Performance Statistics
Rebuilding Time
L Kell
05 January, 2024
Example using ‘FLBRP’
For a stock the ‘FLBRP’ object can be derived, this holds the expected values for the mortality, selectivity, mass, marurity-at-age, and the stock recruitment relationship, and can be used to calculate the expected, i.e. equilibrium values of biomass for any given level of fishing mortality.
Figure 1 Equilibrium curves showing estimates of SSB, recruitment yield and F.
The ‘FLBRP’ object is then used to simulate ‘FLStock’ objects that can be projected through time for a given level of fishing mortality. In Figure 2 the stock is initially at different levels of depletion and then projected for \(F=0\). The stock then recovers, the dashed horizontal line is \(SSB/B_{MSY}\), and the green vertical line is half a geneartion time, and the red vertical line one generation time. For a desired time for recovery the level that the stock should not fall below can be read from the curves, and is summarised in Figure 3.
Figure 2 Recovery from depletion, vertical lines are half (green) and one (red) generation time
Figure 3 Recovery from different depletion levels, red horizontal line is \(B_{MSY}\), vertical lines are half (green) and one (red) generation time
If the management objective is in the case of stock depletion is to close the fishery to recover a stock in a generation time then the biomass limit that should be avoided with high probability is 25% of \(B_{MSY}\). If fishing is still to be allowed under recovery then the biomass limit will have to be higher.
As a performance statistic you could then say there must be a 95% prob that you can rebuild within a given fraction of a generation time with F=0.
Figure 4 Time to recover from different depletion levels, vertical lines are half (white) and one (red) generation time
Impact of M and the SRR
Figure 5 Time series of fishing mortality, with \(F_{MSY}\) (dashed) and \(F_{lim}\) (solid).
Figure 6 Time series of SSB, with \(B_{MSY}\) (dashed) and \(B_{lim}\) (solid).
0: 19.965325: 10.1229 -0.916291 1.91288
1: 19.795713: 10.1627 -0.888221 1.91357
2: 19.768537: 10.1484 -0.872385 1.91251
3: 19.763467: 10.1551 -0.866864 1.91226
4: 19.762547: 10.1525 -0.863914 1.91168
5: 19.762314: 10.1544 -0.862401 1.90855
6: 19.762268: 10.1535 -0.862582 1.90468
7: 19.762238: 10.1533 -0.862163 1.90657
8: 19.762229: 10.1536 -0.862038 1.90713
9: 19.762228: 10.1536 -0.862076 1.90702
10: 19.762228: 10.1536 -0.862075 1.90702
0: 90.016145: 8.77345 -0.916291 1.05279
1: 43.892164: 9.28078 -0.0695157 1.21276
2: 43.836695: 10.1209 -0.609402 1.26451
3: 38.774357: 9.98359 -0.270550 0.923462
4: 37.683715: 10.0659 -0.444244 0.840813
5: 37.617093: 10.0893 -0.456290 0.891391
6: 37.608816: 10.1425 -0.454067 0.870829
7: 37.607875: 10.1336 -0.451616 0.868423
8: 37.607860: 10.1338 -0.452187 0.868684
9: 37.607860: 10.1337 -0.452150 0.868735
10: 37.607860: 10.1337 -0.452152 0.868730
0: 98.134697: 17.0444 -0.916291 -0.105123
1: 66.204427: 17.6782 -0.215462 0.222216
2: 57.386679: 17.6611 -0.438680 0.222004
3: 54.373848: 17.5955 -0.880259 0.256308
4: 52.182700: 17.4034 -0.731071 0.632218
5: 51.700487: 17.4600 -0.750783 0.597805
6: 51.419157: 17.4745 -0.755889 0.530429
7: 51.327330: 17.5123 -0.737745 0.475505
8: 51.309615: 17.5019 -0.747032 0.467676
9: 51.308954: 17.5046 -0.745703 0.465233
10: 51.308767: 17.5057 -0.747009 0.461721
11: 51.308646: 17.5052 -0.745795 0.461631
12: 51.308645: 17.5052 -0.745841 0.461592
13: 51.308645: 17.5052 -0.745839 0.461601
14: 51.308645: 17.5052 -0.745840 0.461602
0: 10.662667: 15.2814 -0.916291 0.215888
1: 1.8827555: 15.5267 -1.04446 0.263769
2: -4.1584956: 15.5871 -1.31877 0.267349
3: -5.7707734: 15.5216 -1.51201 0.222162
4: -6.5514620: 15.6478 -1.53372 0.0570197
5: -6.6401656: 15.6274 -1.55941 0.0850229
6: -6.6502214: 15.6305 -1.57499 0.0891227
7: -6.6518858: 15.6260 -1.57609 0.0862232
8: -6.6531418: 15.6255 -1.57522 0.0915800
9: -6.6532200: 15.6231 -1.57467 0.0964097
10: -6.6533343: 15.6237 -1.57727 0.0978175
11: -6.6534284: 15.6238 -1.57619 0.0964717
12: -6.6534284: 15.6238 -1.57620 0.0965118
13: -6.6534284: 15.6238 -1.57620 0.0965103
0: 51.944736: 11.4870 -0.916291 2.14391
1: 50.127249: 11.6258 -0.652428 2.14344
2: 48.885818: 11.5454 -0.682422 2.13793
3: 48.877253: 11.5051 -0.758061 2.13132
4: 48.730090: 11.5422 -0.736375 2.13089
5: 48.714232: 11.5275 -0.730490 2.12825
6: 48.708318: 11.5342 -0.726225 2.11430
7: 48.706688: 11.5329 -0.731964 2.09940
8: 48.706101: 11.5337 -0.728306 2.09785
9: 48.706056: 11.5334 -0.728784 2.09785
10: 48.706054: 11.5336 -0.729328 2.09789
11: 48.706049: 11.5334 -0.729098 2.09831
12: 48.706049: 11.5335 -0.729057 2.09788
13: 48.706049: 11.5335 -0.729083 2.09810
14: 48.706049: 11.5335 -0.729087 2.09810
15: 48.706049: 11.5335 -0.729088 2.09808
16: 48.706049: 11.5335 -0.729087 2.09807
0: 70.559422: 16.4650 -0.916291 0.901720
1: 50.778048: 16.6680 0.0605462 0.834096
2: 47.838766: 16.8101 -0.143124 0.656977
3: 45.887963: 16.6411 -0.396811 0.645757
4: 45.688468: 16.5973 -0.305013 0.631216
5: 45.653110: 16.5938 -0.337431 0.618891
6: 45.652448: 16.6039 -0.333284 0.622062
7: 45.651859: 16.5999 -0.336147 0.632283
8: 45.651660: 16.5978 -0.334537 0.628430
9: 45.651648: 16.5991 -0.335967 0.625163
10: 45.651619: 16.5995 -0.334981 0.627567
11: 45.651610: 16.5989 -0.335140 0.627416
12: 45.651609: 16.5990 -0.335183 0.627222
13: 45.651609: 16.5990 -0.335173 0.627256
14: 45.651609: 16.5990 -0.335173 0.627254
0: 45.715922: 14.8279 -0.916291 1.26607
1: 43.499074: 15.0945 -0.706194 1.28918
2: 41.716672: 14.9820 -0.716358 1.26789
3: 41.568918: 14.9501 -0.746679 1.25951
4: 41.559130: 14.9588 -0.752947 1.25724
5: 41.556354: 14.9545 -0.756724 1.24793
6: 41.551653: 14.9616 -0.754372 1.23998
7: 41.549079: 14.9587 -0.754459 1.22944
8: 41.546953: 14.9634 -0.754107 1.21955
9: 41.546443: 14.9639 -0.756277 1.20883
10: 41.546367: 14.9642 -0.755346 1.20899
11: 41.546356: 14.9640 -0.755282 1.20999
12: 41.546355: 14.9639 -0.755179 1.21025
13: 41.546354: 14.9639 -0.755227 1.21044
14: 41.546354: 14.9639 -0.755223 1.21033
15: 41.546354: 14.9639 -0.755222 1.21034
0: 268.33933: 11.6247 -0.916291 0.0250076
1: 90.874021: 11.8377 0.0103326 0.334835
2: 83.915219: 12.4485 -0.254936 1.08088
3: 82.438031: 12.5912 -0.0821163 0.683385
4: 82.122344: 12.5948 -0.155777 0.720669
5: 82.100766: 12.5345 -0.155579 0.765341
6: 82.098312: 12.5181 -0.160448 0.767942
7: 82.098247: 12.5178 -0.159771 0.766545
8: 82.098246: 12.5178 -0.159745 0.766449
9: 82.098246: 12.5178 -0.159745 0.766450
0: 78.461633: 16.5552 -0.916291 -0.261546
1: 45.668462: 16.9460 -0.0101555 -0.0998106
2: 42.434501: 17.2165 -0.251476 -0.0316525
3: 42.044121: 17.1927 -0.373741 -0.0920500
4: 41.968374: 17.2318 -0.315044 -0.211159
5: 41.942150: 17.2671 -0.342571 -0.201795
6: 41.937215: 17.2478 -0.338028 -0.177828
7: 41.936904: 17.2294 -0.338237 -0.178239
8: 41.936377: 17.2390 -0.336990 -0.180031
9: 41.936372: 17.2386 -0.337363 -0.179497
10: 41.936372: 17.2386 -0.337295 -0.179580
11: 41.936372: 17.2386 -0.337295 -0.179580
Figure 7
Figure 8
Error in (function (cond) :
error in evaluating the argument 'object' in selecting a method for function 'rebuildTime': subscript out of bounds
Error in (function (cond) :
error in evaluating the argument 'object' in selecting a method for function 'rebuildTime': subscript out of bounds
Error in (function (cond) :
error in evaluating the argument 'object' in selecting a method for function 'rebuildTime': subscript out of bounds
Error in (function (cond) :
error in evaluating the argument 'object' in selecting a method for function 'rebuildTime': subscript out of bounds
Error in (function (cond) :
error in evaluating the argument 'object' in selecting a method for function 'rebuildTime': subscript out of bounds
Error in (function (cond) :
error in evaluating the argument 'object' in selecting a method for function 'rebuildTime': subscript out of bounds
Error in (function (cond) :
error in evaluating the argument 'object' in selecting a method for function 'rebuildTime': subscript out of bounds
Error in (function (cond) :
error in evaluating the argument 'object' in selecting a method for function 'rebuildTime': subscript out of bounds
Error in (function (cond) :
error in evaluating the argument 'object' in selecting a method for function 'rebuildTime': subscript out of bounds
| .id | Beverton & Holt | Steepness Prior | Steepness=0.6 | Steepness=0.9 | Depensation |
|---|---|---|---|---|---|
| bss.27.8ab | 0.455 | 2.015 | NA | 2.068 | 0.328 |
| cod.27.6a | 0.381 | 1.352 | 0.723 | 1.565 | 0.375 |
| her.27.3a47d | 1.216 | 1.165 | 1.192 | 2.252 | 1.285 |
| mac.27.nea | 1.027 | 0.945 | 0.854 | 1.566 | 0.980 |
| mon.27.78abd | 0.987 | 1.264 | 0.532 | 1.155 | 0.835 |
| pil.27.8c9a | 2.267 | 4.416 | 2.838 | 8.067 | 1.696 |
| ple.27.420 | 1.031 | 2.500 | 1.261 | 3.570 | 0.871 |
| sol.27.4 | 1.767 | 0.194 | 0.180 | 0.956 | 1.484 |
| whb.27.1-91214 | 0.771 | 0.989 | 1.125 | 2.096 | NA |
| .id | Beverton & Holt | Steepness Prior | Steepness=0.6 | Steepness=0.9 | Depensation |
|---|---|---|---|---|---|
| bss.27.8ab | 18.010 | -0.200 | NA | -0.338 | 28.834 |
| cod.27.6a | 9.508 | 3.786 | 5.014 | 3.401 | 6.216 |
| her.27.3a47d | 3.102 | 3.517 | 3.385 | -0.249 | 2.333 |
| mac.27.nea | 4.115 | 2.934 | 3.776 | 0.259 | 3.498 |
| mon.27.78abd | 3.614 | 2.175 | 9.059 | 2.610 | 3.956 |
| pil.27.8c9a | -0.038 | -9.249 | -7.736 | -27.194 | 3.643 |
| ple.27.420 | 4.654 | -1.674 | 2.471 | -3.084 | 4.151 |
| sol.27.4 | 2.180 | 29.992 | 31.341 | 4.229 | 12.868 |
| whb.27.1-91214 | 7.136 | 5.951 | 5.179 | 1.439 | NA |
| .id | Beverton & Holt | Steepness=0.9 | Steepness=0.6 | Steepness Prior | Depensation |
|---|---|---|---|---|---|
| bss.27.8ab | 0.383 | 0.724 | 0.722 | 1.000 | 0.969 |
| cod.27.6a | 1.000 | 0.001 | 0.000 | 0.499 | 0.089 |
| her.27.3a47d | 0.001 | 0.030 | 0.000 | 0.028 | 1.000 |
| mac.27.nea | 0.000 | 0.550 | 0.008 | 1.000 | 0.429 |
| mon.27.78abd | 0.429 | 0.906 | 1.000 | 0.005 | 0.762 |
| pil.27.8c9a | 0.045 | 0.001 | 0.000 | 0.023 | 1.000 |
| ple.27.420 | 0.000 | 0.510 | 0.387 | 1.000 | 0.631 |
| sol.27.4 | 0.368 | 0.000 | 0.546 | 0.000 | 1.000 |
| whb.27.1-91214 | 0.351 | 0.721 | 0.218 | 0.598 | 1.000 |
.id s 1
1 bss.27.8ab 0.897 101489.41
2 cod.27.6a 0.793 32580.95
3 her.27.3a47d 0.579 39313400.88
4 mac.27.nea 0.643 5385707.23
5 mon.27.78abd 0.916 110879.42
6 pil.27.8c9a 0.769 25019950.00
7 ple.27.420 0.824 4090214.64
8 sol.27.4 0.605 111825.86
9 whb.27.1-91214 0.548 27321117.97 .id s 1
1 bss.27.8ab 0.897 101489.41
2 cod.27.6a 0.793 32580.95
3 her.27.3a47d 0.579 39313400.88
4 mac.27.nea 0.643 5385707.23
5 mon.27.78abd 0.916 110879.42
6 pil.27.8c9a 0.769 25019950.00
7 ple.27.420 0.824 4090214.64
8 sol.27.4 0.605 111825.86
9 whb.27.1-91214 0.548 27321117.97Fitting depensation (blue) using
sr=fmle(as.FLSR(x, model=bevholtDa), fixed=list(d=1.4),control=list(silent=TRUE))
is equivalent to the bev & holt (red) fitted using
fmle(as.FLSR(stks[[.id]],model=“bevholt”),control=list(silent=TRUE),
There are 3 things of importance i.e.
- scaling, e.g. virgin biomass and MSY
- ratio between BMSY & Virgin
- r and FMSY
As M, wt, mat are all held constant SPR0 is fixed, so the production function is determined by the SRR. i.e. 2 parameters for Bev Holt
In the case of depensation what does steepness mean? All the stuff at low SSB messes up everything on the left-hand limb of the production function r, FMSY, shape, …
I suggest therefore that we fix R0/Virgin, i.e. “a” of the Bev Holt, that way we keep the scale, and explore rates & shape (i.e. differences between targets & limits) i.e.
sr=fmle(as.FLSR(x, model=bevholtDa), fixed=list(d=1.4,a=a),control=list(silent=TRUE))