ISSN 1 1 75-1 584

MINISTRY OF FISHERIES

Te Tautiaki i nga tini a Tangaroa

Stock assessment of paua (Haliotis iris) in PAU 5B and PAU 5D using a new length-based model

Paul A Breen

Neil L. Andrew Terese H. Kendrick

New Zealand Fisheries Assessment Report 2000133 September 2000

Stock assessment of paua (Haliotis iris) in PAU 5B and PAUSD using a new length-based model

Paul A. Breen Neil L. Andrew

Terese H. Kendrick

NIWA PO Box 14 901

Wellington

New Zealand Fisheries Assessment Report 2000133 September 2000

Published by Ministry of Fisheries Wellington

2000

ISSN 1175-1584

0 Ministry of Fisheries

2000

Citation: Breen, P.A., Andrew, N.L., & Kendrick, T.H. 2000: Stock assessment of paua (Haliotis iris) in PAU 5B and PAU 5D

using a new length-based model. New Zealand Fisheries Assessment Report 2000/33.37 p.

This series continues the informal New Zealand Fisheries Assessment Research Document series

which ceased at the end of 1999.

EXECUTIVE SUMMARY

Breen, P.A., Andrew, N.L., & Kendrick, T.H. 2000: Stock assessment of paua (Haliotis iris) in PAU 5B and PAU 5D using a new length-based model.

New Zealand Fisheries Assessment Report 2000/33. 37 p.

A length-based model was used to assess the status of the PAU 5B and PAU 5D stocks. The assessment used Bayesian techniques to estimate model parameters, the state of the stocks, future states of the stocks, and the variability of estimates. Estimates of the modes of the joint posterior distribution (PDM) were used to explore sensitivity of the results to model assumptions and the data used as input; the assessment itself was based on posterior distributions generated fiom Monte Carlo - Markov chain simulation.

The model was applied to four data sets fiom PAU 5B and 5D: standardised CPUE, an independent survey index of relative abundance, and length fiequencies from catch sampling and population surveys.

Model results for PAU 5B suggested a depleted stock, well below BMSY, likely to decrease further at the current level of catch. Although the results were sensitive to some assumptions, particularly those involving growth and the stock-recruit relation, the qualitative conclusions were robust. The four data sets appeared to contain the same information about the stock. Of the parameters estimated, the steepness of the stock-recruit relation was nearly always on the lower bound, and the data did not appear to contain any information about a parameter that described the "hyperstability" of CPUE as an index of relative abundance.

Model results for PAU 5D were sensitive to the data sets used in fitting, and suggested very different states of the stock depending on what data were included. This behaviour, and other indicators that the model had problems fitting the data, led to inconclusive results for PAU 5D.

1. INTRODUCTION

1.1 Overview

Paua (Haliotis iris) supports a valuable fishery in New Zealand, with annual landings of about 1200 t. Legislation requires that New Zealand fisheries be managed so that stocks are maintained at or above Bm, the biomass associated with the maximum sustainable yield (A4SY). The Ministry of Fisheries (Wish) annually advises the Minister of Fisheries whether stocks are at or above Bm and whether current TACCs are sustainable and likely to move stocks toward Bm. The work described here was done by NIWA under contract PAU801 to Wish to provide such an assessment for PAU 5B and 5D. In this report we summarise the results fiom these two stocks for the 1998-99 fishing year.

Except for a biomass estimate based on an experimental fishery in PAU6, on the west coast of the South Island, there has been no recent paua stock assessment. Schiel & Breen (1991) estimated total mortality rates fiom length fiequencies in PAU 7 and compared them with reference points fiom YPR analysis. Except for the early work of Schiel(1989) and a biomass estimate based on an experimental fishery in PAU 6, there has been no other paua stock assessment.

1.2 Description of the fishery

The fishery has been summarked by Annala et al. (1999) and in numerous FARDs (Schiel 1989; and see McShane et al. 1994, 1996 for recent summaries).

In 1995, PAU 5 was divided into three sub-areas, 5A, 5B and 5D (Figure la), each with a TACC of 147.66 t. From 1 November 1997 these areas were further subdivided into 17, 16 and 11 statistical areas respectively (Figure lb). Subdivision of the commercial catch and effort data fiom PAU 5 into these new areas is described by Kendrick & Andrew (2000).

I .3 Literature review

The scientific literature on the population biology and. fisheries for paua has been reviewed by Schiel(1991) and McShane et al. (1994, 1996). Biology of paua was also reviewed by Sainsbury (1982) and Schiel & Breen (1991). Fishery data are thoroughly discussed, for PAU 5, by Kendrick & Andrew (2000); the fishery-independent survey data were described most recently by Andrew et al. (2000a).

Stock assessment methods for abalone fisheries worldwide were reviewed by Breen (1992).

2. DATA

2.1 TACCs, catch, landings and effort data

For information on the paua fishery throughout New Zealand, see Annala et al. (1999).

Commercial catches and standardised CPUE indices for the assessment described here are shown in Table 1 for PAU 5B and Table 2 for PAU 5D. The 1998-99 TACCs were 149 and 148 t for the two areas respectively.

2.2 Other information

The assessment used here also relies on fishery-independent survey estimates of relative abundance (Andrew et al. 2000a). These indices are shown in Tables 3 and 4 for the two areas. The model described below uses estimates of growth, maturity and the length-weight relation. These estimates and their sources are described in the appropriate sections below.

2.3 Recreational and Maori customary fisheries

Although recreational catch estimates are available for paua in PAUS (Teirney et al. 1997, Bradford 1998) and PAU 5D (Annala et al. 1999), no estimate was available for PAU 5B, and no estimated historical trend was available. No estimates of Maori customary catch are available.

2.4 Other sources of fishing mortality

In the past, estimates of illegal harvest for some areas have been provided by Wish. The current levels of illegal harvest are unknown.

Sub-iegal paua may also be subject to handling mortality by the fishery if they are removed fi-om the substrate to be measured. Mortality may originate from wounds caused by removal, desiccation or osmotic stress and temperature stress at the surface, unsuitable habitat when replaced, and predators. Taylor et al. (1994) reported that 14% of paua removed from the reef by commercial divers are undersized and are returned to the reef, but provided no details of the method used to make this estimate or where it was made. Pirker (1992) reported a wide range (10-54%) of undersized animals being captured by a diver in PAU 4. Of these paua, up to 13% were damaged in some way and field estimates suggest up to 80% of these may fall victim to predation by wrasses or starfishes following their return to the reef.

3. MODELS

3.1 Ovewiew

Abalone are difficult to assess because:

they are essentially sedentary after settlement, so the effects of fishing are highly localised;

differences in population dynamics, particularly recruitment, growth and mortality, may also be highly localised;

the behaviour of fishers leads to stability in catch rate as overall abundance decreases, so catch rate is difficult to interpret as an index of abundance;

abalones, including paua, cannot be reliably aged.

In this report we describe a length-based assessment model to assess the PAU 5B and 5D paua stocks. The model is similar to length-based models developed for abalone by Worthington (1997) and for lobsters by Punt & Kennedy (1997) and Starr et al. (1999).

Although catch rate (CPUE) has problems as an index of abundance, we used standardised catch rates which show a declining trend. We consider that declines in an index known to be hyperstable reflect real declines in abundance, which may be steeper than the index. In addition to CPUE, we fit the model to an independent diver survey index. The index is based on stratified random surveys and, while subject to some possible bias (Andrew et al. 2000b), should be fiee of major hyperstability.

The model is also fitted to length frequency data, fiom both catch sampling and population sampling during the diver surveys. There are thus four independent sets of data used in fitting.

The model is fitted to data using Bayesian techniques. Point estimates are compared in sensitivity tests, but the assessment is based on the posterior distributions of parameters and derived parameters of interest.

The model is a stochastic, dynamic, length-based observation-error time series model. It is stochastic because annual variations in recruitment can be estimated as a vector of free parameters. It is dynamic because no equilibrium, other than in the inital length structure, is assumed. Paua are represented in the model as numbers-at-length rather than numbers-at-age. The error is assumed to be observation error rather than process error (see Hilborn & Walters 1992).

The model population is initialised and then driven by observed catches. The model is fitted with robust likelihood to vectors of standardised CPUE, relative abundance indices fiom diver surveys, and observed length frequencies from research surveys and catch sampling. Outputs are the present and projected future states of the stock, estimated using Bayesian methods. The assessment is based on the marginal posterior distributions of the parameters and derived parameters of interest, in turn based on Markov chain - Monte Carlo (mcmc) simulations. Males and females are not modelled separately.

3.2 Model parameters

Model parameters are Rcofi M, h, ql, q2,q3, and a vector of Rdevs.

Rcoflis the natural logarithm ofRO, the average number of recruits to the virgin population. M is the instantaneous rate of natural mortality. The parameter h is the 'steepness' of a Beverton-Holt stock-recruit relation. The relation between recruited biomass and predicted commercial catch rate is determined by ql and 93; between recruited biomass and the diver survey index by q2. Each Rdev modifies the actual model recruitment in a given year.

3.3 Initial conditions

The model contains 50 length 'bins', each encompassing a 2 rnm shell length. The smallest is 70.0 to 71.9 mm; the largest is 168.0 to 169.9 mm. All recruitment enters the first bin, the last bin acts as a 'plus group' bin.

The model is 'burnt in' for 60 years by running it with no fishing to allow numbers-at-length to approach an equilibrium. In each of these years,

where t indexes model years. A growth transition matrix is determined outside the model. From the von Bertalanffy growth parameters L, and K, an expected average annual growth increment is calculated for each length class:

(2) A lk = (L, - lk) (1 - exp(-K))

where k indexes length class. Using these expected increments and an assumed constant standard deviation of increments around the mean, the distribution of increments for length class k is calculated from the normal distribution. The distribution of increments is then translated into the vector of probabilities of transition from length class j to other length classes, for all length classes, to form the matrix G. Negative increments are not permitted, so the transition probabilities for abalone moving fiom a larger size to a smaller one are zero, but zero growth is permitted. The largest size group is effectively a "plus group", so that abalone in this group have a probability of one of remaining in this size group.

In the initialisation calculations, the vectorN, of numbers-at-length for year t is determined fiom & (equation I), numbers in the previous year, survival, and the growth matrix G:

where the prime (') denotes vector transposition and the dot (e) denotes matrix multiplication.

After the burn-in period, the model population is nearly in equilibrium - only the last few length classes are slowly increasing.

At the end of the burn-in, the model calculates the two parameters of the Beverton-Holt stock- recruit curve from the current values of spawning biomass, equilibrium recruitment and steepness, h. Spawning biomass is calculated as

where N , , is the number of abalone in length class k in year t, wk is the mean weight and PMk is the proportion of abalone that are mature in length class k. The two parameters are:

where h is the "steepness", and describes the percentage of virgin egg production that will occur when spawning biomass is 20% of the virgin level.

3.4 Model dynamics

Each year the model calculates the biomass available to the fishery:

where k indexes length classes, Nk, is the number at length k in year t, and Pkr is a switch that determines whether length class k is above minimum legal size in year t.

Exploitation rate, U,, is then calculated fiom model biomass and observed catch, C,:

Exploitation rate is constrained by adding a penalty to the likelihood function when 0.80 is approached. This prevents the model from generating unrealistically high exploitation rates during the minimisations; in the final results the exploitation rates come nowhere near this limit. Survival fiom fishing, SFk,, is calculated as:

The vector of numbers-at-length in the following year is then calculated from the current vector of numbers-at-length, the vector of survival from fishing, the growth transition matrix and natural mortality:

(1 0) N, = SF, Nf-l 'G exp(-M)

Recruitment in year t+l is calculated from spawning biomass and the parameter Rdev, :

where Rdev, is the recruitment residual for year t and o~ is the calculated standard deviation of the recruitment residuals. All recruitment enters the first length class in the model.

3.5 Model predictions

The model predicts CPUE, Pmdl, and relative survey abundance, IF^^,, from the model biomass B"golt:

It predicts numbers-at-length from the numbers of abalone in each length class:

Predicted numbers-at-length are normalised to proportions-at-length, fmdkI, in two sets: one beginning at 126 mm shell length, for comparison with lengths observed in catch sampling, and one beginning at 96 mm shell length, for comparison with lengths observed in the population surveys. Predicted numbers-at-length are zero for lengths below 126 and 96 rnm respectively. For each set:

3.6 Model fitting

Predictions are fitted to observed CPUE and population survey indices, Pbst and IS"^^^ respectively, with maximum likelihood and Bayesian techniques. These indices are fitted by minimising the negative log-likelihoods:

where 8 is the parameter vector, q and q s are the standard deviations of the observation error for CPUE and population surveys. These are summed for all years with observations. Values for q and an were assumed - these act as relative weights (weight is inversely proportional to the value). The populations surveys were thought to have more accuracy than CPUE, so q was set to 0.50 and q s to 0.25.

The "robust normal likelihood" formulation proposed b Fournier et al. (1990) is used to fit K model predictions to observed length compositions, pO h, . The variance is assumed to be multinomial and is weighted by the effective sample size, 5:

summed across all k for each year t and where i2 is the number of size bins. The 0.01 term reduces the influence of outliers. The O.l/Q term prevents the variance from tending to zero as the predicted value tends to zero, reducing the influence of observed outliers with small predicted probability (Fournier et al. 1990).

To obtain 5, observed proportions-at-length were weighted by the square root of the number measured for catch samples, and by twice the square root of numbers measured for population samples. This reflected greater confidence in the sampling procedures for population sampling than for catch sampling.

3.7 Priors and bounds

Bayesian priors were established for all parameters. Some were uninformative, incorporated simply as uniform distributions with upper and lower bounds. Where the prior is considered to be informative and lognormal, as for M, the contribution to the total negative log-likelihood function for each such parameter is:

where 4 and a, are the mean and standard deviation of the prior distribution for parameterp and x,, is the current value of parameter p.

The contribution to the negative log-likelihood function from the lognormal prior forRdevs, with mean 0, is:

where am is the calculated standard deviation of Rdevs.

Table 5 shows the prior distribution and bounds used for each parameter.

The prior for M was based on a consideration of published natural mortality rate discussions for temperate species of abalone (Shepherd & Breen 1992). The c.v. was set to give a distribution between 0.05 and 0.20, believed to be the reasonable range. For the non-informative prior distributions, the bounds were set arbitrarily wide except as follows.

For steepness, we considered low values to be as likely as high values because of the biology of abalone. Abalones may have short dispersal distances (e.g. Prince et al. 1987); their populations may demonstrate Allee effects at low density (e.g., Clavier 1992); and abalones are considered susceptible to recruitment collapse.

The literature supports the concept of hyperstability in CPUE from abalone fisheries (e.g., Breen 1992) because of the way abalone are distributed, their sedentary habit and the way fishers operate. This parameter was therefore bounded at just over 1.0 and at an arbitrarily low value.

3.8 Biological assumptions

The length-weight relation was taken from Schiel & Breen (1991) and was:

where wt is weight (kg) at length I (mm). Growth was based on two sets of tag return data from Waituna Bay on the west coast of Stewart Island (Andrew et al., unpub. data). In both sets, individuals were measured and tagged with small plastic numbers glued to the shell and replaced on the bottom. Tagged abalones were recovered almost a year later and their shells measured again. Growth increments were scaled by the factor 365ldays at liberty, and a von Bertalanffy growth curve fitted with non-linear least squares. This gave the parameters L, = 150 mm and K = 0.252. The residuals of this fit suggested that the standard deviation of the increment-at-length was 4 mm. These values were used to create the growth transition matrix G.

The lag between spawning and recruitment to the model at 70 mm was assumed to be 3 years. Some data are available on juvenile growth from studies near Wellington and some modes are available from juvenile surveys at Stewart Island (Schiel, unpub. data).

It was assumed that maturity was knife-edged at 92 mm shell length. McShane & Naylor (1995) suggested that 50% maturity is attained at 75-95 mm shell length. Recruitment to the fishery was assumed to be knife-edged at 126 mm shell length. Length measured by catch sampling and population surveys is slightly different from length as measured by the fishery - in the latter, organisms attached to the shell can be included in the measurement. The model excluded observations below 126 mm in catch samples.

It was assumed that selection by population surveys was knife-edged at 96 mrn shell length. Trials with the early model showed a sensitivity to this assumption. The relevant factors are first, the length at which paua become fully emergent from cryptic habitat, considered to be smaller than 96 mm (J.R. Naylor & N.L. Andrew, unpub. data), and second, the size at which recruitment to the first length class in the model has become realistically distributed among model length classes. The assumed value of 96 mm was a length at which results were reasonably stable to the choice of size. The model ignored observations below this size.

3.9 Forward projections

The model makes forward projections by using parameter estimates obtained from fitting or mcmc simulation, and using the dynamics equations in conjunction with specified catch and MLS for the period of projection. Projections were made to the beginning of 2004.

In the sensitivity tests, based on the PDM estimates of parameters, the projections were deterministic with and recruitment was calculated from

In the mcmc simulations, recruitment was stochastic, calculated from:

The value of opmj,, was assumed to be 0.6 or the same as (TR, whichever was greater. RND is a random normal deviate with mean zero and standard deviation of unity.

3.10 Model indicators

In addition to model parameters, derived parameters such as population size and exploitation rate were calculated and their posterior distributions summarised. These parameters were as follows.

Virgin biomass, Bo, was estimated as the recruited biomass at the end of the burn-in period:

Bm was estimated with forward simulations, with no recruitment variability, to determine Fo.] taking estimated steepness into account. The model did this for each maximum likelihood estimate and each mcmc simulation. B m was then the equilibrium biomass associated with the F0.1 estimate.

The indicators for current and projected biomass, B1999 and B2004, were the values for B'"ga1~999 and B'"gd2W4 Spawning biomass indicators for the virgin, current and projected populations were called So , S1999 and S2004; and were taken from B'P""1974, P-1999 and B'pown2004.

Ratios of the population indices were used to estimate how depleted the current and projected population was, where it stood relative to Bmy, and whether it would increase or decrease by 2004.

Exploitation rates in 1999 and 2004 were taken from U1999 and U2OO4.

For the sets of mcmc simulations, three additional indicators were the percentages of runs in which projected biomass was less than 20% Bo, in which projected biomass was greater than 1999 biomass, and in which projected spawning biomass was greater than the 1999 spawning biomass.

3.1 1 Data fitted by the model

The model was driven by a vector of observed catches C fiom 1974 through 1998 (see Tables 1 and 2). For PAU SB, the model was fitted to four data sets: standardised CPUE from 1984 through 1998 (see Table I), independent survey estimates for 1994, 1995, 1996 and 1998 (see Table 3), length frequencies from catch samples in 1992, 1993, 1994, 1998 and 1999, and length frequencies from population samples for 1989,1994,1995 and 1998. For PAU 5D, the model was fitted to standardised CPUE from 1984 through 1998 (see Table 2), independent survey estimates for 1993, 1995 and 1998 (see Table 4), length frequencies from catch samples in 1992, 1993 and 1994, and length frequencies fiom population samples for 1992, 1994,1997 and 1999.

Although recreational catch estimates are available for paua in PAUS (Teimey et al. 1997, Bradford 1998) and PAU 5D (Annala et al. 1999), no estimate was available for PAU SB, and no estimated historical trend was available. No estimates of Maori customary catch are available and the scale of illegal catches is unknown. Given these uncertainties, in this first attempt to assess paua stocks with the length based model no non-commercial catch estimates were incorporated.

3.1 2 Assessment procedure

The PDM estimates of the parameters served as the starting point for Monte Carlo-Markov chain (mcmc) simulations. Two million simulations were made, using the mcmc capability of ADModelBuilderTM. Of these, 5000 were sampled and results from them were saved. Posterior distributions of parameters and indicators formed the basis of the assessment. These were summarised by the mean, median, 5 and 95 percentiles of the distributions.

The PDM estimates were also used as the basis of comparisons in sensitivity tests. Sensitivity tests were used to explore sources of uncertainty not incorporated into the assessment procedure described above, such as parameter estimates constrained by a bound, the growth estimates, and uncertainty about the observed catch vector. Sensitivity to different data sets used in fitting was examined by removing the four data sets from the estimation procedure one at a time and then two at a time as described below.

4. MODEL RESULTS

4.1 Results from PAU 5B

4.1.1 PAU 5B PDM fit

The PDM fit of the model to CPUE and the survey index are shown in Figure 2, the fits to length frequency data are shown in Figure 3, and the estimated parameter and indicator values are shown in Table 6.

In this fit, the model reproduces some of the observed downward trend of the survey index, and reproduces the short upturn and general downward trend of the CPUE index. It does not fit

particularly well to either index, but clearly this is not possible. The observed CPUE trend is flat fiom 1994, while the survey index is steeply downward over the same period. The model fit is a compromise between these.

Similarly, the model captures the main features of the length frequency data (Figure 3), but does not fit any data set particularly well. Again, the model fit is a compromise. Both catch sampling and population s w e y data are available for 1994. In the observed catch sampling length frequency, the fiequencies increase in the range 126-136 mm; in the population length frequency they decrease; the model cannot reproduce both.

In this fit, the estimated steepness, h, is on the lower bound of 0.40 (see Table 6); q3 and at least one Rdev are on the upper bounds. Because of this, sensitivity to these bounds was examined.

The fit suggested that the population was 25% of the original population and 55% ofBmY; it suggested that both recruited and spawning biomass would be lower, given average recruitment, than the 1999 values if current catches continue to 2004 (see Table 6).

4.1.2 Sensitivity

Sensitivity of the results to the data sets is shown in Table 7. Results from the fit described above are termed the "base case". In the next four columns (tests s l to s4), results are shown fiom fits with one of the data sets removed, and the model fitted to the remaining three. Then results are shown (tests s5 and s6) from the model fitted to length frequencies only (both abundance indices removed) and to abundance indices only (both length fiequencies removed).

When single data sets were removed, some parameter estimates changed significantly - for instance, q3 was much reduced in test sl, A4 was much reduced in test s3. In the indicators, however, only test s4 showed results that were substantially different fiom the base case: the population appeared to be more depleted and showed a much higher current exploitation rate than in the base case. In this case only (of all the sensitivities examined), the projected population increased fiom the 1999 values.

Results also changed substantially in tests s5 and s6. Results fiom fitting the length frequency data only suggested a healthier stock (less depleted, a larger proportion ofBmY and with a lower current exploitation rate) than the fit made with abundance indices only.

In each of these sensitivity trials, B1999 was less than Bm. In all but one, the projected recruited and spawning biomass was smaller than B1999 or Sd004 respectively.

Figure 4 shows the fit to length frequency data from test s3, where the catch sampling length frequency data were not fitted. The fit to population length frequency data is greatly improved. Conversely, Figure 5 shows the fit from test s4, where the population length frequency data were not fitted. The fit to catch sampling data is much better than in the base case. The predicted population length frequency shows the strong waves of recruitment used by the model to attempt to fit the other three data sets.

Sensitivity to the growth parameters is shown in Table 8. Two trials were conducted (s7 and s8) as follows. First, L, was arbitrarily assumed to be 140 mm in test s7 and 160 mrn in test s8; in the base case the value is 150. Second, the Brody coefficient, K, was fitted to the growth data using

the fixed value of L, . Finally, for each of these tests the growth transition matrix G was recalculated fiom the new values of assumed L, and K.

Sensitivity to these different values for growth was large. With the reduced value ofL, , the results suggested a stock just above Bm and with a low current exploitation rate. With the increased L, , results suggested a badly depleted stock with a high exploitation rate. In both tests, however, the projected biomass decreased fiom 1999 levels at the current level of removals.

Sensitivity to steepness was tested in two ways, tests s9 and s10, also shown in Table 8. In test s9, the lower bound of steepness was set at 0.21 rather than 0.40 as in the base case. At this value, there is little surplus recruitment at low spawning biomass levels. The base case estimate ofh was on this lower bound. In test s10, the value of steepness was fuced at the higher value of 0.70. This is the value suggested by Francis (1993) as the value that should be used in risk simulations when steepness is unknown.

In test s9 the estimate of h went to the new lower bound. 73e results suggested a more depleted stock with a higher exploitation rate than in the base case; the decrease in biomass during the projection period was dramatic. Conversely, the higher estimated h suggested a less depleted stock than in the base case, a lower current exploitation rate and slower decreases over the projection period.

As in Table 7, the estimated biomass was less than Bnap (except in test s7) and the projected population was less than the 1999 population.

At least one Rdev parameter was usually on the upper bound of 2.3. Two tests (sl la and s l l ) were made. In slla, the bounds on Rdevs were reduced to -0.2 and 0.2, corresponding with recruitment limited to 82-122% of the expected recruitment. In s l l , the bounds were increased to -5 and 5, corresponding with very large variation in recruitment fiom less than 1% to 148 times the expected recruitment.

Neither test had any substantial effect on the results (Table 9). In test sll, the maximum Rdev value was actually less than values observed in the base case.

Three tests, s15 to s17, were made to test the sensitivity to the assumed catch trajectory. Catch data before 1995 were available only for the larger area PAU 5. Because the new areas straddle the old statistical reporting areas (see Figure la), assumptions were necessary about the proportion of catch reported in areas 25 and 30 that came fiom Stewart Island between 1983-84 and 1994-95. The sensitivity of the results was tested by making two alternative assumptions. For test s15 it was assumed that PAU 5B accounted for a constant 48% of the PAU 5 landings between 1983-84 and 1994-95. For test s16 it was assumed that PAU 5B catches comprised all of the catch from areas 27 and 29 plus 75% of the catches fiom areas 25 and 30. The resulting catch vectors are compared with the base case in Figure 6. In test s17, the catch for 1986 was interpolated between the 1985 and 1987 catches. This explored a suggestion fiom the Shellfish Working Group that 1986 reported catch, associated with the first year of the Quota Management System, showed lower than normal catches for many species, reflecting poor data capture by the new system.

Results (see Table 10) were qualitatively similar in all tests to those from the base case. The greatest change was in the $16 results, where Bl999/BrClSy increased from 55% to 68%. In all cases, the 1999 biomass was below Bmy and projected biomass was less than 1999 levels.

4.1.3 Posterior distributions

Figure 7 shows the sequential trends of the 5000 parameter values forM, h, 93 and RcofSfrom the 5000 sampled mcmc simulations. The parameter 93 was evenly distributed between the bounds on its prior, while M and RcoH occupied a narrow range of the space between their bounds. Although h was restricted to the lower end of the range between the bounds in its prior, this parameter displayed a different behaviour from the others during the mcmc simulations: its value wandered with a long-period tendency through the simulations, whereas the others varied much more quickly.

Posterior distributions for the major parameters and indicators are shown in Figure 8. The posterior for M is centred well away from the mean of the prior distribution, which was 0.10. Sensitivity to this prior will be described below. The posterior for 93 suggests that the data contained little information about the value of this parameter. The posterior forh was strongly concentrated near the lower bound of the prior.

Posterior distributions are summarked in Table 11. Most PDM estimates (see Table 6) were reasonably close to either the mean or median of the posterior distribution. The two parameters relating CPUE and biomass, q l and 93, were exceptions to this.

The ratio of BmY to BO (see Figure 8, Table 11) indicates how large the optimum population size might be as a percentage of virgin biomass. The distribution of estimates for PAU 5B is tightly limited between 42% and 46%.

Current biomass, BI999, was usually less than Bm. The 90% confidence interval for B1999/BMSY was 25% to loo%, reflecting considerable uncertainty in the result. However, because the chance of this ratio being 100% or greater is only 5%, the qualitative conclusion that current biomass is less than Bmy is supported. Similarly, the posteriors support the conclusion that biomass is likely to decrease in the next 5 years at average recruitment and current catches.

4.1.4 Sensitivity to the priors on M and 93

Two tests were made, at the request of the Shellfish Working Group, of the sensitivity of results to the prior distributions for M and 93.

The prior on M was changed to a uniform prior with bounds 0.05 to 0.20. The model was fitted and a set of mcmc simulations made as described above. The posterior distributions of M from the base case and sensitivity test are shown in Figure 9. There was very little effect on the posterior distribution of M and on other posteriors.

The prior on 93 was changed to a uniform prior with bounds from 0.99 to 1.025. The model was fitted and a set of mcmc simulations made as described above. The effect was measured by examining the posterior distribution of the ratio B1999/Bo (Figure 10). Again, there was very little effect of this change.

4.2 Results from PAU 5D

4.2.1 PAU 5D PDM fit

PAU 5B data gave much less trouble in fitting than the PAU 5D data. In the latter, the abilitjr of the model to converge was dependent on initial values, the suite of parameters being estimated and the order in which parameters were initially estimated (the estimation phasing). None of these problems was observed in the initial explorations with PAU 5B data.

For PAU 5D the assessment was conducted with a fixed value for h of 0.60. This was done to obtain reasonable convergence.

The PDM fit is shown in Figures 11 and 12; parameter estimates and indicators are shown in Table 12. The model results do not fit the two abundance indices well. Fits to the observed length frequency data sets are similar to those seen in PAU 5B (see Figure 3).

The assessment results suggest a stock currently above Bm. However, they also suggest that this stock would decrease over the projection period at average recruitment levels with current catch levels. In contrast to the assessment for PAU 5B, these conclusions are not robust, as will be seen in the sensitivity trials described below.

4.2.2 PAU 5D sensitivity

Sensitivity of the results to the data sets is shown in Table 13. Results from the fit described above are termed the "base case". In the next four columns (tests Dsl to Ds4), results are shown from fits with one of the data sets removed and the model fitted to the remaining three. Then results are shown (tests Ds5 and Ds6) fiom the model fitted to length frequencies only (both abundance indices removed) and to abundance indices only (both length frequencies removed).

Removal of either abundance index (tests Dsl and Ds2) made the assessment more optimistic (see Table 13): the population was less depleted than in the base case and was a higher percentage of Bm. In both tests the population was still projected to decline by about 25%. Removal of the catch sampling data (test Ds3) gave essentially the same result. When the population length frequencies were removed, however (test Ds4), the assessment changed dramatically: these results suggest a badly depleted population, well below BMy, which would also be projected to decrease with current catches.

Of the tests with both abundance indices (test Ds5) or both length frequencies (test Ds6) removed, only the former converged properly. This test gave a result similar to Dsl (see Table 13).

Based on the major change caused by removal of one of the data sets, the poor fit to abundance indices, and the other problems with fitting, the PAU 5D assessment was suspended at this point.

5. DISCUSSION

For PAU 5B, the model provided a reasonably robust assessment of the stock. The assessment and nearly all sensitivity trials suggest a depleted stock, well belowBm, likely to decline further at the current catch level.

These conclusions are robust to the two data types: they remain the same when the model is fitted only to abundance indices or only to length frequencies. Success of the assessment is due partly to having two abundance indices, both showing contrast and the same trend. Many abalone scientists are skeptical of assessments based on CPUE, but the independent survey index lends credence to the strong trend seen in CPUE. The data on proportion-at-length are also reasonably consistent with each other and with the abundance index data (see Table 7).

Although hyperstability of the CPUE index in abalone fisheries is widely recognised (Breen 1992), the data for PAU 5B appear to contain no information on this parameter (see Figure 8). Although it is possible that standardised CPUE has a linear relation with abundance in this fishery, it is more likely that hyperstability exists and is confounded with other parameter estimates. If hyperstability does exist, then the model results presented above, in whichq3 is close to one, are conservative, because the trend in CPUE would be less steep than the true decline in abundance at high biomass. The sharper downward trend in the survey index (see Figure 2) suggests that this is a possibility.

For PAU 5D the model was much less successful. When data sets were removed sequentially from the fitting, they appeared to have substantially different signals (see Table 13). Catch samples and population samples may not be representative of the area fished; the survey index may also not be representative of the area fished. Much less population sampling has been conducted in PAU 5D compared with 5B, mainly for logistic reasons.

Model results for PAU 5B were sensitive to some of the assumptions. They were most sensitive to the assumed growth rate: a change of 10 mm in L, and the concomitant change in K, caused substantial change to the estimate, for instance, ofBI99dBm (see Table 8). Growth data from PAU 5B are very sparse, and work has been initiated to address this problem. In later versions of the model, growth parameters were estimated within the model as part of the fitting procedure, by including the growth increment data as an additional data set and adding a contribution to the negative log-likelihood function. The Shellfish Working Group recognised this sensitivity to growth as an uncertainty that should be improved for future assessments.

Model results were also sensitive to the assumed value for steepness. The data suggest that steepness is low (see Figure 8), and in maximum likelihood fits the estimatedh was always on the lower bound (see Table 8). If steepness is actually higher, then the model results are too pessimistic.

Model results were not very sensitive to the other assumptions tested. The uncertainty around the catch vector from PAU 5B (see Figure 6) did not lead to much uncertainty in model results (see Table 10).

The model assumes that PAU 5B is a unit stock, that abalones have the same growth and mortality characteristics in all parts of the stock, and that CPUE responds to abundance as if abundance were homogeneous within statistical areas. These are obviously over-simplifications. Growth, for instance, is likely to vary over small distances (Day & Fleming 1992). However, any attempt to reduce the assumed stock size to include more biological realism will encounter problems with fishery data, which have been collected at larger scales. Any quantitative assessment must balance these competing scales. Future work in PAU 5B may be able to divide that area into three in the short term, but further reductions would require longer time series of small-scale fishery data.

6. ACKNOWLEDGMENTS

Thanks to David Fournier and David MacDonald for their help in developing the model, and to Dave Gilbert for comments on a previous draft of this report.

This work was funded by the Ministry of Fisheries, contract PAU80 1.

7. REFERENCES

Andrew, N.L., Naylor, J.R., Gerring, P., & Notman, P.R. 2000a: Fishery independent surveys of paua (Haliotis iris) in PAU 5B and PAU 5D. New Zealand Fisheries Assessment Report 2000/3. 21 p.

Andrew, N.L., Naylor, J.R., & Gerring, P. 2000b: A modified timed-swim method for paua stock assessment. New Zealand Fzkheries Assessment Report 2000/4.23 p.

Annala, J.H., Sullivan, K.J., & O'Brien, C.J. 1999: Report fiom the Fishery Assessment Plenary, April 1999: Stock assessments and yield estimates. Ministry of Fisheries, Wellington. 430 p. (Unpublished report held in NIWA library, Wellington).

Bradford, E. 1998: Harvest estimates from the 1998 National Marine Recreational Fishing Surveys. New Zealand Fisheries Assessment Research Document 98116.27 p.

Breen, P.A. 1992: A review of models used for stock assessment in abalone fisheries. pp. 253- 275 In Shepherd, S.A., Tegner, M.J. & Guzman del Proo, S. (Eds.) Abalone of the world: Biology, fisheries and culture. Blackwell Scientific, Oxford.

Clavier, J. 1992: Fecundity and optimal sperm density for fertilization in the ormer (Haliotis tuberculata L.). pp. 86-92 In Shepherd, S.A., Tegner, M.J. & Guzman del Proo, S. (Eds.) Abalone of the world: Biology, fisheries and culture. Blackwell Scientific, Oxford.

Day, R.W. & Fleming, A.E. 1992: The determinants and measurement of abalone growth. pp. 141-168 In Shepherd, S.A., Tegner, M.J. & Guzman del Proo, S. (Eds.) Abalone of the world: Biology, fisheries and culture. Blackwell Scientific, Oxford.

Elvy, D., Grindley, R., & Teirney, L. 1997: Management Plan for Paua 5. Otago Southland Paua Management Working Group Report. 57 p. (Held by Ministry of Fisheries, Dunedin).

Fournier, D.A., Sibert, J.R., Majkowski J., & Hampton, J. 1990: MULTIFAN a likelihood-based method for estimating growth parameters and age composition from multiple length frequency data sets illustrated using data for southern blufin tuna (Thunnus rnaccoyiz). Canadian Journal of Fisheries and Aquatic Sciences 47: 30 1-3 17.

Francis, R.I.C.C. 1993: Monte Carlo evaluation of risks for biological reference points used in New Zealand fishery assessments. pp. 221-230 In Smith, S.J., Hunt, J.J., & Rivard, D. (Eds.) Risk Evaluation and Biological Reference Points for Fisheries Management Canadian Special Publication of Fisheries and Aquatic Sciences 120.

Hilborn, R. & Walters, C.J. 1992: Quantitative fisheries stock assessment. Chapman & Hall, New York. xv + 570 p.

Kendrick, T.H. & Andrew, N.L. 2000: Catch and effort statistics for paua (Haliotis iris) in PAU5 and a summary of standardised CPUE indices of stock abundance. New Zealand Fisheries Assessment Report 2000/(in press).

McShane, P.E., Mercer, S.F., & Naylor, J.R. 1994: Spatial variation and commercial fishing of the New Zealand abalone (Haliotis iris and H australis). New Zealand Journal of Marine and Freshwater Research 28: 345-355.

McShane, P.E., Mercer, S.F, Naylor, J.R., & Notman, P.R. 1996: Paua (Haliotis iris) fishery assessment in PAU 5, 6, and 7. New Zealand Fisheries Assessment Research Document 9611 1.35 p. (Unpublished report held in NIWA library, Wellington).

McShane, P.E. & Naylor, J.R 1995: , Small-scale spatial variation in growth, size at maturity, and yield- and egg-per-recruit relations in the New Zealand abalone Halitis iris. New Zealand Journal of Marine and Freshwater Research 29: 603-612.

Pirker, J.G. 1992: Growth, shell-ring deposition, and mortality of paua Qlaliotis iris M a w ) in the Kaikoura region. Unpublished M.Sc. thesis, University of Canterbury. 165 p.

Prince, J.D., Sellers, T.L., Ford, W.B., & Talbot, S.R. 1987: Experimental evidence for limited dispersal of haliotid larvae (Haliotis; Mollusca: Gastropoda). Journal of Experimental Marine Biology and Ecology 106: 243-263.

Punt, A. & Kennedy, R.B. 1997: Population modelling of Tasmanian rock lobster, Jasus edwardsii, resources. Marine and Freshwater Research 48(8): 967-980.

Sainsbwy, K.J. 1982: Population dynamics and fishery management of the paua, Haliotis iris. 1. Population structure, growth, reproduction and mortality. New Zealand Journal of Marine and Freshwater Research 16: 147-1 6 1.

Schiel, D.R. 1989: Paua fishery assessment 1989. New Zealand Fishery Assessment Research Document 8919: 20-p. (Unpublished report held in NIWA library, Wellington).

Schiel, D.R. 1991: The paua (abalone) fishery of New Zealand. pp. 427437 In Shepherd, S.A., Tegner, M.J. & Guzman del Proo, S. (Eds.) Abalone of the world: Biology, fisheries and culture. Blackwell Scientific, Oxford.

Schiel, D.R. & Breen, P.A. 1991: Population structure, ageing, and fishing mortality of the New Zealand abalone Haliotis iris. Fishery Bulletin 89(4): 681-69 1.

Shepherd, S.A. & Breen, P.A. 1992: Mortality in abalone: Its estimation, variability and causes. pp. 276-304 In Shepherd, S.A., Tegner, M.J. & Guzman del Proo, S. (Eds.) Abalone of the world: Biology, fisheries and culture. Blackwell Scientific, Oxford.

Stan; P.J., Bentley, N., & Maunder, M.N. 1999: Assessment of the NSN and NSS stocks of red rock lobster (Jasus ehardsii) for 1998. New Zealand Fisheries Assessment Research Document 99134.45 p. (Unpublished report held in NIWA library, Wellington).

Taylor, J., Schiel, D., & Taylor, H. 1994: The first cut is the deepest. Wounding, bleeding and healing in the black-foot paua (Haliotis iris). Sedood New Zealand 2(1): 47-49.

Teirney, L.D., Kilner, A.R., Millar, R.E., Bradford, E., & Bell, J.D. 1997: Estimation of recreational catch from 199 1-92 to 1993-94 New Zealand Fisheries Assessment Research Document 9711 5.43 p. (Unpublished report held in NIWA library, Wellington).

Worthiigton, D.G. 1997: Demography and dynamics of the population of blacklip abalone, Haliotis rubra, with implications for management of the fishery in NSW, Australia. Unpublished PhD thesis, Macquarie University, Sydney, Australia. 203 p.

Table 1: Commercial catch (kg) and standardised CPUE estimates for PAU 5B. The vectors "s15" and "s16" are catch vectors made to test sensitivity to assumptions used in estimating the PAU 5B catches between 1983-84 and 1994-95 -see text for details. "Std CPUE" refers to the year effect from CPUE standardisation (Kendrick & Andrew 2000)

Year 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998

Base Case 110 588 104 614 83 257

115 128 173 399 181 979 225 212 272 657 180211 230 350 287 058 178 981 112 206 199 915 214 989 191 494 198 741 220 590 196 049 170 720 164 253 156 168 131 462 140 732 142 806

Std CPUE

1.8979 2.1775 2.1440 2.2909 1 S844 1.6194 1 AM6 1.2145 1.1799 1.0732 1.0194 1.0149 0.923 1 1 .oooo 1.0134

Table 2: Commercial catch (kg) and standardised CPUE estimates for PAU 5D

Year 1974 1975

Catch 53 168 50 295 40 028 55 350 83 365 87 490

108 275 131 085 86 640

110 745 137 962 93 536 63 636

122 838 142 928 137 594 154 212 184 734 177 133 166 392 172 707 177 200 181 706 189 907 172 508

Std CPUE

Table 3: Fihery-independent survey estimates of relative abundance for PAU 5B. Data are the mean number of emergent paua per timed search (Andrew eta& 2000a)

Year Survey, 1994 50.1 1995 36.9 1996 28.7 1997 1998 17.6

Table 4: Fishery-independent survey estimates of relative abundance for PAU 5D. Data are the mean number of emergent paua per timed search (Andrew et al. 2000a)

Year Survey 1993 97.2 1994 1995 85 1996 1997 1998 55.8

Table 5: The prior distributions and bounds used for each parameter in the model

Parameter Rcoff(In virgin recruitment)

M (natural mortality) h (steepness)

In(ql) (In of catchability for CPUE) 43 (hyperstability parameter)

h(q2) (In of catchability for Survey) Rdevs (recruitment residuals)

Type of prior

uniform lognormal

uniform uniform uniform uniform

lognormal

Lower bound

5 .O 0.1 0.4

-50.0 0.2

-30.0 -2.3

Upper bound Mean cv. 50.00 0.50 0.1 0.5

0.975 1 .oo

1 .O25 1 .oo 2.30 0.0 0.6

Table 6: The best-fit point estimates for the major parameters and indicators for PAU 5B

Estimated values 13.971 0.120 0.400

-13.833 -10.129

1.022 2.300 0.224 2500 637

25.5% 55.3% 33.7% 3 1.8% 22.6%

Table 7: Sensitivity of the results to the data sets for PAU 5B

Parameter Rcofl

M h

Wql) Wq2)

93 sigmaR

mmRdev f

ERate FO. I

BO (0 Bcurrent (0

depletion Bcurr/Buru

B2004/ Burp Scurr/SO

S2004/SO

Base 13.97 0.12

0.400 -13.83 -10.13

1.02 1 .O3 2.30

-994.3 0.224 0.060 2499 637

25.5% 55.3% 33.7% 3 1.8% 22.6%

no CPUE 13.99 0.121 0.400

-25.00 -10.16

0.61 1.04 2.30

-995.4 0.214 0.063 2510 668

26.6% 59.2% 37.8% 33.0% 24.1%

Table 8: Sensitivity to the growth parameters and to steepness for PAU 5B

Parameter Base s7 s8 Linf

K

Rcof M h

W9l) W 2 )

93 sigmaR

m d d e v f

ERate FO. I

BO (0 Bcurrent

s5 no CPUE

no IS 14.12 0.124 0.400

-2 1.99 -16.00

0.61 1.10 2.27

-996.7 0.161 0.066 2720

886 32.5% 73.2% 56.1% 38.8% 31.8%

s9 steepness

s6 no LF

data 13.74 0.108 0.568

-12.95 -9.54 1 .oo 1.22 2.2 1 20.3

0.800 0.090 2390

179 7.5%

18.2% 11.7% 14.5% 11.8%

s10

Table 9: Sensitivity to Rdevs for PAU 5B. In test s l l , the bounds were increased to -5 and 5, and in sl la, the bounds oo Rdevs were reduced to -0.2 and 0.2

Parameter Rcof

M h

4 1 ) w.92)

q3 sigmaR

mmRdev f

ERate FO. 1

BO (0 Bcurrent (0

depletion Bcurr/ Bm Bprojl Bm

scurr/so Sproj/SO

Base 13.97 0.12

0.400 -13.83 -10.13

1.02 1 .O3 2.30

-994.3 0.224 0.060 2500

637 25.5% 55.3% 33.7% 31.8% 22.6%

s l l Rdev

bounds 5 13.73 0.097 0.400

-13.28 -10.02

0.99 1.06 2.10

-99 1.3 0.255 0.048 2860 56 1

19.6% 43.1% 19.5% 24.6% 14.2%

s l la Rdev

bounds 0.2 13.98 0.111 0.400

-13.95 -10.19

1.02 0.19 0.20

-973.1 0.204 0.056

2906770 699844 24.1% 53.1% 35.5% 29.9% 22.1%

Table 10: Sensitivity to the assumed catch trajectory for PAU 5B

Parameter Rcof

M h

W q O W-?)

43 sigmaR

mmrRdev f

ERate FO. 1

BO (0 Bcurrent (0

depletion Bcurr/ Bm Bproj/ Bmr

Scurr/SO Sproj/SO

Base 1'3.97 0.12

0.400 -13.83 -10.13

1.02 1 .O3 2.30

-994.3 0.224 0.060 2500

637 25.5% 55.3% 33.7% 3 1.8% 22.6%

s15 series 1

Elvy et al. 14.19 0.134 0.400

-12.68 -10.30

0.92 1 .oo 2.30

-991.8 0.186 0.072 253 1

770 30.4% 68.1% 5 1.4% 37.5% 30.6%

s16 s17 series 2 series 3

Working Group 14.01 0.107 0.400

-12.81' - 10.26 0.93 1.10 2.30

-99 1.8 0.185 0.054 3178 773

24.3% 53.8% 38.9% 29.9% 23 .O%

1986 interpolated 13.99 0.120 0.400

-13.88 -10.15

1.02 1.02 2.30

-994.4 0.2 18 0.060 2546 656

25.8% 55.9% 35.1% 32.1% 23 -2%

Table 11: Summary of posterior distributions for the major parameters and indicators for PAU 5B

Parameter Rcof

M h

W 9 l ) ln(92)

q3 BO (t)

B m ~ r (t) Bcurrent (t)

Bproj (0 so (9

Scurrent (t) SP~O? (t)

Erate Erateproj

f depletion

BCW/ Bm Bproj/current

Bmsy/BO Bproj/ Bmy ScurrentXSO

Mean 14.2

0.134 0.445

-8.8 -10.2 0.65 2701 1194 70 1 544

3345 1107 948

0.263 0.404

-980.1 25.1% 56.9% 69.9% 44.2% 43.6% 32.0%

Median 14.2

0.135 0.439

-9.0 -10.2 0.67 2664 1176 630 455

3288 1006 753

0.227 0.3 19

-980.4 23.8% 53.8% 72.0% 44.3% 38.7% 30.8%

Table 12: The best-fit point estimates for the major parameters and indicators for PAU 5D

Parameter Rcoff

M h

W 9 l ) MZ)

93 sigmaR

m m W f

ERate FO. I

BO (0 Bcurrent (0

depletion Bcurr/ Bmy Bproj/ Bm

Scurr/SO Sproj/SO

Estimated values .13.639

0.120 0.600

-12.966 -9.222 0.990 1.285 2.598

-750.572 0.159 0.109 1790 902

50.5% 124.5% 83.0% 56.6% 43.1%

Table 13: Sensitivity of the results to the data sets for PAU 5D

Parameter Rcof

M h

Wil l ) Wq2)

93 s i p &

mmRdev f

ERate FO. 1

BO (9 Bcurrent (0

depletion Bcwr/BUrY Bproj/Bm

scwr/so Sproj/SO

Base 13.78 0.133

0.6 -13.02 -9.26 0.99

1.01 1 2.30

-75 1.9 0.155 0.114 1710 92 1

53.8% 125.9% 87.7% 60.8% 47.6%

s l No cpue

14.06 0.148

0.6 -12.80 -9.56 1.01

1.142 2.30

-7612 0.112 0.144 1860 1270

68.5% 171.4% 129.9% 73.2% 61.1%

s3 No shed

13.66 0.106

0.6 -13.45, -9.49 0.99

1 . O X 2.30

-389.6 0.115 0.094 2270 1240

54.5% 134.0% 108.6% 6 1 .O% 50.9%

I I

I a-, Awarua Point -.

PAU 5A Fiordland I"

CaswelI Sound 031

$' Cape Providence

Figure la. Area PAU 5 and the new subdivision boundaries creating three areas PAU 5A, PAU 5B and PAU 5C.

Figure 1 b. The locations of new statistical reporting areas within PAU 5.

survey

CPUE

5 0.0 1970 1975 1980 1985 1990 1995 2000

year

Figure 2. The maximum likelihood fit of the model to the survey index (upper) and CPUE (lower) for PAU 5B. In each figure, the squaresindicate observations and the line indicates the model predictions.

0.14 -

). a 0.04.

137 147 1 s 127 167 87 107 117 in is7 147 im 167

shell length (mm) 0.12 -

W 1999

0 L- .--- .

0.12 : 007 -

shed samples survey

127 137 147 157 167

shell length (mm)

3 0.1'

c

Figure 3. The maximum likelihood fit of the model to length frequency data for PAU 5B Catch samples are on the left; population samples on the right. In each figure, the squares indicate observations and the line indicates the model predictions.

0.08. .. mu 1992 0.05 .

m - 0.02.

0 r 127 137 147 157 167

97 107 117 127 137 147 157 167

0.12 I shed samples

127 137 147 157 167

om.

-.--

am. 1994

(II

'. am' 0.01 .

.

survey I

shell length (mm)

127 137 147 157 167

shell length (mm)

Figure 4. The fit to length frequency data fiom PAU 5B fiom sensitivity test s3, where the model was not fitted to catch sampling length frequency data. Catch samples are on the left; population samples on the right. In each figure, the squares indicate observations and the line indicatk the model predictions.

shed samples survey m 1 I

0.1 , o m .

shell length (mm)

shell length (mm)

Figure 5.7'he fit to length frequency data from PAU 5B from sensitivity test s4, where the model was not fitted to population length frequency data. Catch samples are on the left; population samples on the right. In each figure, the squares indicate observations and the line indicates the model predictions.

Fishing year

Figure 6. The catch vectors resulting fiom two alternative asmptions about PAU 5B catches from 1983-84 to 1994-95. The heavy solid line shows the base case, the dashed line shows the catch assumed in test s15, and the lighter solid line shows the catch assumed in test s16.

12 C I

0 1000 2000 3000 4000 5000

sequential mcmc simulation

Figure 7. Sequential trends of the 5000 parameter values for M, h, q3 and ROcoflfiom 5000 sampled mcmc simulations using base case assumptions for PAU 5B.

0.04 0.18 0.33 0.48 0.62 0.77 0.92 1.06 1.21 1.35 1.50 parameter value

parameter value

Figure 8. Posterior distributions for the major parameters and indicators for PAU 5B.

Figure 9. Paua 5B: Posterior distributions of Mfiom the base k e (heavy line) and from a sensitivity test with a d o r m prior distribution of Mwith bounds 0.05 to 0.20. In the base case, the prior distribution was lognormal, with a mean of 0.10 and CV. of 0.50.

0 0.2 0.4 0.6 0.8 1

depletion

Figure 10. Paua 5B: Posterior distributions of Bls9y/Bo (depletion) fiom the base case (heavy line) and fiom a sensitivity test with a uniform prior distribution of parameter q3 with bounds 0.990 to 1.025. In the base case, the uniform prior had bounds 0.200 to 0.025.

CPUE

survey

1970 1975 1980 1985 1990 1995 2000 year

Figure 1 1. The maximum likelihood fit of the model to the CPUE (upper) and survey index (lower) for PAU 5d. In each figure, the squares indicate observations and the line indicates the model predictions.

survey

1992

shed samples 0.08 . 0.07 .

1992 0.06 .

0.05 - 0.04 .

g 0.04 - 0.02 -

137 147 157

she11 length (mm)

97 107 117 in 137 147 157 167

shell length (mm)

Figure 12. The maximum likelihood fit of the model to length frequency data for PAU 5D. Catch samples are on the left; population samples on the right. In each figure, the squares indicate observations and the line indicates the model predictions.