Estimating historical commercial rock lobster (Jasus edwardsii) catch inside Australian State territorial waters for marine protected area assessment: the binomial likelihood method

Abstract

The rock lobster (Jasus edwardsii) fishery of South Australia is the State’s most valuable fisheries resource with an export value exceeding ~AU$100 million. The fishery operates primarily inshore (<60 m), and overlaps with a series of marine protected areas (MPAs) currently proposed for State territorial waters. As a result, the need to quantify the impact of proposed MPAs on commercial landings of rock lobster within territorial waters is an integral part of the MPA assessment process. Removing fishing effort displaced by MPAs prevents a corresponding increase in exploitation outside protected zones. We describe a binomial likelihood method that utilises historical commercial catch data to estimate catch totals of rock lobster inside South Australian State waters. Lobster catches per km² showed a high level of spatial variation with estimated historical lobster catch in State waters spanning approximately three orders of magnitude. The method identified key areas where high lobster catch (up to 500 kg/km²) overlapped with State waters. Binomial likelihood outputs have particular application to the estimation of net catch loss in situations where fishery buy-back or financial compensation are a considered option as part of the MPA implementation process.

Appendix: binomial likelihood

In this Appendix, the statistical algorithm of the binomial likelihood estimation is outlined. The objective was to estimate the proportion of catch that fell inside State territorial waters. As noted in the Methods, uncertainty in this estimated proportion will be inferred from its reported yearly variation. The sample size of 8 fishing seasons of pot sampling data, 1993/1994 to 2000/2001 is a ‘small sample’ by conventional definitions (Cochran 1977).

Let the estimate for the yearly average catch inside each MFA-State sub-block be denoted $\hat{\bar{C}}_{ < 3}$, where the “<3” subscript indicates the catch is inside the 3 nm limit of State waters. The ‘hat’ (^) indicates that the quantity is an estimate (not input data) and the bar that it is a yearly average. Upper case ‘C’ will denote catch totals (either input data from catch logs, or the final estimates themselves) while lower case ‘c’ will denote pot-sampled catches. This mean catch (by weight) in each sub-block will be estimated from (1) sampled catches inside each MFA-State sub-block (c _<3(y,b)), (2) sampled catches in each overall MFA block (c(y,b)), and (3) the catch-log-reported total catch in each year and MFA block (C(y,b)), where “b” designates the MFA block under consideration, and “y” the year. Bars over the catch symbols indicate time averages over the years of sampling and catch-log history, respectively.

An independent estimate of catch proportion taken inside State territorial waters is required for each MFA block. Applying a maximum likelihood estimate, we maximised a binomial likelihood to estimate the probability, ‘p _c ’, that a unit of catch is taken from the sub-block. The negative log of the binomial likelihood is written:

$$ - \log L_{{p_{C}}}(b) = \sum\limits_{y[c[y,b]\,>\,0]\;=\;1993}^{2000}{\left\{{-c_{<3} (y,b) \cdot \log \left[{p_{c}(b)} \right]\;-\;\left({c(y,b)-c_{<3}(y,b)}\right) \cdot \log \left[ {1-p_{c} (b)} \right]} \right\}} $$

(0.1)

Note that this negative-log-likelihood sum will exclude years for each block when no pot samples were obtained. The parameters to be estimated are {p _c (b), b = 1, 21}, for the 21 blocks with 100 or more sample pot lifts. One parameter value of catch proportion in sub-block b ($\hat{p}_{c} (b)$) was estimated for each block, by numerically minimising $- \log L_{{p_{C} }} (b)$ using the AD Model Builder parameter estimation software (http://otter-rsch.com/admodel.htm).

The estimate of average historical yearly catch from each sub-block ($\hat{\bar{C}}_{ < 3} (b)$ in Table 2) was obtained as the product of the yearly average of logbook catches in each whole block ($\bar{C}(b)$) times the estimated probability of catch being taken from inside each sub-block ($\hat{p}_{c} (b)$):

$$ \hat{\bar{C}}_{ < 3} (b)\,\doteq\,\bar{C}(b) \cdot \hat{p}_{c} (b) . $$

(A.1)

The reported catch total $\bar{C}(b)$ is assumed given without error. This is assumed because we have no knowledge to the contrary and because the fishery management, usually government-agency-reported, catch totals have legal standing.

The asymptotic approximation for the standard error of the estimate of $\hat{p}_{c} (b)$, denoted $SE\left[ {\hat{p}_{c} (b)} \right]$, was obtained numerically as the inverse of the second derivative with respect to the parameter (P _c (b)) of the minus log likelihood at the maximum. This implicitly approximates the binomial by a normal in the neighbourhood of the likelihood maximum, generally a satisfactory approximation for estimating confidence intervals (Cochran 1977; Rice 1995). Denoting the percentage error for the $\hat{p}_{c} (b)$ estimate as CV_SE, ${\text{CV}}_{\text{SE}} \left[ {\hat{p}_{c} (b)} \right] = {\text{SE}}\left[ {\hat{p}_{c} (b)} \right]\;/\hat{p}_{c} (b)$. The standard error of yearly average catch in each sub-block becomes

$$ {\text{SE}}\left( {\hat{\bar{C}}_{ < 3} (b)}\right)\,\doteq\,\hat{\bar{C}}(b) \cdot {\text{CV}}_{\text{SE}}\left[{\hat{p}_{c} (b)} \right].$$

(A.2)