FISHERIES/11/SWG-PEL/75 MRM IWS/DEC11/OMP/P3 ssessment of the South frican anchovy resource using data from 1984 1: posterior distributions for the two base case hypotheses C.L. de Moor and D.S. Butterworth Correspondence email: carryn.demoor@uct.ac.za bstract The operating model (OM) for the South frican anchovy resource has been updated from that used to develop OMP-8 given four more years of data and a revised time series of commercial catch. The OM with results at the posterior mode has alrey been presented (de Moor and Butterworth 11). The posterior distributions for the two base case hypotheses are similar in many respects except for parameters relating to the stock recruitment relationship. Introduction The operating model of the South frican anchovy resource has recently been updated to be used in developing and simulation testing OMP-1. The full model description is given in ppendix of de Moor and Butterworth (11). Two base case hypotheses have been chosen, one assuming a constant ult natural mortality over time, cstm, and the other allowing for random effects about annual ult natural mortality, HS. de Moor and Butterworth (11) present results at the posterior mode for these two base case hypotheses as well as for a range of robustness tests. In this document the posterior distributions for these two base case hypotheses are presented. Bayesian Estimation The objective function consisting of the negative log likelihood (equation (.7) of de Moor and Butterworth 11) ded to the negative of the log prior distributions was minimised using D Model Builder (Otter Research Ltd. ) to fit the model to the observed data and estimate the parameters at the posterior mode. glossary defining all parameters is given in ppendix C of de Moor and Butterworth (11) and the prior distributions utilised are listed in Table 1. The posterior probability distributions were estimated using Markov Chain Monte Carlo (Gelman et al. 1995) in D Model Builder. The length of the chain, the thinning and burnin applied are given in Table 1. Results presented in this document are based on a random sample of 5 from the remaining chain. Convergence of the chains was tested using the BO (Bayesian Output nalysis) package (Smith 3). Results and Discussion The posterior medians, means and CVs of key model parameters and outputs for HS and cstm are given in Table. The posterior distributions of key model parameters and outputs are plotted in Figure 1 and the annual biomass posterior distributions are plotted in Figure. MRM (Marine Resource ssessment and Management Group), Department of Mathematics and pplied Mathematics, University of Cape Town, Rondebosch, 771, South frica. 1
FISHERIES/11/SWG-PEL/75 MRM IWS/DEC11/OMP/P3 The posterior distribution of the maximum median recruitment, and consequently that of carrying capacity, under cstm is centered on a lower number than under HS. The variance about the stock recruitment curve is estimated to be slightly larger under cstm compared to HS. The main difference between the biomass distributions for the two base case hypotheses are during the peak years of 1 to 3. This can also be seen in Figure 3 which plots the median and 95% posterior interval over time. This 1 3 period corresponds with a dramatic change from low to high ult mortality under HS, compared to a constant value over all years for cstm (Figure 4). difference between HS and cstm is also evident in the numbers at age in November each year, with a much higher peak in the late 199s to early s in HS compared to cstm (Figure 5). s was evident at the posterior mode, the loss to predation during the past dece as estimated by HS is much higher than that estimated by cstm (Figure 6). References de Moor, C.L., and Butterworth, D.S. 11. ssessment of the South frican anchovy resource using data from 1984-1: results at the posterior mode. Department of griculture, Forestry and Fisheries Document FISHERIES/11/SWG-PEL/66/ 9pp. Gelman,., Carlin, J.B., Stern, H.S. & Rubin, D.B. 1995. Bayesian Data nalysis. Chapman & Hall. 55pp. Otter Research Ltd.. n Introduction to D Model Builder Version 4: For Use in Nonlinear Modeling and Statistics. Otter Research Ltd. (http://www.otter-rsch.com/) Smith, B.J. 3. Bayesian Output nalysis Program (BO) Version 1. User s Manual.
FISHERIES/11/SWG-PEL/75 MRM IWS/DEC11/OMP/P3 Table 1. list of the model parameters and their prior distributions (see ppendix of de Moor and Butterworth (11) for further details). Where population numbers are concerned, the units are billions. Parameter ( ) ~ U[ 1,.7] Description ln k N Log of the constant of proportionality associated with the acoustic survey estimate of ult anchovy biomass from the November survey ln ln ( k ) ~ U[ 1,.7] Log of the constant of proportionality associated with the acoustic survey r estimate of recruit survey numbers from the recruit survey ( ) ~ U[ 1,.7] k p ( λ ) ~ U[,1 ] r ( ) ε y ~ N, σ r, y = 1984,..., yn 1 ( σ ) ~ U[.4,1 ] r [,5] Log of the constant of proportionality associated with the proportion of 1- year-olds in the November survey dditional variance associated with the recruit survey nnual lognormal deviation of anchovy recruitment N 1983, a ~ U, a =, 1 Numbers at age in the initial year [,.1] N 1983, a ~ U, a =, 3 (, ) ~ η y N σ σ ~ U[.,.5] p ~ U [,1] σ p ln( b K ~ U[.9,1] a ) ~ U ~ U[,1 ] [,7.] Standard deviation in the residuals about the stock recruitment curve Numbers at age in the initial year. These are set effectively at zero because there are insufficient data to determine these values with reasonable precision. Normally distributed error used in calculating the annual residuals about ult natural mortality Standard deviation in the annual residuals about ult natural mortality nnual autocorrelation coefficient in annual residuals about ult natural mortality Standard deviation associated with the estimated proportion of 1-year-olds in the November survey Log of the maximum median recruitment in the hockey stick stock recruitment curve The biomass above which median recruitment is not impaired in the hockey stick stock recruitment curve as a proportion of carrying capacity 3
FISHERIES/11/SWG-PEL/75 MRM IWS/DEC11/OMP/P3 Table. The MCMC chain length, thinning and burn-in used to get a sample from the posterior distribution for the robustness tests. The posterior means and CVs of key model parameters and outputs are also shown. Biomasses are given in thousands of tons and numbers in billions. Parameters fixed for MCMC runs are given in bold (initial testing showed very little movement in the chain from the posterior mode) cstm Total chain length 1 4 Thinning 3 3 Chain excluded (eg for burn-in) 1 3 Length of chain used for posterior 3 5 Parameter Median Mean CV Median Mean CV HS k N 1.7 1.8.1 1.7 1.8.13 k R.78.79.14 1. 1.3.14 k p.86.87.5.95.94.6 ( ) λ.11.1.49...43 r ( ) σ.9.9 N/.85.91.35 p N, 1983 174 178.18 18 189.3 N1983, 1 N, 19 133.8 146 149.38 1983.5.5.58.5.5.55 N1983, 3.5.5.57.5.5.58 σ.45.44.1 N/ N/ N/ ρ.59.57.31 N/ N/ N/ η 1.4.4.65 N/ N/ N/ a 93 96.6 474 551.45 b 4318 438.3 6 3146.55 σ r.66.67.18.75.76.17 K 739 759.6 3798 4418.45 η -.56 -.57.63 -.5 -.53.79 9 s cor.16.17.59.17.17.61 B Nov 168 174.11 151 16.1 N 1, 1 N, 159 16. 89 93.34 1 4 4.34 57 58.7 N, 3 1 8.84 9.57.55 4 41.3 N,4+ 1 1.4 1.39.86 16 16.1 B 1,N 183 17.18 451 495.18 4
FISHERIES/11/SWG-PEL/75 MRM IWS/DEC11/OMP/P3 k N k r k p λ R σ.5 1 1.5.5 1 1.5.5 1 1.5.5 1 1.5..3.4.5 ρ η 1 (in billions) a (in 't) b b/k..4.6.8 1-3 - -1 1 5 1 15 1.5 1 K (in 't) σ r B 1,N vg(b 91-99 ) (in 't) η 9 5 1.5 1 1.5 1 1 3 4 - -1 1 s cor - -1 1 Figure 1. Posterior distributions for key model parameters and outputs for the two base case hypotheses HS (solid lines) and cstm (dotted lines). The prior distributions for model parameters estimated are shown by the thin dashed lines. The prior distribution for η 1 is not plotted as it depends on the distribution of σ, as (, ) ~ η y N σ. 5
FISHERIES/11/SWG-PEL/75 MRM IWS/DEC11/OMP/P3 B 1984,N B 1985,N B 1986,N B 1987,N B 1988,N B 1989,N B 199,N B 1991,N B 199,N B 1993,N B 1994,N B 1995,N B 1996,N B 1997,N B 1998,N B 1999,N B,N B 1,N B,N B 3,N Figure. The posterior pdfs of annual November biomass from the two base case hypotheses HS (solid lines) and cstm (dotted lines) note that in some cases where these are virtually identical, the latter is not visible as it is covered by the former. 6
FISHERIES/11/SWG-PEL/75 MRM IWS/DEC11/OMP/P3 B 4,N B 5,N B 6,N B 7,N B 8,N B 9,N B 1,N Figure (continued). a) B(y) (in 't) 9 8 7 6 5 4 3 1 b) B(y) (in 't) 9 8 7 6 5 4 3 1 c) B(y) (in 't) 7 6 5 4 3 1 1983 1988 1993 1998 3 8 1983 1988 1993 1998 3 8 1983 1988 1993 1998 3 8 Figure 3. The posterior median and 95% probability intervals of the annual anchovy November 1+ biomass. The posterior medians for HS (solid) and cstm (dotted) are plotted together in c). 7
FISHERIES/11/SWG-PEL/75 MRM IWS/DEC11/OMP/P3 M (y) (in year -1 ) 4 3.5 3.5 1.5 1.5 1983 1988 1993 1998 3 8 Figure 4. The posterior median and 95% probability intervals for ult natural mortality for HS. a) 3 b) 3 c) 5 N(y,) (in billions) 5 15 1 5 N(y,) (in billions) 5 15 1 5 N(y,) (in billions) 15 1 5 1983 1988 1993 1998 3 8 1983 1988 1993 1998 3 8 1983 1988 1993 1998 3 8 Figure 5. The posterior median and 95% probability intervals of the annual numbers of anchovy at age in November for a) HS and b) cstm. The posterior medians for HS (solid) and cstm (dotted) are plotted together in c). 8
FISHERIES/11/SWG-PEL/75 MRM IWS/DEC11/OMP/P3 Loss to Predation (in 't) 14 1 1 8 6 4 1983 1988 1993 1998 3 8 b) Loss to Predation (in 't) 14 1 1 8 6 4 1983 1988 1993 1998 3 8 c) Loss to Predation (in 't) 1 8 6 4 1983 1988 1993 1998 3 8 Figure 6. The posterior median and 95% probability intervals in anchovy loss to predation for a) HS and b) cstm. The posterior medians for HS (solid) and cstm (dotted) are plotted together in c). 9