ZWEIFEL and LASKER: PREHATCH AND POSTHATCH GROWTH OF FISHES 



and letting 1=\t-T,J 



we have the symmetric relationship 



Oy = a 



opt 



e"'opt' 



(5b) 



Substituting Equation (5b) for Equation (5a) and 

 treating T^,p^ as an unknown parameter, a six 

 parameter model was fitted to the growth data 

 with the results shown in Table 2. 



Table 2.-Growth in length of yolk-sac lan'ae of the Pacific 

 sardine at several temperatures. 



'From Lasker (1964). 



^Calculated from Equations (5) and (5b) with parameters L^ = 

 3.716, K = 0.4872, a , = 1.8523, m = 3.3878, ft = 0.0490, and 



opt 



19.38. 



Growth From Fertilization to Hatch 



Coincident to the investigation of early larval 



growth, a study of the incubation times for the 

 sardine showed that they also could be character- 

 ized by a Laird-Gompertz curve. The fitting of 

 Equation (5a) with uj- being incubation time 

 showed no bias at any point along the curve 

 (Figure 4). Unlike the posthatch growth curves, 

 however, no evidence of a temperature optimum 

 was found, i.e., incubation time did not increase at 

 high temperatures. One possible explanation is 

 that larvae which expire cannot be included and 

 hence mortality introduces a negative bias in the 

 estimate of average or median incubation time. 



The question arises whether changes in growth 

 rates occur at hatching, i.e., is there a single curve 

 from fertilization to onset of feeding? It can be 

 shown that under the Laird-Gompertz model 

 w^here growth is approaching a common asymp- 

 tote from a common origin, i.e. fertilization, the 

 incubation time It is simple multiple of the decay 

 rate a j. From Equation (5) we may solve for the 

 time to hatch Ij at size L^ to obtain: 



,4 



K 



K-\n{L„/L,) 



,/a. 



Since incubation times were not available for all 

 temperatures used in the growth experiment, the 

 sardine curve from Figure 4 was used to convert 

 all data taken at temperatures less than optimum 

 to time from fertilization and fitted to Equation 

 (5). 



The results for sardines indicated an increasing 

 size at hatch with increasing temperature which 

 was not evidenced by the observed data and an 

 overestimate of size at temperatures less than 

 14°C. It was thus concluded that a change in 

 growth rate occurs at hatch, the more noticeably at 

 extreme temperatures and that the prehatch curve 

 must be estimated separately. 



The parameters of the prehatch growth curves 

 were obtained by fitting the equation 



L„ =l^e^(^-e-"TiT) 



(6) 



to only data obtained less than 12 h following 

 hatch. The average estimated hatching size was 

 3.73 mm and the asymptotic limit was 6.13 mm. 

 The plot for several selected temperatures is 

 shown in Figure 5. Laird (1965a) has shown that 

 the length scale may be standardized and logically 

 simplified by expressing size relative to the 

 asymptotic limit. Biological events such as 



615 



