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Fishery Bulletin 94(4), 1996 



constrain an analysis more accurately than will many 

 years of relative abundance indices. 



Population dynamics are described by Schnute's 

 ( 1985, (o > 0) or Deriso's ( 1980, ft) = 0) delay-difference 

 equation: 



B, =(1 + p)s t _ 1 B t _ 1 - ps t _ x s t _ 2 B t _ 2 + 

 Rt-pa>s t -iRt-i. 



For the case when t=2, 



B 2 = ( 1 + p)s 1 B 1 - psjS B +R 1 - pcos l R l 



(2) 



(3) 



B t and R t are the population biomass and recruit- 

 ment biomass at the beginning of year t, and s t is 

 total survival during year t (note B t includes R t ). 



The delay-difference equations allow for variable 

 recruitment. In the data analysis and simulations 

 performed in this paper the recruitment process is 

 assumed to have constant mean, i.e. E(R t ) = R x for 

 all t. The delay-difference Kalman filter method al- 

 lows for variable recruitment by considering variable 

 recruitment to be process error about the mean re- 

 cruitment. The nonlinear least-squares estimation 

 method allows only for measurement error, there- 

 fore the recruitment process is assumed to be strictly 

 constant with R t = R 1 for all t. 



However, both the nonlinear least-squares and 

 Kalman filter methods can be easily generalized to 

 have mean recruitment as a function of stock biom- 

 ass in earlier years. In this case E(R t ) = flB t _ k ), where 

 k > 1 (Kimura, 1988). 



Estimating parameters with nonlinear 

 least squares 



To fit the delay-difference equation with nonlinear 

 least squares, we must provide biomass projections 

 {B t \, from Equations 2 and 3, using as parameters to 

 be estimated, initial values (B l ,R l ). If B, is assumed 

 to be virgin biomass and R x is in equilibrium with 

 Sj, the expected (i.e. equilibrium) recruitment line 

 (ERL) follows from Equation 2 (Kimura, 1985): 



fl,=fl,{[l-pexp(-,flf)][l-exp(-M)]} 



/ 



[l-pojexp(-M)]. 



(4) 



The ERL is a straight line through the origin of the 

 (B V R X ) plane. 



The initial parameters (B,,B,) maybe parameter- 

 ized by either of two assumptions: DSj is virgin bio- 

 mass or 2) B, is not virgin biomass. If Bj is virgin 



biomass, R 1 is determined from Equation 4 (or con- 

 versely, B x can be determined from R x ). B 2 can then be 

 projected from Equation 3, with B = B 1 and s = exp(- 

 M). Projections of biomass for t>l requires {s t , t = l, ..., 

 n\. These survivals s, = expf-M-F,), are obtained by 

 iteratively solving the catch equations for F t (Kimura 

 and Tagart, 1982): 



c t =B t F t (l-exp(-M-F t ))/(M + F t ) (5) 



in a sequential manner (relative to t). If B Y is not 

 virgin biomass, B } and Bj are independent parameters 

 to be estimated. The initial projection value B = Bj 

 will be used, but with s = exp (-M-F ) if information 

 concerning F is available. In either case, further pro- 

 jections of biomass can be made by using Equation 2. 

 In the simulations to be presented, we assume that 

 the initial population is a virgin biomass (i.e. Equa- 

 tion 4 holds). This allows us to initiate the biomass 

 time series using the results in Appendix 1 and to fit 

 the simulated data without substantial model bias. 

 However, the fits to the simulated data with both 

 nonlinear least squares and the Kalman filter will 

 not assume virgin biomass. Instead, the delay-dif- 

 ference equation is fit to a time series of relative 

 abundance data by varying three parameters: B V R V 

 and a catchability coefficient A. We assume the 

 catchability coefficient scales the biomass projections 

 from the delay-difference equation IB,) to the expected 

 values of observed relative abundance indices \y t \ (i.e. 

 E(y t ) = AB t ). For nonlinear least squares, if abundance 

 indices y t have coefficient of variation cv t , parameters 

 can be estimated by minimizing 



SS = £[ln(y ( ) - ln(AB, )f Icvf 



(6) 



We used a lognormal formulation because of its su- 

 perior numerical stability in quasi-Newton estima- 

 tion algorithms. Parameters were estimated in two 

 stages: initial estimates were found by a direct search 

 over a grid of values for B 1 ,R l , then refined by using 

 quasi-Newton methods. 



The survey catchability A can be either estimated 

 or fixed depending on whether y, is thought to be a 

 relative or absolute index of abundance. Differenti- 

 ating Equation 6 with respect to A and solving the 

 normal equation shows that A can be estimated by 



^expl^Jlnfx/B^/c^J/^fl/ct;, 2 ]} (7) 



so that at this stage searches for A are not necessary. 

 However, the final minimization of Equation 6 with 

 quasi-Newton methods should be based on all three 



