722 



Fishery Bulletin 94(4). 1996 



cific salmon species of the genus Oncorhynchus, "re- 

 cruitment is maximum at some intermediate stock 

 size." The three hypotheses hold for the four salmon 

 species in Table 2 and for all life-stage transitions. 



Discussion 



Could our results be an artifact of the necessary re- 

 lationship between recruitment and subsequent 

 spawner abundance? It has been argued that the 

 observed relationships between spawners and re- 

 cruitment is a by-product of the high autocorrelations 

 present in some spawner and recruitment series. 

 Consider a simple example in which recruitment is 

 unrelated to the spawner abundance but is com- 

 pletely determined by the environment. Let recruit- 

 ment be a first-order autoregressive process with 

 autocorrelation parameter p, i.e. the correlation of 

 recruitment with lag t is pf. For a semelparous spe- 



cies, or a heavily-exploited species with relatively 

 little survival after reproduction, the amount of vari- 

 ance in recruitment that would be "explained" by this 

 process is p 2 ", where a is the age at maturity. For 

 example, if the age at maturity is a = 4 and the 

 autocorrelation in recruitment is p = 0.4, then only 

 0.066% of the variance would be explained by 

 spawner abundance. Thus, this mechanism will be 

 important only when there is high environmental 

 autocorrelation in recruitment and low age at matu- 

 rity. However, when we restricted our analysis to data 

 series with estimated autocorrelation less than 0.4, 

 the observed patterns remained (Fig. 5). Alternative 

 cutoff values for the degree of autocorrelation pro- 

 duced similar results. We conclude that our results 

 are not caused by autocorrelation in recruitment and 

 the necessary relationship between recruitment and 

 subsequent spawner abundance. 



In each of our three tests, the hypothesis that there 

 is no practical relationship between spawners and 



