Reference Point Analysis
Western Mediterranean Red Mullet Example
L Kell
25 June, 2021
The aim of this vignette is to demonstrate how to evaluate the factors that affect productivity and hence reference points using Western Mediterranean red mullet as an example. To do this we devlop a simulation model (i.e. an Operating Model) conditioned on the latest assessment that can be used to simulate stocks under a variety of future conditions. However, unlike in Management Strategy Evaluation there is no feedback, where a Management Procedure is used to set a catch quota to update the Operating Model.
At the third Workshop on Guidelines for Management Strategy Evaluations (WGMSE3) performance statistics or summary metrics were defined as set of statistics used to evaluate the performance of Management Procedures against specified pre-agreed management objectives, and the robustness of these Management Procedures to uncertainties in resource and fishery dynamics of concern to stakeholders and managers. These are properties of the simulated system e.g. foregone catch relative to \(MSY\), or the level of a stock at which recruitment is impaired. There are two main ways to calculate the performance statistics, namely i) using equilibrium assumptions (e.g. Sissenwine and Shepherd (1987) for age based OMs); or ii) though stochastic simulation by projecting at \(F=F_{MSY}\) or F=0 (e.g. Carruthers et al. (2016), De Moor, Butterworth, and De Oliveira (2011)). The later approach is preferable where environmental forcing or resonant cohort effects impact on productivity.
The ICES reference points can be calculated using the msy package which can be found on github at github.com/ices-tools-prod/msy.
Installation
To run the the code in this vignette a number of packages need to be installed, from CRAN and the FLR website, where tutorials are also available.
Libraries
FLR
The FLR packages can be installed from www.flr-project.org
library(FLCore)
library(ggplotFL)
library(FLBRP)
library(FLasher)
library(FLife)
library(diags)
library(mydas)Packages
The examples make extensive use of the packages of Hadley Wickham. For example plotting is done using ggplot2 based on the Grammar of Graphics.1 Grammar is to specifies the individual building blocks and allows them to be combined to create the graphic desired2.
While ‘dplyr’ is a grammar of data manipulation, providing a consistent set of verbs that help solve the most common data manipulation challenges, while ‘plyr’ is a set of tools to split up a big data structure into homogeneous pieces, apply a function to each piece and then combine all the results back together.
library(ggplot2)
library(plyr)
library(dplyr)Results
Figure 1 summarises the ICES \(PA\) and \(MSY\) reference points (\(B_{lim}\), \(B_{pa}\), \(B_{trigger}\), \(F_{lim}\), \(F_{pa}\), \(F_{MSY}\)) by plotting them on the equilibrium curves and compares them to \(MSY\) and \(F_{0.1}\) based reference points; a stock recruitment relationship with a steepness of 0.99 was assumed. The plots allow the \(F\) and \(B\) reference points to be checked for consistency with each other. \(B_{trigger}\) is an overestimate of \(B_{MSY}\). The yield v SSB curve is shown in Figure 3.
The time series and reference points from the last assessment are shown in Figure 3. A main feature is that all the reference points imply a similar yield.
Productivity and reference points depend on the assumed life history parameters and selectivity-at-age, these are summarised in Figure 4. The changes in the parameters will affect productivity and the reference points as shown in Figure 5, where to explore the impact of the assumptions about the biology and selectivity on \(MSY\), estimates of \(F_{MSY}\), \(B_{MSY}\) and \(F_{MSY}\) are plotted by year in Figure 5. The blue line is when only selectivity was allowed to vary, green when only the biological parameters and red both. \(SPR_0\) is shown in Figure 6.
Figure 7 plots the production function, i.e. yield v SSB; the veritical lines are \(B_{lim}\) and \(B_{pa}\). The trajectory should cycle anti-clockwise, since if the catch is greater than production then the stock should decline, while if the catch is less that production it should increase. The fact that the stock and catch continues to increase in the recent period implies that production is driven by process error.
Process error and surplus production are plotted in Figure 8, it can be seen that in the recent period production has on average been larger than expected, i.e. process error is positive.
To explore this further the stock recruitment relatiionship is evaluated, Figure 9 is an unconstrained fit using a Beverton and Holt stock recruitment relationship. Residuals are plotted in Figure 10.
Figure 11 shows fits to the stock recruit pairs for different values of steepness, there is not a lot of information to the left of the data to estimate steepness, it appears that the fit is driven more by the upward trend in the data.
The conseqeunces for reference points is explored in Figure 12, these show that as steepness increases \(MSY\) increases and \(B_{MSY}\) decreases, therefore \(F_{MSY}\) will increase. The consequences for the ICES reference points is that if steepness is low then all the reference points are below \(MSY\) while if steepness is high they are above. The choice of steepness is therefore critical to the robustness of ICES advice. Figure 13 overlays the stock trajectory over the production functions.
Projections are shiwn in Figure 14 for the reference case and a shock scenario. Next summary outputs and performance metrics are showm.
Figure 14 shows some potential indicators
Reference Points
Figure 1. Reference points and equilibrium curves for Beverton and Holt stock recruitment relationship with a steepness of 0.99.
Figure 2. Yield v. SSB curve wirg reference points for Beverton and Holt stock recruitment relationship with a steepness of 0.99.
An object of class "FLPar"
params
Btrigger Fmsy Blim Bpa Flim Fpa lFmsy uFmsy
123.512 0.777 88.885 123.512 1.075 0.725 0.573 0.945
F05
0.777
units: t f t t f f f f t t t An object of class "FLPar"
quant
refpt harvest ssb yield rec
virgin 0.00e+00 5.79e+02 0.00e+00 1.00e+04
msy 4.11e+00 1.99e+01 1.31e+02 9.38e+03
crash 1.06e+01 3.70e-06 3.93e-04 2.54e-02
f0.1 4.42e-01 2.12e+02 1.01e+02 1.00e+04
fmax 2.45e+01 -1.46e+00 -4.75e+04 -2.99e+06
spr.30 5.87e-01 1.73e+02 1.08e+02 9.99e+03
Btrigger 8.87e-01 1.24e+02 1.16e+02 9.96e+03
Fmsy 7.77e-01 1.38e+02 1.14e+02 9.97e+03
Blim 1.27e+00 8.89e+01 1.21e+02 9.91e+03
Bpa 8.87e-01 1.24e+02 1.16e+02 9.96e+03
Flim 1.07e+00 1.04e+02 1.19e+02 9.94e+03
Fpa 7.25e-01 1.46e+02 1.13e+02 9.98e+03
lFmsy 5.73e-01 1.76e+02 1.08e+02 9.99e+03
uFmsy 9.45e-01 1.17e+02 1.17e+02 9.95e+03
F05 7.77e-01 1.38e+02 1.14e+02 9.97e+03
units: NA 
Figure 3. Time series, with ICES PA and MSY reference points, along with MSY estimated with a Beverton and Holt stock recruitment relationship.
Productivity
Figure 4. Weigth, selectivity, maturity and M-at-age.
Figure 5. Time series of MSY benchmarks
Figure 6. Time series of \(SPR_0\) benchmarks
Figure 7. Production function, with ICES reference points and stock trajectory.
Figure 8. Time series of surplus production and process error.
Stock and recruitment
Figure 9. Stock recruitment fit for Beverton and Holt
Figure 10. Recruitment residuals, with regimes.
Steepness
Figure 11. Stock recruitment relationship by steepness.
Figure 12. Production functions by steepness.
Figure 13. Production functions.
Projections
Figure 14. Projection at MSY.
Summaries
year iter ssb stock rec catch catchjuv fbar
20 2021 1 249.9610 567.8770 12284.74 118.0958 0.08308894 0.4422255
21 2022 1 256.5236 566.1401 10985.68 122.1443 0.07430267 0.4422255
22 2023 1 246.9077 574.3214 13678.04 118.8999 0.09251267 0.4422255
23 2024 1 265.3151 643.5406 17256.92 124.7826 0.11671878 0.4422255
24 2025 1 311.1831 690.7050 13499.80 144.4389 0.09130710 0.4422255
25 2026 1 303.9837 748.3804 20912.56 146.5291 0.14144402 0.4422255
swt cwt sage cage
20 0.03173228 0.08779144 0.4412548 1.533116
21 0.03408110 0.08992479 0.4860946 1.567352
22 0.03039395 0.09314786 0.4086289 1.616521
23 0.02772010 0.08781407 0.3600292 1.529730
24 0.03317692 0.08383066 0.4757342 1.471377
25 0.02727767 0.09132694 0.3487153 1.592366Performance metrics
iter amplitude responsiveness biolrisk recoverRate recoverSpeed
1 1 0.6670383 2027 0 TRUE 2025
2 1 0.6985860 2026 0 TRUE 2021
3 1 0.6797383 2026 0 FALSE 2021
4 1 0.7056091 2026 0 TRUE 2023
5 1 0.6894998 2026 0 TRUE 2022
6 1 0.6697426 2026 0 TRUE 2023
Figure 15. Potential indicators.
[1] "Mullus barbatus"Software Versions
- R version 4.0.3 (2020-10-10)
- FLCore: 2.6.16
- FLPKG:
- Compiled: Fri Jun 25 10:07:11 2021
- Git Hash: 71adb46
Acknowledgements
References
Carruthers, Thomas R, L.aurence T Kell, Doug DS Butterworth, Mark N Maunder, Helena F Geromont, Carl Walters, Murdoch K McAllister, et al. 2016. “Performance Review of Simple Management Procedures.” ICES J. Mar. Sci. 73 (2). Oxford University Press: 464–82.
De Moor, Carryn L, Douglas S Butterworth, and José AA De Oliveira. 2011. “Is the Management Procedure Approach Equipped to Handle Short-Lived Pelagic Species with Their Boom and Bust Dynamics? The Case of the South African Fishery for Sardine and Anchovy.” ICES Journal of Marine Science 68 (10). Oxford University Press: 2075–85.
Sissenwine, MP, and JG Shepherd. 1987. “An Alternative Perspective on Recruitment Overfishing and Biological Reference Points.” Can. J. Fish. Aquat. Sci. 44 (4). NRC Research Press: 913–18.
Wilkinson, L. 1999. The Grammar of Graphics, Springer. doi 10.1007/978-3-642-21551-3_13.↩