Birectilinear Recruitment Curves to 

 Assess Influence of Lake Size on Survival of Sockeye Salmon 

 ( Oncorhynchus nerka) to Bristol Bay and Forecast Runs 



By 



RALPH P. SILLIMAN, Fishery Biologist 



Bureau of Commercial Fisheries 



Biological Laboratory 



Seattle, Washington 98102 



ABSTRACT 



Comparison of the sizes of lakes and the sizes of sockeye salmon runs to Bris- 

 tol Bay shows that the two variables are closely related. Birectilinear reproduction 

 curves express quantitatively the dependence of small returns on escapement num- 

 bers and of large returns on lake capacity. Comparison of "hindcasts" from the 

 birectilinear curves with published forecasts for 1961-67 showed that those from 

 the birectilinear curves were closest to the actual returns. This situation changed 

 in 1968-69. A composite of birectilinear return estimates and "probability tree" age 

 allocations is worth considering. 



INTRODUCTION 



Essential to forecasting of Bristol Bay sock- 

 eye salmon ( Oncorhynchus nerka ) runs, and to 

 salmon forecasting generally, is the establish- 

 ment of a relation between the number of 

 spawners escaping to the streams and the num- 

 ber of adult salmon returning to the fishery 

 and to the streams as the progeny reach ma- 

 turity. This relation is studied by fishery 

 biologists as a topic within the more general 

 field of stock-recruitment relations (Ricker, 

 1954). It would appear desirable, where possi- 

 ble, to have the stock-recruitment curve re- 

 flect features of the reproductive biology of 

 the species in question, rather than simply 

 to furnish an empirical fit to the observed 

 data. This need holds particularly for the 

 Bristol Bay sockeye salmon, where the em- 

 pirical points have great random variability 

 and do not closely define any type of mathe- 

 matical curve. 



The purpose of this report is to study 

 the available escapement-return data for 

 the Bristol Bay sockeye salmon and to de- 

 rive curves that reflect the influence of the 

 size of the lake on survival of young. Such 

 curves may be useful in forecasting future 

 runs. 



THEORY OF BIRECTILINEAR APPROACH 



Comparison of the sizes of the fresh-water 

 and oceanic environments of the sockeye salmon 

 suggests that the former may be the limiting 

 factor to size of run. Furthermore, it seems 

 reasonable to assume that when escapements 

 of spawners are small the numbers of their 

 progeny will vary with size of the escapement, 

 as the progeny would not fully use the capacity 

 of the stream-lake systems. When escapements 

 are large, space or food supply or both may 

 beconne limiting. This latter idea is supported 

 by the findings of Burgner (1964), who dis- 

 covered an inverse relation between numbers 

 of fish in escapennents and average weights of 

 progeny in the Wood River Lakes. He also re- 

 ported a previous finding that ocean survi- 

 val is positively correlated with smolt size. 

 Together, these findings support the thesis that 

 survival varies with lake size, provided one 

 adds the reasonable (in my view) assump- 

 tion of some positive relation between lake 

 area and total food supply. Rounsefell (1946) 

 and Ryder (1965) also recognized the influ- 

 ence of size on productivity in north-tem- 

 perate lakes by expressing production in 

 pounds per acre. Rounsefell and Stringer 

 (1943) demonstrated a general relationbetween 



