

 may provide more appropriate models of fish 

 growth. In particular, the simple von Bertalanffy 

 growth model has no inflection point and hence 

 curves such as the generalized von Bertalanffy, 

 Gompertz, or logistic must be used when an in- 

 flection in the growth trajectory is evident. 



Laird et al. (1965) have presented a Gompertz- 

 type mathematical model of growth based on the 



Manuscript accepted March 1976. 



FISHERY BULLETIN: VOL. 74, NO. 3, 1976. 



609 



