of actual realized size. Comparison of the ex- 

 trapolated curves for the sardine and anchovy, 

 Figures 5 and 7, shows that for the same temper- 

 ature and relative to the asymptotic size, hatching 

 occurs later for the anchovy, but jaw development 

 and first feeding occur at about the same time. 



In summary, each growth cycle may be repre- 

 sented by an equation of the form 



L = L„e^''^-'""^" 



with 



or 



with 



an 



o„e '"(i-'-/^^' 



«r = «opt^ "1' 



\T- T. 



,(\-.-Ph 



.1 



OJlt 



when a temperature optimum exists. The time 

 required to attain a given size S is 



t^ = 



^='^ [kTi^J/"^ 



which has the same form as the original equation. 



Most of the data available were from studies of 

 two species, Sardinops sagax and Engranlh mor- 

 dax, so that generalizations must be made with 

 caution. Nevertheless, incubation times for sever- 

 al other species fit the model well. 



Finally, it seems worthwhile to repeat that 

 every growth problem becomes at last a specific 

 one depending on many factors known or un- 

 known, measureable or not. For example, time of 

 fertilization will often not be known and age 

 determinations will be inexact. Further, Hunter 

 and Lenarz' have shown that egg size is a mea- 

 surable and probably important factor in growth 

 and survival of anchovy larvae. For feeding larvae, 

 the quantity and quality of food is critical. Egg 

 size appears to afl'ect growth by a simple scale 

 factor, all events being shifted up or down in 

 proportion to the egg size. Variation in food may 

 result in many "artificial" cycles when nutritional 

 and caloric requirements are not met. Neverthe- 

 less, it seems clear that at least the early growth of 

 many fishes may be described in terms of genet- 

 ically determined but dynamically changing 

 growth rates as defined by the Laird-Gompertz 

 growth function. 



'Hunter, J., and W. Lenarz. 1974. A discussion on the adaptive 

 values of variation of fish egg sizes. Unpubl. manuscr., 7 p. 

 Southwest Fisheries Center, Tiburon Laboratory, National 

 Marine Fisheries Service, NOAA, Tiburon, CA 94920. 



P^ISHERY BULl.KTI.N: VOL. 74, NO. 3 



ACKNOWLEDGMENTS 



We express our appreciation to David Kramer 

 and Pete Paloma of the National Marine Fisheries 

 Service for making unpublished data available to 

 us, to Michel Coirat for her diligent eff"orts in the 

 laboratory, to Lorraine Downing for her typing 

 skill and patience with mathematical formulae, 

 and to John R. Hunter for his advice on this work. 

 Special thanks are due to Fisherii Bulletin 

 reviewers for checking our mathematics and for 

 constructive criticism of the manuscript. 



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620 



