## 2 Materials and Methods
### General Description of the LBB Method  
The length-based Bayesian biomass estimator method is an innovative method presented for estimation of size structure and stock status only using length-frequency analysis, in which all relevant parameters are simultaneously analyzed with a Bayesian Monte Carlo Markov chain (MCMC) approach [12]. It is assumed that the growth in length or width of species follows [PERSON]'s [16] growth equation, in the form given by [PERSON] and [PERSON] [17], i.e. :  
\[W_{t}=W_{inf}\Big{[}1-e^{-K(t-t_{0})}\Big{]} \tag{1}\]  
where \(W_{t}\) is the width at age t, \(W_{inf}\) is the asymptotic length, \(K\) is the rate at which \(W_{inf}\) is approached, and \(t_{0}\) is the theoretical age at zero length [12]. With fully selected by the gear for the crabs, the curvature of the catch in number-at-width curve is a function of total mortality (\(Z=M+F\)) relative to \(K\). This curve is expressed by the equation:  
\[N_{W}=N_{W_{start}}\Bigg{(}\frac{W_{inf}-W}{W_{inf}-W_{start}}\Bigg{)}^{Z/K} \tag{2}\]  
where \(N_{W}\) is the number of survivors with width \(W\), \(N_{Wstart}\) is the number at length \(W_{start}\) through full selection, from which all individuals entering the gear are retained by the gear, and \(Z/K\) is the ratio of the total mortality rate to somatic natural growth rate [12]. When \(Z/K\) is equal to \(M/K\) and \(W_{start}\) is zero, \(N_{Wstart}\) becomes 1 in the unfished state. Equation (2) can be then simplified to:  
\[P_{W/W_{inf}}=\Bigg{(}1-\frac{W}{W_{inf}}\Bigg{)}^{\frac{M}{K}} \tag{3}\]where \(P_{W}/W_{inf}\) is the probability to survive to length \(W/W_{inf}\), which is a function of the \(M/K\) where \(P_{W}/W_{inf}\) is the probability to survive to length \(W/W_{inf}\), which is a function of the \(M/K\) ratio [12]. The widths that are affected by partial selection are a function of the fishing gear (it is assumed to be a trawl or other gear with a selection curve similar to a trawl), as described by Equation (4):  
\[Sw=\frac{1}{1+e^{-\alpha(W-W_{c})}} \tag{4}\]  
where \(S_{W}\) is the individual proportion retained by the gear at width \(W\), and \(\alpha\) describes the steepness of the ogive [18; 19]. The width corresponding to the gear keeping a certain probability \(p\) can be obtained by  
\[W_{p}=\frac{\alpha W_{c}-\log\left(\frac{1}{p}-1\right)}{\alpha} \tag{5}\]  
where \(W_{p}\) is the width with probability \(P\) of being retained by the gear and \(W_{c}\) and \(\alpha\) are as defined above [12]. The \(P\) value is equal to 0.01, 0.5 or 0.95. The parameters of the selection ogive can be estimated at the same time as \(W_{inf}\), \(W_{c}\), \(\alpha\), \(M/K\) and \(F/K\) by fitting:  
\[N_{W_{i}}=N_{W_{i-1}}\left(\frac{W_{inf}-W_{i}}{W_{inf}-W_{i-1}}\right)^{\frac {M}{K}+\frac{F}{K}S_{W_{i}}} \tag{6}\]  
And  
\[C_{W_{i}}=N_{W_{i}}S_{W_{i}} \tag{7}\]  
where \(W_{i}\) is the number of individuals at width \(I\) and \(W_{i-1}\) is the number at the previous width, whereas C is the number of individuals vulnerable to gear damage; the other parameters are as mentioned above [12]. By dividing both sides of Equation (7) by the corresponding sums, the samples from different years are normalized and their numbers become compatible, i.e. :  
\[\frac{C_{w_{i}}}{\sum\ C_{W_{i}}}=\frac{N_{W_{i}}S_{W_{i}}}{\sum\ N_{W_{i}}S_ {W_{i}}} \tag{8}\]  
Fitting Equation (8) to LF data provides estimates of \(M/K\) and \(F/K\), which could be combined to give \(F/M=(F/K)/(M/K)\). LBB estimation was implemented in Bayesian Gibbs sampler software JAGS, and the statistical language R was used to fit the observed proportional length \(P_{Wi}\) with its expected values \(\hat{P}_{L_{i}}\). According to Equation (8), the model predicted width distribution \(\hat{P}_{W_{i}}\) can be expressed as:  
\[\hat{P}_{W_{i}}=\frac{\hat{N}_{W_{i}}}{\sum\ \hat{N}_{W_{i}}} \tag{9}\]  
where \(\hat{N}_{W_{i}}\) is a function of the estimable population dynamic parameters \(W_{inf}\), \(M/K\) and \(F/K\) (Equation (7)) and the selectivity parameters \(W_{c}\) and \(\alpha\)[12].