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The assessment method being explored in this case study is a length-structured approach of the size-transition matrix type with the population model structure based on the catch-at-size analysis (CASA) of Sullivan et al. (1990) . In brief, the population is described by a vector of numbers-at-length which is projected forwards in time using a size transition matrix obtained from a stochastic growth model with externally estimated parameters. All population dynamics processes (e.g. recruitment, fishing mortality) are assumed to be dependent on length rather than age. Parameter estimates and historical trends in stock abundance (annual recruitment, mean and standard deviation of recruitment distribution, selectivity parameters and temporal component of fishing mortality) are then obtained by fitting the model to the observed data.

Male and female Nephrops are known to grow at different rates and in addition have somewhat different behaviour with females spending longer in their burrows when incubating eggs. These differences in seasonal burrow emergence patterns result in different levels of fishing mortality for males and females. The population model is therefore structured by sex, maturity and also season and carries out the assessment of males and females simultaneously. Males and females are linked by assuming identical numbers recruit to the male and female populations and also that the fishing mortalities fluctuate temporally in a similar way. The reduced fishing mortality on the female population due to reduced burrow emergence is modelled by introducing a seasonally dependent ‘catchability’ parameter (fixed from year to year) which calculates mature female fishing mortality as a proportion of the male fishing mortality. A fuller description of the model can be found in the case study report.

Length-based assessment models of the type being investigated here assume that growth is represented by as stochastic model, the stochasticity coming from the variation in individual growth rate which for Nephrops is known to be quite high (Tuck et al., 1997). If enough data are available (e.g. tag-recapture data) then a size transition matrix can be derived empirically. However this is generally not the case and the transition matrix has to be derived from an assumed probability distribution of growth increments for individuals in each size class which is parameterized by a mean (calculated using von Bertalanffy) and standard deviation.

Following the onset of sexual maturity, it is known that the growth of female Nephrops is considerably slower than males. To account for this, the ICES Working Group on Nephrops stocks has introduced the concept of a ‘combined’ growth curve, whereby immature females follow the same growth curve as males until they reach the size at 50% maturity when they switch to a much slower growth curve. Hence there are two sets of von Bertalanffy parameter values for female growth.

Natural mortality is dependent on sex and maturity, with mature females having a lower natural mortality rate than the rest of the population due to their reduced burrow emergence while incubating eggs. Fishing mortality is assumed to be separable into a temporal component and length dependent ‘selectivity’ curve which takes the form of a logistic function parameterized by a slope and length at 50% maximum selectivity (estimated within the assessment model). For the immature component of the population, fishing mortality is assumed to be independent of sex. However, the reduced burrow emergence of mature females implies that they are protected somewhat from the fishery at particular times of the year and therefore have lower fishing mortality than the males at this size. This is modelled by incorporating a ‘catchability’ multiplier into the expression for female fishing mortality which is estimated within the model.

Recruitment is assumed to occur across a number of length classes. It is assumed to be separable into a size distribution (fixed over time) and an estimate of total annual recruitment. The size distribution of recruits is assumed to be Gaussian with a mean of 10 mm which is the approximate size of individuals when they settle to the seabed as juvenile Nephrops. The total annual mortalities are estimated within the assessment model. Recruitment is assumed to be split equally between males and females and occurs in the first season of the year.

The population is assumed to be initially in equilibrium at the start of the catch-at-length data time series, with estimated temporal fishing mortality multiplier and total recruitment. The equilibrium selection pattern is assumed to be the same as that for the years for which catch-at-length data are available. If historical estimates of total catch (in weight) are available, an ‘equilibrium’ value can be included into the model to help estimate the initial equilibrium conditions, although this feature is not used in the current implementation.

The length-based assessment method described here can make use of a wide range of input data. The basic requirement is for catch-at-length data in the form of a frequency distribution and total numbers caught. Additionally the method can make use of a range of other fishery dependent and independent indices. In this case we focus on making use of UWTV survey biomass abundance estimates as auxiliary information. Although UWTV surveys have only been conducted for around 12 years, the generated TV survey series are constructed for the whole of the time period of the assessment i.e. from the start of the catch-at-length data which runs from 1981 to 2002. The OM was conditioned using annual data and therefore the assessment model is run using an annual time-step.

A variety of distributional assumptions can be made about the input data (normal, lognormal, multinomial) and if available CVs can be input to provide appropriate weighting for the data. In addition, user defined weights can be implemented to provide greater weight to one particular data source over others.

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