MSY and evaluating trade-offs for anchovy fishery
39. Constrained optimization of fishing effort
39. Constrained optimization of fishing
species i by fleet j, and is also the Ecopath base fishing mortality rate for i,j). Suppose that for each species i, there is a target or maximum allowable fishing mortality rate FTARGETi, summed over all fleets that cause fishing mortality (landings and/or discards) on i. Suppose that we assign an “importance weight” vj to effort by fleet j, where vj reflects the relative value of increasing (or maintaining) effort for fleet j because of its contribution to overall long term ecosystem value and/or its legal entitlement to fish. Then the linear programming optimiza- tion can be formulated simply, as, maximize,
by varying the efforts Ej subject to the constraints,
That is, try to make the efforts Ej as large as possible without exceeding EOPTj while allowing efforts of at least EMINj and not allowing the sums of qijEj to exceed target fishing rate FTAR- GETi for any species i. An alternative formulation for recognizing fishery development rate constraints would be to replace the first set of nf constraints (Eq. 2) with EMINj < Ej < EMAXj. Where the EMIN and EMAX are allowed to vary from year to year by limited increments from the previous year’s values, and are not allowed to exceed EOPT.
Figure 1 illustrates the kind of complicated solutions that can arise from this optimization, even for a very simple case where two fishing fleets pursue just two species, with one fleet having higher catchability for one of the species and the other fleet having higher catchability for the other species.
Figure 1. Graphical representation of the linear programming problem. Each line represents a constraint (vertical and hori- zontal lines are the EOPT constraints, sloped lines are the FTARGET constraints.
The lines in Figure 1 represent effort levels that exactly meet the constraints; efforts must be to the left and below each line in order to be feasible. Thus the feasible effort combinations are only those in the polygon from the graph origin out to the first constraint lines met. Since efforts are to be as large as possible, the solution has to lie along one of those first lines met, and in fact has to be on one of the three vertices marked A, B, C.
• Effort combination A represents Fleet 1 being severely restricted, but Fleet 2 operating at its EOPT.
• Effort combination C represents Fleet 2 being severely restricted but Fleet 1 operating at its EOPT, and
• Effort combination B represents a “balanced” policy choice where both fleets are restricted to below EOPT in order to “share the burden” of avoiding exceeding either of the two FTARGET species constraints.
Which of these three combinations will be chosen (solve the linear programming maximiza- tion) depends on the value weights vj in Eq. 1. If v1 is much higher than v2, combination C will be chosen, combination A will be chosen if v2 is larger, and setting the two v’s equal will more likely lead to the balanced combination B. Note also in Figure 1 that as the FTARGET con- straints are “relaxed” (increased so the sloped lines move upward and to the right), it becomes more likely that the optimum effort combination will lie near EOPT for both fleets; likewise, as these constraints are “tightened” (reduced so the sloped lines move down and to the left), it becomes more likely that the EOPTs will not be in the feasible region so that the optimum solution will lie either with a mixed effort combination or with one or another of the fleets shut down entirely.
Using the linear programming formulation, it is simple to evaluate the cost, in terms of lost total value, of introducing more restrictive constraints on species harvest rates. In the overall management strategy evaluation setup for Ecosim, the only other way to evaluate this cost is to do policy runs with and without “weakest stock” constraints on fleet quotas, where all fleets are assumed to share equally in reductions needed to meet such constraints. The linear pro- gramming solution may well demonstrate that such equal sharing of the conservation burden is in fact far from optimum.
The linear programming formulation can also be used to demonstrate potential increases in fishery value from selective fishing practices that change the species-specific catchabilities qij. For example, if q12 and q21 (catchability of species 1 by fleet 2 and of species 2 by fleet 1) could be greatly reduced in the Figure 1 example, the slopes of the two FTARGET constraint lines would decrease/increase so as to move the solution toward higher total efforts (move point B up and to the right, closer to the EOPT1-EOPT2 intersection) and thus higher total value.
The key to getting useful results from the linear programming exercise is to make wise choice of the fleet value weights vj. One objective option for doing this each year (assuming a man- agement strategy where biomass and perhaps catchability qij estimates are being updated reg- ularly) is to set each weight to be
where Pij is the landed price for species i by fleet j and Bi is the current estimated biomass of
species i. Using this formula, vj is just the sum over species of predicted catches per effort times prices, so that vjBj represents the (short term) predicted total value of landings by fleet j and the overall linear programming objective function just becomes the predicted total landed value of all catches.
An option in the management strategy evaluation interface allows users to replace the compli- cated older EwE and CEFAS rules for setting quotas to limit mortality rates to acceptable lev- els, to instead use annual LP optimization each year during each MSE simulation run so as to limit fishing efforts rather than quotas. The basic idea is just to replace the single Ej values in the equations above with time-varying values calculated for each year and group using annual values for the FTARGETi target fishing rate for each group i for that year, calculated from the estimated biomass of the group and the group’s harvest control rule that specifies how target fishing rate for the group should vary with stock size. This option can lead to complex policy changes over time, especially when the FTARGETi decrease with decreases in biomasses Bi.
Notes
1. Murawski, S.A. and J.T. Finn. 1986. Optimal effort allocation among competing mixed-species fish- eries, subject to fishing mortality constraints. Can. J. Fish. Aquat. Sci. 43: 90-100. https://doi.org/
10.1139/f86-010
Tutorial: Stock recovery scenarios
Learning Objectives
• Get experience with how to consider the combined impact of fisheries, envi- ronmental conditions, and food web structure in a simple ecosystem model
• Get experience with a more complex procedure for time series fitting
The cod population in Anchovy Bay has been depleted, and there is concern for its recovery.
Assessments in the 1980s indicated overexploitation to be the cause of the population decline, and the trawl fishery for cod was closed in 1990, despite concerns for the socio-economic consequences. It was some comfort, however, that the stock was predicted to rebound within a few (cod) populations, perhaps in a decade or so.
By 2010, the cod population, however, had shown little sign of recovery. In this tutorial, we use a previously constructed, (but slightly modified) ecosystem model of Anchovy Bay to evaluate alternative hypotheses for why the cod population has not recovered to its 1970-level in spite of a strong reduction in fishing pressure.
Download the Anchovy Bay Cod Recovery.ecomdb database along with the cod recovery.csv time series file from this zip file. Open the EwE software and load the Anchovy Bay model, the Ecosim scenario, and import the cod recovery time series file. Then run Ecosim, and note the Summed Squared residuals (SS, on the run screen, top left corner). You can scroll through the groups, to see the trajectory for each (along with the groups contribution to the SS). Also, open the Group plot form (Ecosim > Output > Ecosim group plots), and examine the plots for each of the functional groups/species. You will notice that the fits to time series are pretty poor – which shouldn’t be surprising as we are only starting the fitting process now.
For the fitting, we will consider the combined impact of fisheries, environmental conditions, and food web structure. We do that in a semi-structured manner.
Predictions of the impact of fisheries will, in any model, depend on density-dependent factors.
In Ecosim, the most important factor is the vulnerability multiplier. Vulnerability multipliers express how much the predation pressure that a given predator causes on its prey can be increased if the predator was to increase to its carrying capacity. If the predator is at carrying capacity, it cannot increase the predation pressure on its prey (that’s what being at carrying capacity means), so the vulnerability multiplier should be 1. If, on the other hand, the preda-
tor has been depleted, the vulnerability multipliers should be higher. The default setting for vulnerability is 2, i.e. a predator can at most double the predation mortality it’s causing on its prey.
If you examine the Ecosim group plots after the first run, you’ll notice for seals that the time series indicate a strong increase (7x) in the seal population, while Ecosim indicates less than a doubling. Why? The default vulnerability is part of the problem. Try increasing the vulner- ability multipliers for seals as consumers, (Ecosim > Input > Vulnerabilities, click the column heading for column 2 (i.e. for cod as consumer), enter, e.g., 10 in the Set input box in the top right corner, and click Apply).
Run Ecosim again, and check the trajectory for seals. Better? You can also try to lower the vulnerability multipliers, e.g., to 1.1 and see what happens. By the way, the vulnerability multi- pliers scale from 1 to infinity, it can never be lower than 1, that would mean that the predator had exceeded its carrying capacity in the Ecopath baseline). Most of the ‘action’ is in the 1-20 range, as you increase the vulnerabilities beyond that it gradually has less and less impact.
Note that the Ecosim time series fitting may come back with very high vulnerability multipli- ers – it may be that changing the multiplier from, e.g., 100 to 100,000 decreases the SS a tiny tiny bit. If that happens, the best is to manually reduce the multipliers and check if it makes any difference in the SS.
For cod, we know that it has been exploited as a target fishery in Anchovy Bay for more than a century, so it would not have been close to its carrying capacity in 1970, (the year for which the Anchovy Bay ecosystem model was constructed, which provides reference points for vul- nerability multipliers and other settings). So, to improve the fit, how should you change the vulnerability setting for cod as a consumer? Try it.
The above goes to show that vulnerability multipliers are not ‘nuisance’ parameters, they have a clear interpretation that makes sense from an ecological perspective, and could in principle be estimated independently of the ecosystem model. The main hurdle for this, however, is that while carrying capacity is on old and well-founded concept, it changes, every day, so it would be difficult, (but perhaps not impossible) to estimate it independently – this factor is indeed what most single-species assessment estimate (Bt/Bo) – though with little basis in reality.
Therefore, our best option is to use constraints in our model, to estimate the density-depen- dent vulnerability multipliers. We can do this using ‘observations’, made accessible to the model through time series files. In our case (cod recovery.csv), it is rather restricted what we have: seals have increased, cod declined and have not recovered, whiting have increased a bit, and shrimp catches have increased. The principle here is: the more information we have, the more constraints this pose for the model. Therefore, the more data, the more difficult the fit- ting becomes, but the more confident we can be about the model behavior.
To use the time series for fitting, go to Ecosim > Tools > Fit to time series. Click Search groups with time series, and Search. Ecosim will now run a time series fitting, trying to find vulnera- bility multipliers that minimizes the SS. You will likely see some reduction in SS, but nothing
spectacular. Next, go back to Ecosim, Output, Run Ecosim, and make a run. Examine the group plots. You will likely find that the fit for seals is good, but not the fit for cod, which likely have recovered to the 1970-level as a result of the lower fishing pressure in Anchovy Bay since 1990.
Also, shrimp catches don’t even get near to matching the level from the time series.
In conclusion for where we are so far, time series fitting to evaluate the impact of fishing pro- vides some information, but doesn’t explain why cod hasn’t recovered. Is it the environment that has changed then? Reset the environmental forcing function by going to Ecosim > Input >
Forcing functions, click the 1: Fitting function, and click Reset.
Next question is: Is it the environment then? To evaluate this, we need information about how the environmental productivity of Anchovy Bay has changed since 1970, with the most impor- tant indicator being primary production. Unfortunately, such long-term information is hardly ever available as oceanographers tend to run their models for short time periods only. There is indeed a gap between oceanography and fisheries, and we need to fill it.
In lieu of environmental productivity data, we can ask Ecosim to estimate a ‘primary produc- tion anomaly’ (PPA), i.e. how might relative primary production have to have changed over time to fit the time series better. First, go to Ecosim > Input > Forcing function > Apply FF (pro- ducer), click the spreadsheet cell for phytoplankton, select 1: Fitting, and click the arrow to the right to apply this forcing function. This only means that you’ve associated primary pro- ductivity with the forcing function. Go to Ecosim > Tools > Fit to time series, and click Search groups with time series, click Vulnerability Search (i.e., checked), click Anomaly Search. Then click the Search, Anomaly Search tab. You should now see the 1: Fitting forcing function on the form. Next, increase the Spline points on the form, e.g., to 8, and click Search. Ecosim now starts a search, resetting vulnerability multipliers and evaluating the combined effects of den- sity dependence (vulnerabilities) and environmental productivity changes (primary produc- tion anomaly).
The SS will likely have decreased somewhat (keep track!), but not a lot, so what has it done?
Go back to Ecosim > Output > Run Ecosim, and make a new run – the vulnerability multipliers and primary production anomaly from the search have been transferred there already. Exam- ine the Ecosim group plots, check the estimated vulnerabilities and the primary production pattern. You’ll find that there isn’t much improvement for cod. Why?
One part of the answer is that the time series fitting puts the same weight on all of the time series as entered. There’s a weight attributed to each, and the time series file we read in had a weight of 1 for all. If you really want the search to prioritize cod, you could give the cod bio- mass time series a higher weight, perhaps 10 or even 100. The downside is that you’d be twist- ing your ecosystem model in the direction of a single-species model.
What then?
Perhaps cod reacts differently to environmental change than the ecosystem overall? To eval-
uate this, let’s consider how temperature impacts cod (they like cooler water), and fortunately temperature is usually one of the time series we may obtain from the oceanographers.
If you look in Ecosim > Input > Forcing functions, you should find a T bottom time series. Let’s apply this to cod. Go to Ecosim > Input > Functional responses, here there should be a Temp cold environmental response function, which we’ll use for cod. Click Ecosim > Input > Func- tional response > Apply functional responses, click the cell intersecting Cod juv. with T bottom, and transfer Temp cold to Applied responses. Do the same for Cod ad. Does it improve the fit? Some, but cod is likely still not recovering to the 1970-level.
Fit to time series again, use the Search group with time series option, but add shrimp as con- sumer to the fitting (select a non-used color, and click the column heading for group 10).
Examine the fit. You’ll likely find that seals increase more in Ecosim than in the time series.
The search has chosen a higher vulnerability for seals in order to get more increase and there- fore more predation pressure on cod, to help keep that group down. If you examine the diet composition for seals, you’ll see that cod is a very minor component, but this represents a high predation pressure on cod.
Next, examine the diet compositions. You’ll see the whiting does not eat cod. That’s unlikely to be correct, so try including it. For instance, by letting juv. whiting take 0.005 juv. cod (and change the proportion of zooplankton in the juv. whiting diet to 0.9. For ad. whiting change the proportion of juv. cod to 0.03, for ad. cod to 0.01, and for benthos to 0.34. Run Ecosim, then do a new fit to time series, again with fitting for groups 2, 4, 6, 10, i.e. groups with reference time series including shrimp, (which has a catch time series).
How does this look? Does cod recover now?
You likely have a pretty good fit now, examine it, each group, vulnerability multipliers, … Think about how you got the fit. Also, try to get the predictions to break down again. Play! But reflect on what you’re doing and notice what effect you see for different scenarios.
Even if you now have a good fit, try one more thing. Redo the time fit, but this time also include an Anomaly search for a primary production anomaly (using spline points, still 8 perhaps). Do the search, and when it’s done compare to the derived forcing function (1: fitting) to the forc- ing function that was actually used when constructing the model (3: True PP). You’ll likely find some resemblance between the two shapes, but also that the (1: fitting) shows much more variation. The reason for this is that the search criteria is chasing observations, and can do that without penalty as the time series we use in this example have very little constraints. You can get an idea about this by going back to Ecosim and do a run. You’ll likely see some strange things happen in between years with observations
Primary production (PP) should preferably not change over time with more than perhaps +/- 20% or so, and in this case the change was likely much more. Try setting the PP Variance to 0.01, (which will provide a much lower prior for the sampling). This likely caused much less variation in the PP anomaly plot.