378 



Fishery Bulletin 92(2). 1994 



Extensions to the model 



A great strength of the model presented here is the 

 ease with which it can be extended and modified. 

 Such extensions can include, for example, modeling 

 fisheries divided by space, time, or gear type; ana- 

 lyzing data series including some years of no effort, 

 as would occur during a closure; analyzing data se- 

 ries with years of missing or highly uncertain effort 

 data; incorporating changes in catchability within 

 the data series, perhaps after periods of closure or 

 following regulatory changes; and tuning the model 

 to fishery-independent estimates or indices of popu- 

 lation biomass. 



Missing data 



Gaps in the effort and yield time series do not present 

 a problem to these dynamic production model analy- 

 ses. Years with no effort (and therefore no catch) can 

 easily be treated by defining the residual to be zero. 

 Although such years do not influence parameter es- 

 timation directly, the time lag during the years of 

 closure carries information that is incorporated in 

 fitting the model, and an estimate of population bio- 

 mass for each missing year is made according to the 

 logistic growth model. In contrast, years of closure 

 contribute no information to production models that 

 assume equilibrium conditions. 



A slightly more difficult problem is the correct 

 treatment of years in which effort is known to have 

 existed, but for which the data are missing or highly 

 uncertain. In such a case, the framework presented 

 here can be used to estimate, simultaneously with 

 the other parameters, effort levels for a limited num- 

 ber of such years within the series. As in any estima- 

 tion scheme, the total number of estimated param- 

 eters should be kept reasonably small in comparison 

 to the number of years of nonzero data. If residuals 

 are constructed in effort (rather than yield) the esti- 

 mation of missing effort becomes trivial, as a pre- 

 dicted effort is computed for each year during pa- 

 rameter estimation. 



More than one data series 



Another simple extension of the basic estimation 

 framework is analysis of stocks fished by two or more 

 different gear types, either in the same years or se- 

 rially For convenience, I refer to these as different 

 fisheries on the same stock. To define the situation 

 more precisely, there are J different fisheries, indexed 

 by j = (1, 2, ..., J). The effort applied by fishery,/ in 

 period T is f-, the catchability coefficient of that fish- 

 ery is o , and the yield in period x is Y z . All q are 



assumed time— invariant. The total instantaneous 

 fishing mortality in period x is 



Fr^Qjfjr- (ID 



Biomass and yield projections can be computed by 

 Equations 4a, 4b, 6a, and 6b as before. The estimated 

 yield from fishery./' in period t is 



v - -LlLLLv 

 JT F, " 



(12) 



where Y is the total yield in period x. During pa- 

 rameter estimation, a residual is obtained for each 

 fishery having nonzero effort in period t. The contri- 

 bution to the objective function for each period is thus 

 composed of a sum of terms, one for each fishery with 

 nonzero effort. In addition, the individual fisheries 

 may carry different statistical weights to reflect vary- 

 ing levels of confidence in the data from each fish- 

 ery. Inverse-variance weighting can be approximated 

 by iteratively examining the mean-squared error 

 (MSE) from each series, weighting, and re-estimat- 

 ing the parameters. 



Model tuning 



If an external series of population biomass estimates 

 is available, it can be incorporated into the analysis 

 in a procedure analogous to tuning an age-structured 

 analysis. The external estimates are compared to the 

 population estimates derived within the production 

 model and the residuals incorporated in computa- 

 tion of the objective function. Rivard and Bledsoe 

 (1978) suggested this possibility, but did not pursue 

 the idea, and it has also been described by Hilborn 

 and Walters ( 1992). The external biomass series need 

 not be continuous, but may contain missing values; 

 the series' contribution to the objective function is 

 set to zero for years with missing values. An exter- 

 nal index of biomass can be used similarly, with the 

 cost of estimating one more parameter (the 

 catchability associated with the index). 



The model formulation involved in tuning the 

 model is similar to that used when fitting more than 

 one fishery. As in that situation, each year's contri- 

 bution to the objective function consists of a sum of 

 terms. Here, the sum includes a term from each fish- 

 ery and a term for each biomass— estimate or index 

 series. For a maximum-likelihood solution, the com- 

 ponents should carry statistical weights inversely 

 proportional to their variances. 



Varying catchability 



In many situations, catchability is thought to change 

 relatively suddenly, perhaps because more efficient 



