80 



Fishery Bulletin 94(1), 1996 



weight is about -0. 18 and is due mainly to metabolic 

 processes, but 2) within individual species the slope 

 is about —0.37 and is related to ecological factors such 

 as the relative sizes and distributions of predators 

 and prey. 



Therefore, the primary hypothesis to be tested in 

 this study is that M scales as W~° - 31 for Pacific 

 salmon. 



Much less is known of the probable numerical value 

 of the parameter c. However, an upper limit can be 

 placed on c if some assumptions are made about the 

 value of the weight exponent of growth rate in Pa- 

 cific salmon. The instantaneous rate of growth of 

 salmonid fishes, G (d _1 ), decreases with body weight 

 according to the allometric relationship G = aW~ h , 

 where a is the instantaneous growth rate (d _1 ) at a 

 weight of 1 g, and b is a dimensionless exponent 

 (Brett and Shelbourne, 1975; Elliott, 1975; Jobling, 

 1983 ). The biomass of a brood year of salmon is usu- 

 ally assumed to increase continuously with age, at 

 least until the brood year is old enough to be fished. 

 Therefore, if b ~ x, then c must have an absolute value 

 that is lower than a. Otherwise, the biomass of a 

 brood year would not increase with age, and may even 

 decrease with age, thereby extinguishing the brood 

 year before it was able to reproduce. To assess the 

 validity of the assumption that b ~ x, I briefly re- 

 viewed the history of growth experiments with salmo- 

 nid fishes. 



Previous research on allometry of G 



Estimates of b have been published for seven spe- 

 cies of salmonid fishes: five species of Pacific salmon 

 (pink, coho, and sockeye salmon, as well as chinook 

 salmon [O. tshawytscha] and steelhead trout), brook 

 trout [Salvelinus fontinalis], and brown trout (Salmo 

 trutta). Reported values of b have ranged from 0.20 

 to 0.50 with a mean of 0.37 (SD=0.08, n = 19) (Iwama 

 and Tautz, 1981 ). With one exception, all values were 

 estimated from juvenile salmonids weighing 0.3 to 

 350 g that were reared in hatcheries or laboratory 

 aquaria. The exception was Parker and Larkin ( 1959 ) 

 who reported that b ranged from 0.20 to 0.33 for sexu- 

 ally mature steelhead captured from the Chilliwack 

 River, B.C., and from 0.37 to 0.44 for chinook salmon 

 captured in their final year of sea life off southeast 

 Alaska. 



Brett and Shelbourne ( 1975 ) were the first to iden- 

 tify a universal value of b; they concluded that the 

 average value for salmonids was 0.41 (SD=0.06, 

 n-10). Elliott ( 1975) reported exponents ranging from 

 0.28 to 0.33 for juvenile brown trout, which led Brett 

 (1979) to suggest that the range of exponents for 

 salmonids may be wider than that reported by Brett 



and Shelbourne ( 1975). Iwama and Tautz (1981) ar- 

 gued that the best value of b for the purpose of mod- 

 eling growth of hatchery-reared salmonids was closer 

 to 0.33 than 0.41 because the line of W 1/,! versus time 

 was linear with temperature over the temperature 

 range commonly found in fish hatcheries and because 

 W 1/3 is easily converted to length. Jobling ( 1983) re- 

 ported that b for Atlantic cod (Gadus morhua) and 

 seven other nonsalmonid fishes weighing 0.9 to 150 

 g ranged from 0.33 to 0.63 with a mean of 0.41 

 (SD=0.06, «=52) and argued that a universal slope 

 of 0.4 was applicable to all species offish. Wangila 

 and Dick ( 1988) found that the average value of b for 

 two strains of rainbow trout, the freshwater form of 

 steelhead trout, was 0.31 but that slopes varied sig- 

 nificantly among strains and rearing temperatures, 

 suggesting that a universal value of b must be em- 

 ployed with caution. 



In summary, the best estimate of b for salmonids 

 is the grand mean of all exponents or 0.37 (Iwama 

 and Tautz, 1981). The value of b for sockeye salmon, 

 the species of salmon on which the most experimen- 

 tal work has been conducted, ranges from 0.39 to 0.49 

 with a mean of 0.41 (SD=0.05, n=4). This shows that 

 b is indeed close to the most probable value of.r, which 

 in turn supports the hypothesis that c is a constant 

 fraction of a, i.e. c = act. 



Therefore, the second hypothesis to be tested by 

 this study is that a has a value greater than zero 

 but less than one. 



Allometry of survival 



To test the hypotheses that x = 0.37 and c = aa, a 

 mathematical model that relates smolt-adult sur- 

 vival to smolt weight and adult weight is required. 

 Such a model can be created by using size as an in- 

 dex of age in the simple allometric model of natural 

 mortality. The first step in developing the model is the 

 adoption of a growth model that can be easily integrated 

 over time so that age and size can be interchanged. 



Over periods of days, growth in weight of salmon 

 can be accurately modeled with the well-known ex- 

 ponential model: 



dW/dt=GW. 



(li 



However, over longer periods of time, G decreases 

 with increasing weight. Substituting G = aW~ h into 

 Equation 1 leads to 



dW/dt=aW 



i i, 



(2) 



which can be integrated over time to give the growth 

 model: 



