Stewart; Defining migration rates of Parophrys vetulus 



471 



Stock assessments of west coast flatfish (and other 

 groundfish species) have been based on one of two as- 

 sumptions about latitudinal movement: no movement at 

 all (multiple isolated stocks) or complete mixing (single 

 stock models). In a Bayesian context, these two oppo- 

 site assumptions fix the magnitude of movement rates 

 before analysis and therefore can be considered highly 

 informative priors. The goal of this study was to provide 

 a generalized method with which to develop informative 

 priors on movement rates based on quantitative analy- 

 sis of historical tagging data, thereby adding a third 

 choice of prior for use in stock assessments. 



Materials and methods 



Model development 



A model very similar to those used in other mark-recap- 

 ture studies (e.g., Hilborn, 1990; Hampton and Fournier, 

 2001) was developed to predict the number of tags 

 returned from multiple tag-release events (tags released 

 in one spatial area over a short period of time; hereafter 

 referred to as a data set) by projecting each population 

 forward in time. Predicted returns are tracked by month 

 and for each spatial area. The predicted number of tags 

 present (N ) in each data set {d), month {t), and area (i) 

 are given by 



N,,,,,=Y.{^,,^,.Aj^:)i 



12 



(1) 



where F^ = the average fishing mortality rate for each 

 data set; 

 M - the average natural mortality rate; and 

 Q = the instantaneous rate of tag loss (ongo- 

 ing "attrition" due to fouling or mechanical 

 failure; often referred to as "type-2 tag loss" 

 in traditional terms; Beverton and Holt, 

 1957). 



The subscript j denotes all possible source areas, and 

 the proportion of individuals moving from each area to 

 another in any month is given by the matrix P,. The P, 

 matrix (area x area for month t) includes nonzero values 

 in only the first off-diagonals, and a variable number 

 of parameters within each diagonal depending on the 

 movement hypothesis to be explored. Instantaneous 

 rates are divided by 12 because they are applied on a 

 monthly basis. Predicted numbers in the first month are 

 the reported tag releases (alternately, type-1 tag loss, 

 those tags that are shed immediately after tagging, 

 could be included by multiplying the initial releases by 

 1 - type-1 tag-loss rate). The predicted recoveries (i?) 

 by data set, month, and area are then 



Rd.u=<PN^^,^, 



yF^+M + a 



(F^+M+il) ' 



1-e 12 



(2) 



where (p = the reporting rate of captured tags during 

 the time period over which tag recoveries 

 occurred. Given these dynamics, the tagged 

 population and predicted recoveries for all 

 data sets available may be projected forward 

 simultaneously. 



The major departure from previous models is the use 

 of a single average fishing mortality rate for each data 

 set (an approach that reflects a lack of direct effort or 

 of fishing mortality information). If only a single data 

 set is analyzed in this manner, it is clear that any het- 

 erogeneity in fishing mortality over time or space could 

 result in substantially biased estimates of movement 

 rates. However, if multiple tagging events are analyzed 

 simultaneously, and there is no consistent relationship 

 between location of tag releases and areas of increased 

 fishing mortality, this potential source of bias may be 

 reduced. Were information on the spatial and temporal 

 variability in fishing mortality available, it would be 

 simple (and recommended) to extend the notation fur- 

 ther to either input mortality rates directly into the 

 analysis or to estimate them from relative effort. 



Variability in observed recoveries is caused by many 

 factors, including schooling behavior, heterogeneous dis- 

 tribution of fishing effort, tag loss or tag reporting over 

 time and space, and by the stochastic nature of very low 

 recovery rates. Because of the many potential sources 

 of extra-model error, a likelihood function that allows 

 for substantial variation among observations is desired. 

 The negative binomial likelihood is the logical choice for 

 this type of tagging data (e.g., Cormack and Skalski, 

 1992; Hampton and Fournier, 2001). If each tag-release 

 group is assumed to be independent, the full likelihood 

 (L) of the observed recoveries iR) is given by 



L{Ra,..\K<..k) = 



(3) 



a+R. 





•D! 



k-l)\ 



-^d.t.l ( 



l + - 



R. 



d.t.i 



R 



d.t.i 



4-1 



where R = the predicted recoveries in a data set, month, 

 and area; and 

 k = the overdispersion (variance) parameter. 



The negative binomial asymptotically approaches the 

 Poisson distribution as the value of the overdispersion 

 parameter moves to infinity (Bishop et al., 1988). 



A common problem with historical data is that only 

 summarized reports are available for analysis. Where 

 tag recoveries have been aggregated across time or 

 space, the model predictions and the original observa- 

 tions are no longer on an equivalent scale. This prob- 

 lem is easily dealt with by aggregating the predicted 

 recoveries to match the observed recoveries, while still 

 maintaining the same predictive model structure. How- 

 ever, this method creates different types of comparisons 

 within the likelihood (monthly recoveries compared to 



