Maunder and Walters: Integrating environmental time series into stock assessment models 



97 



Appendix I: description of simulator 

 and estimator 



The following is a description of the model equations used 

 for the data simulator and for the estimator. The model is 

 run from an unexploited state at the start of the fishery 

 for 20 years. The model includes 10 age classes, where the 

 10th age class is a plus group. 



Dynamics 



Af,,i = floexp(j8/,+ff +a) 



a = In 



^exp(ff +/}/,,) 



(I.l) 



(1.2) 



A^,.. = (^,-i,o-i(l- ",-A-i')'^""" ' f0Ta<A (1.3) 



Simulation 



f,, ~NiO, a^„) 



/„ ~NiO,o']) 



cCPUE _ A/If) <t'-^ 1 



(1.9) 

 (1. 10) 

 (1.11) 



CPUE^, = qB^. exp (fC'''''^ - Q.5a\piJE^ (112) 



where q = thecatchability coefficient for the CPUE index; 



a, = the standard deviation for the variation in the 



recruitment index; 



eCPUE - ^hg observation error in the CPUE index; and 



a(~.piJi- = the standard deviation of the error in the CPUE 



index. 



D.. 



Multinomial 



I^r. 



n = 50 



(1.13) 



--w,,,, 





• B. 



By=^N,,s,w, 



(1.5) 



(1.6) 



C numbers — AT n a 



(1.14) 



where D,^ = the number of individuals of age a in the 

 catch-at-age same in year y; and 

 n = the number in the catch-at-age sample; and 

 a^py,^ = the catch in numbers of age a individuals in 

 yeary. 



Estimation 



where Ny,a = the numbers in age class a at the beginning 



of yeary; 

 i?Q = the average recruitment; 

 e^ = the recruitment anomaly for year y; 

 a-jj = the standard deviations for the recruitment 



anomalies; 

 M„ = the age specific natural mortality rate; 

 A = the maximum age used in the analysis; 

 u,, = the exploitation rate in yeary; 

 s^ = the selectivity to the fishing gear for age a 



individuals; 

 C^, = the total catch in weight for yeary; 

 B . = the exploitable biomass for yeary; and 

 iL\^ = the weight for an individual of age a. 



The likelihood values can be calculated by using the fol- 

 lowing equations: 



-lnL(0|/) = ^ 



ln(crr 



.) + 



(ln(CP[7£;,,)-ln((7B,,)) 



-lnL(0|D) = -^D,„ln(p,„) 



Pv.» 



"V "" y-t num 



(1.15) 



(1.16) 



(1.17) 



Initial conditions 



N,„ =7?oe " for l<a<A 



A^u 



1-e 





(1.7) 



(1.8) 



The penalities (priors) on the annual recruitment anoma- 

 lies can be calculated by using the following equation: 



-In Prior (?"! = 



'-I 



14" 



(1.18) 



