of temperature for Stages II- VH. The slopes of the 

 resulting straight lines (Fig. 2) were compared using a 

 multivariate general linear hypothesis model 

 (Morrison 1967), which regresses a vector of obser- 

 vations (development time of each stage) against 

 temperature. The hypothesis of parallel slopes was 

 not rejected (P > 0.05); therefore, the relation be- 

 tween age and temperature is probably independent 

 of the stage of development. 



To facilitate estimating age (time after fertilization, 

 in hours) of walleye pollock embryos, we generated a 

 contour plot (Fig. 3) from the generalized predictive 

 Equation (1) over the temperature range of 2°-l 1°C 

 for development Stages II- VII. For both contour plot 

 and generalized predictive Equation (1), the es- 

 timates of age of walleye pollock embryos can be 

 made more precise by refining the staging scheme 

 (Table 1) into fractions of stage development. The 



2 1.0 6.0 8.0 10.0 



TEMPERATURE °C 



FIGURE 2.— Age-temperature relations of walleye pollock embryos. 

 Stages II- VII. 



FIGURE 3.— Contours of predicted development time (h) of walleye 

 pollock embryos as related to stage of development and water tem- 

 perature (' C). 



qualitative estimates of stage fractions are quan- 

 tified by adding or subtracting the proportion of 

 stage development to the stage number. 



Similarity of the age -temperature relations among 

 stages implies that embryo development is predict- 

 able regardless of whether water temperatures fluc- 

 tuate or remain constant. We examined this 

 implication mathematically by simulating stage of 

 embryo development after 200 h given a mean tem- 

 perature of 6°C. Examples of temperature variation 

 used were 1) 100 h of embryo development at 4°C 

 followed by 100 h at 8°C, 2) 100 h at 8°C followed 

 by 100 h at 4°C, 3) 67 h at 2°C followed by 133 h at 

 8°C, and 4) 133 h at 8°C followed by 67 h at 2°C. We 

 compared the simulations with 200 h of embryo de- 

 velopment at constant 6°C. For various fluctuating 

 temperatures, the mean of the simulations predicting 

 stage of development was 6.08 with a standard error 

 of 0.05. The value 6.08 corresponds closely to the 

 computed stage of development of 6.06 for embryos 

 reared at a constant temperature of 6°C. The stan- 

 dard error of 0.05 transforms into a standard error of 

 3.5 h in terms of development time and is similar to 

 the standard errorof 3.03 h of the generalized predic- 

 tive Equation (1). The similarity of the standard 

 errors (in hours) shows that temperature fluctuations 

 exert a negligible decrease in accuracy of the 

 generalized predictive Equation (1) and that an es- 

 timate of embryo development time based on mean 

 temperature has the same reliability as an estimate 

 based on a constant temperature. It should be noted, 

 however, that mean development times were 

 simulated and that estimates of mean development 

 time based on empirical data are needed to verify the 

 implication of the age-temperature relations. 



We further substantiated that the mean of fluctuat- 

 ing temperatures could be used to estimate develop- 

 ment time by comparing results from our generalized 

 predictive equation with the development time ob- 

 served by Hamai et al. (1971). Hamai et al. collected 

 walleye pollock adults near Hokkaido, Japan, and 

 reared the embryos from these fish at three different 

 temperature ranges: 7.8°-14.5°C, 5.1°-10.6°C, and 

 0.0°-6.7°C. Only the temperature range 5.1°-10.6°C 

 (x = 6.6° C) and stage at completion of epiboly (our 

 late Stage IV) were comparable with our data. We 

 determined time to completion of epiboly over the 

 temperature range 5.1°-10.6°C from their figure 3 

 (100 h). In our experiment, predicted development 

 time to completion of epiboly at 6.6°C was similar (92 

 h) to time for completion of epiboly observed by 

 Hamai et al. 



The only other study on embryonic development of 

 walleye pollock embryos comparable with ours is 



893 



