The parameter of the best model, model 2, are estimated with 4 Monte Carlo Markov Chains (MCMC) using the R package BayesianTools.
The traces of the MCMC show that the MCMC converged after about 20,000 iterations.
The output of the MCMC looks like this:
| par1 | par2 | par3 | par4 | LP | LL | LPr | |
|---|---|---|---|---|---|---|---|
| 1 | 1.582301 | 0.6822409 | 0.08536556 | 0.03436937 | -5104.798 | -5101.415 | -3.382696 |
| 2 | 1.582301 | 0.6822409 | 0.08536556 | 0.03436937 | -5104.798 | -5101.415 | -3.382696 |
| 3 | 1.582301 | 0.6822409 | 0.08536556 | 0.03436937 | -5104.798 | -5101.415 | -3.382696 |
| 4 | 1.723482 | 0.6332424 | 0.08708028 | 0.03938038 | -5104.685 | -5101.302 | -3.382696 |
| 5 | 1.723482 | 0.6332424 | 0.08708028 | 0.03938038 | -5104.685 | -5101.302 | -3.382696 |
| 6 | 1.723482 | 0.6332424 | 0.08708028 | 0.03938038 | -5104.685 | -5101.302 | -3.382696 |
The posterior distributions of the parameters are uni-modal and symmetric. There is a strong negative correlation between fishing mortality (catchability, par1) and natural mortality (par2)
The posterior distributions of the parameters of model 2 were compared to Gaussian distributions with the same mean and standard distributions: the empirical and theoretical distributions are in good agreement.
Samples from the posterior distribution were used to calculate the distribution of the stock biomass and spawning stock biomass. Those data look like
| par1 | par2 | par3 | par4 | log.lik | rec1 | rec2 | rec3 | rec4 | |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1.582301 | 0.6822409 | 0.08536556 | 0.03436937 | -5101.415 | 0 | 0 | 0 | 51876.79 |
| 2 | 1.582301 | 0.6822409 | 0.08536556 | 0.03436937 | -5101.415 | 0 | 0 | 0 | 51876.79 |
| 3 | 1.582301 | 0.6822409 | 0.08536556 | 0.03436937 | -5101.415 | 0 | 0 | 0 | 51876.79 |
| 4 | 1.723482 | 0.6332424 | 0.08708028 | 0.03938038 | -5101.302 | 0 | 0 | 0 | 49372.36 |
| 5 | 1.723482 | 0.6332424 | 0.08708028 | 0.03938038 | -5101.302 | 0 | 0 | 0 | 49372.36 |
| 6 | 1.723482 | 0.6332424 | 0.08708028 | 0.03938038 | -5101.302 | 0 | 0 | 0 | 49372.36 |
| rec5 | rec6 | rec7 | rec8 | rec9 | rec10 | rec11 | rec12 | rec13 | |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 942969.5 | 2151690 | 1606584 | 1519344 | 2575502 | 5068420 | 3652790 | 3469817 | 2724337 |
| 2 | 942969.5 | 2151690 | 1606584 | 1519344 | 2575502 | 5068420 | 3652790 | 3469817 | 2724337 |
| 3 | 942969.5 | 2151690 | 1606584 | 1519344 | 2575502 | 5068420 | 3652790 | 3469817 | 2724337 |
| 4 | 897497.7 | 1958160 | 1454550 | 1364195 | 2294326 | 4446068 | 3170434 | 2978055 | 2334480 |
| 5 | 897497.7 | 1958160 | 1454550 | 1364195 | 2294326 | 4446068 | 3170434 | 2978055 | 2334480 |
| 6 | 897497.7 | 1958160 | 1454550 | 1364195 | 2294326 | 4446068 | 3170434 | 2978055 | 2334480 |
| rec14 | rec15 | rec16 | rec17 | rec18 | rec19 | rec20 | Biomass1 | Biomass2 | |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 3293877 | 4466441 | 3406995 | 1826319 | 3029519 | 3765319 | 7575271 | 125.9936 | 86.17279 |
| 2 | 3293877 | 4466441 | 3406995 | 1826319 | 3029519 | 3765319 | 7575271 | 125.9936 | 86.17279 |
| 3 | 3293877 | 4466441 | 3406995 | 1826319 | 3029519 | 3765319 | 7575271 | 125.9936 | 86.17279 |
| 4 | 2819025 | 3816542 | 2895193 | 1552425 | 2587540 | 3240658 | 5944767 | 113.9433 | 78.06261 |
| 5 | 2819025 | 3816542 | 2895193 | 1552425 | 2587540 | 3240658 | 5944767 | 113.9433 | 78.06261 |
| 6 | 2819025 | 3816542 | 2895193 | 1552425 | 2587540 | 3240658 | 5944767 | 113.9433 | 78.06261 |
| ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |
| Biomass11 | Biomass12 | Biomass13 | Biomass14 | Biomass15 | SSB1 | SSB2 | SSB3 | |
|---|---|---|---|---|---|---|---|---|
| 1 | 216.5568 | 196.6916 | 187.9461 | 264.6694 | 336.3326 | 30.33500 | 28.02512 | 22.9917 |
| 2 | 216.5568 | 196.6916 | 187.9461 | 264.6694 | 336.3326 | 30.33500 | 28.02512 | 22.9917 |
| 3 | 216.5568 | 196.6916 | 187.9461 | 264.6694 | 336.3326 | 30.33500 | 28.02512 | 22.9917 |
| 4 | 189.6608 | 172.8766 | 167.0109 | 224.2023 | 283.3116 | 27.85006 | 25.72940 | 21.1083 |
| 5 | 189.6608 | 172.8766 | 167.0109 | 224.2023 | 283.3116 | 27.85006 | 25.72940 | 21.1083 |
| 6 | 189.6608 | 172.8766 | 167.0109 | 224.2023 | 283.3116 | 27.85006 | 25.72940 | 21.1083 |
The estimated trend in recruitment looks like (same as estimate with maximum likelihood)
The estimated trend in biomass looks like (same as estimate with maximum likelihood)
The estimated trend in Spawning Stock Biomass looks like
And the estimate distribution of Spawning Stock Biomass in 2018/19 is much larger than SSB at MSY (62 tonnes): the probability that the stock is over-fished (SSB < SSBmsy) is 0
The trajectory of the stock along fishing effort (x-axis) and estimated Spawning Stock Biomass (y-axis) is shown on the Kobe plot below. At the beginning of the time series, in 2004/05, the stock received more fishing effort than sustainable (Emsy = 700 boat-days, vertical line): over-fishing was occurring on a stock already over-fished (SSB below SSBmsy indicated by the horizontal line). As fishing effort decreased and larger mesh size were introduced, over-fishing ceased and the stock recovered and ceased to be over-fished. In 2018/19 the NSW sea garfish fishery is not over-fished nor over-fishing is occurring.
