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Appendix 1 —Independent process errors 



We simulated independent production process errors 

 using a three-parameter gamma probability distribution 

 (Johnson et al., 1994). The gamma probability density 

 function for independent (no autocorrelation) r^, values in 

 one simulation (denoted as s) was 



P(/;,Ja,,/3„7) = 



(r. 



,a, -1 -(r^.-yllp. 



p:'naj 



(1) 



where ViaJ is the gamma function and the parameters of 

 the gamma distribution are a^ >0, jS^ >0 and y. 



The expected ( mean ) value for r^, values from the three- 

 parameter gamma distribution is P^ a^ + yand the variance 

 is P^oig. The parameter /defines a minimum value for the 

 distribution of r^, values; therefore we set y= -M (see text). 

 There were too little data to directly estimate the lower- 

 bound 7 for either of the species in our analysis because 

 years with negative production occurred infrequently for 

 stocks in our analysis. The three-parameter gamma dis- 

 tribution has a single mode at y-n pJa-\) if a, >1. When 

 a, <1, the probability distribution function declines mono- 

 tonically as r^ increses from the minimum value at r^=y. 



