McGurk: Allometry of marine mortality of Pacific salmon 



W = (W b +abt 



1/6 



(3) 



where W Q = initial weight (g). Equation 3 was first 

 introduced by Parker and Larkin ( 1959) to describe 

 marine growth of chinook salmon and steelhead 

 trout. Elliott ( 1975) and Iwama and Tautz ( 1981 ) used 

 the model to describe successfully the freshwater 

 growth of several species of salmonids. Apart from 

 being based on an allometric growth argument, this 

 model has the important advantage of easier inte- 

 gration over time in comparison with other three- 

 parameter growth models that have been applied to 

 fish. 



Mortality of Pacific salmon can also be modeled as 

 a power function of body weight: 



dNldt = -MN = -cW~ x N, 



(4) 



where N is the population number and c has units of 

 d" 1 (Ursin, 1967; Peterson and Wroblewski, 1984; 

 McGurk, 1993). Substituting Equation 3 into Equa- 

 tion 4 and integrating over the duration of sea life 

 gives an expression for smolt-adult survival, s, in 

 terms of body weight: 



log (> (s) = log (> (A' , /iV ) 



-(c/(a(6 -*)))( 



W 



W, 



fc-x 1 



(5) 



where N = the total number of adults that return to 

 their natal stream (the sum of catches in all com- 

 mercial and recreational fisheries plus the number 

 of fish that escaped the fisheries and were counted 

 on the spawning grounds ); N = the number of smolts; 

 W = adult weight (g); and W Q = smolt weight (g). Ag- 

 gregating parameters in Equation 5 gives a simple 

 expression that is suitable for nonlinear regression: 



\og e (s) = -(a/p)(W li -Wf), 



(6) 



where a = c/a and p = b - x. Both a and P are dimen- 

 sionless numbers. 



In this paper, Equation 6 was used to regress log,(s) 

 on W and W and thereby estimate values of a and p. 

 I tested two hypotheses about the values of a and /J: 

 1) P is a small number that is not significantly dif- 

 ferent from zero because b ~ x ~ 0.37, so P = b - x = 0; 

 and 2) a has a value significantly lower than 1.0 but 

 greater than zero because a = c/a and c < a. 



In fitting Equation 6 to the available smolt-adult 

 survival data, it was necessary to pool survival and 

 size information from five species of Pacific salmon 

 because, with the exception of sockeye salmon, the 

 range of mean survivals and mean sizes within a 

 single species was too narrow to estimate reliable 



values of a and P for particular species. Therefore, I 

 assumed that there are average values of c and x for 

 Oncorhynchus species which have practical utility 

 over different time and space scales (species, popu- 

 lation, and brood year) and that overlaid on this av- 

 erage allometry is species-specific, population-spe- 

 cific, and brood-year specific variability which can- 

 not as yet be estimated given the information avail- 

 able in the primary literature. 



Materials and methods 



Sources of data 



Survival I searched the primary scientific literature 

 for well-documented records of smolt-adult survival 

 for wild populations of Oncorhynchus. Records de- 

 rived entirely from secondary sources were not in- 

 cluded because of difficulties in obtaining informa- 

 tion on body sizes. A few records reported in primary 

 sources were also excluded for the same reason. Popu- 

 lations of pink, chum, and coho salmon that were 

 composed predominately of hatchery production were 

 excluded because of controversy over whether hatch- 

 ery fish have the same smolt-adult survival as wild 

 fish of the same body size. The exceptions to this rule 

 were estimates of smolt-adult survival calculated 

 from populations of sockeye enhanced by stocking of 

 young fry into nursery lakes (e.g. Leisure Lake, Alaska: 

 Koenings and Burkett, 1987; Koenings et al., 1993). 

 These fish spent a minimum of one year after release 

 in a natural nursery system before migrating to sea. 



Smolt-adult survivals were collected only for brood 

 years, defined as the production of adult fish of all 

 ages from a single year's spawning in a single nurs- 

 ery system. To account for multiple ages of return, 

 the number of returning adults of the zth brood year, 

 N was summed over all ages as 



N, 



X<c„+£„), 



(7) 



where C it = the catch of the ;th brood year at an age 

 of return of/ years; and E = the escapement of the 

 /th brood year at an age of return of r years. To ac- 

 count for multiple ages of smolting, the number of 

 smolts was summed over all smolt ages. For all popu- 

 lations of pink and chum salmon, there was only one 

 smolt age because the newly emergent fry immedi- 

 ately went to sea. The same situation applied to some 

 coho and sockeye populations that were predomi- 

 nantly of a single smolt age. For example, 95% of 

 Chilko Lake sockeye smolts migrate to sea at an age 

 of 1+ yr (Henderson and Cass, 1991). The exception 



