77 



AbStfclCt.— Rearing experiments 

 have shown that instantaneous growth 

 rate, G (d" 1 ), of juvenile salmonids 

 scales with body weight, W (g), as G = 

 aW~ h , where b has an average value of 

 0.37. Research on nonsalmonid fishes 

 has shown that instantaneous natural 

 marine mortality rate, M (d _1 ), also 

 scales with body weight as M = cW~ x , 

 where .r has an average value of 0.37. 

 Therefore, if b - x ~ 0, then c < a . These 

 two hypotheses were tested for Pacific 

 salmon with data on smolt-adult sur- 

 vival, s, smolt weight, W Q (g). and adult 

 weight, W(g), taken from the scientific 

 literature. A nonlinear regression of 

 survival on weight was developed, on 

 the basis of allometric marine growth: 

 log^ls) = -(a//3)(W r/J -W ( / J ), where a = c/a 

 and fl = b - x. The regression model ex- 

 plained 33% of the variance in mean 

 log p (s) of sockeye salmon (Oncorhyn- 

 chus nerka) with parameter values 

 (±1SD) of a= 0.226 ±1.171 and = 0.120 

 ±0.990. The model explained 68% of the 

 variance in the pooled mean log f (s) of 

 pink (O. gorbuscha), chum (O. keta), 

 coho (O. kisutch), and sockeye salmon, 

 as well as steelhead trout (O. mykiss), 

 with parameter values (±1SD) of a = 

 0.528 ±0.490 and /3 = -0.053 ±0.221. 

 The near-zero estimates of /3 and the 

 fractional estimates of asupport the hy- 

 pothesis that x - 0.37 and c < a. There- 

 fore, the best estimate of M for Pacific 

 salmon is M = 0.528aW" 037 , or, since a 

 = G/W-° 37 , M = 0.528G. These survival- 

 size and mortality-size relationships 

 may be used to make preliminary esti- 

 mates of survival and mortality for wild 

 populations of Pacific salmon. 



Allometry of marine mortality of 

 Pacific salmon 



Michael D. McGurk 



Triton Environmental Consultants Ltd 



120-1351 I Commerce Parkway. 



Richmond, British Columbia, Canada V6V2L1 



Manuscript accepted 14 July 1995. 

 Fishery Bulletin 94:77-88 (1996). 



There are few accurate estimates of 

 instantaneous natural marine mor- 

 tality rate, M (d _1 ), for the seven 

 species of Pacific salmon (genus 

 Oncorhynchus), despite the impor- 

 tance of this information for recon- 

 structing stock histories (Pacific 

 Salmon Commission, 1992) and for 

 modeling the dynamics of salmon 

 populations (Ricker, 1962, 1976; 

 Walters et al., 1978). For example, 

 Ricker's (1976) review of marine 

 mortality of Pacific salmon identi- 

 fied only three estimates of monthly 

 M for the last year of sea life, which 

 had no known bias and small sam- 

 pling errors. No new estimates of 

 marine M of Pacific salmon have 

 been added to the primary scientific 

 literature in the last 15 years (Groot 

 and Margolis, 1991), although many 

 estimates of smolt-adult survival 

 have been reported (Table 1). 



The lack of new estimates of M is 

 probably due to the difficulty of cal- 

 culating it. Current methods of 

 population enumeration provide, at 

 most, an estimate of the number of 

 smolts that go to sea and an esti- 

 mate of the total number of adults 

 that return 1-5 years later. They do 

 not cover the 1-5 years of sea life 

 during which marine mortality oc- 

 curs. It is possible to calculate an 

 average marine M for a brood year 

 by dividing the natural logarithm 

 of smolt-adult survival by the av- 

 erage duration of sea life. However, 

 such estimates of average M cannot 

 accurately predict the decrease in 

 abundance of a salmon population 

 for time periods shorter than a 



salmon's entire sea life, as is re- 

 quired by cohort analysis and by 

 simulation models with time steps 

 of days or weeks. This is because M 

 of fishes is not constant over a life 

 stage but decreases with increasing 

 body size. Average marine M under- 

 estimates M of smolts and overesti- 

 mates M of adults. The degree of 

 bias is proportional to the duration 

 of sea life and can be significant for 

 salmon that spend more than one 

 year at sea. 



To model the marine dynamics of 

 a salmon population, a model of the 

 way in which M varies with body 

 size is required. Following Ursin 

 (1967), Peterson and Wroblewski 

 ( 1984), Dickie et al. ( 1987), McGurk 

 ( 1993), and others, I assume that M 

 follows the allometric rule: M = cW~ x , 

 where W = body weight (g), c = M 

 (d~') at a weight of 1 g, and x is a 

 dimensionless exponent. In this 

 paper, I provide the first empirically 

 derived estimate of the weight ex- 

 ponent x for the marine life stages 

 of five of the nine species of the ge- 

 nus Oncorhynchus: pink (O. gor- 

 buscha), chum (O. keta), coho (O. 

 kisutch), and sockeye (O. nerka) 

 salmon, and steelhead trout (O. 

 mykiss ). The estimates are based on 

 an analysis of the weight depen- 

 dency of published estimates of 

 smolt-adult survival for wild popu- 

 lations. I also provide a method for 

 estimating the coefficient c for Pa- 

 cific salmon from the instantaneous 

 marine growth rate of salmon. 



Before describing the methods 

 used to estimate x and c, it is useful 



