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Parameter uncertainty of a dynamic multi-species size spectrum1
model2
Michael A. Spence1,2,*, Paul G. Blackwell1and Julia L. Blanchard3
3
1School of Mathematics and Statistics, University of Sheffield, Sheffield, UK4
2Department of Animal and Plant Sciences, University of Sheffield, Sheffield, UK5
3Institute for Marine and Antarctic Studies, University of Tasmania, 20 Castray6
Esplanade, Battery Point. TAS. 70047
*Corresponding author: +44 114 222 4714, m.a.spence@sheffield.ac.uk8
1
Abstract9
Dynamic size spectrum models have been recognized as an effective way of describing how10
size-based interactions can give rise to the size structure of aquatic communities. They are11
intermediate complexity ecological models that are solutions to partial differential equations12
driven by the size-dependent processes of predation, growth, mortality and reproduction13
in a community of interacting species and sizes. To be useful for quantitative fisheries14
management these models need to be developed further in a formal statistical framework.15
Previous work has used time-averaged data to “calibrate” the model using optimization16
methods with the disadvantage of losing detailed time series information. Using a published17
multi-species size spectrum model parameterized for the North Sea comprising 12 interacting18
fish species and a background resource, we fit the model to time series data using a Bayesian19
framework for the first time.20
We capture the 1967-2010 period using annual estimates of fishing mortality rates as21
input to the model and time series of fisheries landings data to fit the model to output.22
We estimate 38 key parameters representing the carrying capacity of each species and back-23
ground resource as well as initial inputs of the dynamical system and errors on the model24
output. We then forecast the model forward to evaluate how uncertainty propagates through25
to population- and community-level indicators under alternative management strategies.26
Key words: Multi-species models; Bayesian Statistics; Uncertainty quantification;27
Ecosystem-based management28
2
1 Introduction29
There are a number of ecological models that can be applied to answer marine management30
questions (Plag´anyi et al 2014). An emerging class of marine ecosystem models is size spectrum31
models (Benoˆıt and Rochet 2004; Law et al 2009; Blanchard et al 2009). Size spectrum models32
are models of intermediate complexity and are formulated around the McKendrick von Foerster33
partial differential equation. Conceptually they are based on very simple ecological assumptions34
(Andersen and Pedersen 2009) about how the role of individual body size in a food web (“big35
individuals eat small individuals”) gives rise to community abundance (and biomass) size spectra36
(Hartvig et al 2011). Size-based predation leads to growth and mortality, which drive changes37
in the abundance of organisms along the size spectrum. Maturation is also size-dependent38
and once an individual reaches maturation size, it produces offspring (Hartvig et al 2011)39
that enter the model at the smaller sizes. Food for the smallest sized organisms is provided40
by a background community (representing phytoplankton, zooplankton and benthos) which is41
modeled as an external size-structured resource that is not driven by predation but instead42
follows semi-chemostat logistic growth (Andersen and Pedersen 2009; Roos et al 2008).43
Size spectrum models are increasingly being used to help us understand the structure of44
marine ecosystems and establish abundance baselines of marine communities and their responses45
to the potential effects of fishing and climate change (Benoˆıt and Rochet 2004; Blanchard46
et al 2009, 2012; Law et al 2009; Jacobsen et al 2013; Maury and Poggiale 2013; Woodworth-47
Jefcoats et al 2013; Law et al 2013). Several approaches exist spanning a wide range of model48
complexity: simple community models, trait-based models and more detailed multi-species49
models (Scott et al 2014). The generic community- and trait- based models have been used50
to develop theory (Benoˆıt and Rochet 2004; Andersen and Pedersen 2009; Hartvig et al 2011),51
to examine the community responses to fishing mortality and selectivity, and as a test-bed for52
evaluating indicators of the ecosystem effects of fishing (Rochet and Benoˆıt 2011; Zhang et al53
2014; Law et al 2013; Jacobsen et al 2013).54
Both size and species identity are important for fisheries management and the development55
of methods to parameterize trait-based models for real multi-species fish communities has been56
a recent focus of research, particularly for testing indicators and management strategies at both57
population and community levels. Blanchard et al (2014) parameterized and calibrated a trait-58
based model for 12 species in the North Sea using fisheries survey data and stock assessment data59
to determine whether meeting management targets for exploited North Sea populations would60
3
be sufficient to meet proposed Marine Strategy Framework Directive targets for biodiversity61
and food web functioning (including the “Large fish indicator”).62
Although trait-based models can be parameterized for real systems based on either the63
literature or statistical analyses of fisheries datasets, there are inevitably parameters that are64
uncertain that have to be estimated by fitting the model to data. For multi-species models to65
be useful for tactical management they need to be developed and tested in a formal statistical66
framework (Plag´anyi et al 2014). Uncertain parameters for the Blanchard et al (2014) multi-67
species size spectrum model included Rmax, the maximum recruitment for each species, and68
κ, the background food resource’s carrying capacity. To estimate these parameters the model69
was ‘calibrated’ to time-averaged spawning stock biomass (SSB) and landings data using time70
averaged fishing mortality from 1985-1995, by minimizing the sum of squared errors between71
the model and the data to find a single best parameter set. The model was cross-validated72
with survey data and then forced with time-varying fishing mortalities and scenarios to eval-73
uate whether single species FM SY management targets (the fishing mortality that lead to the74
maximum sustainable yield) would lead to recovery in food webs and biodiversity in the North75
Sea. Stochasticity was incorporated in the recruitment stage. Although the model produced76
realistic growth rates and species size distributions, some of the time series fits to SSB and77
landings were poor. This is partially due to the fact that time series data were not fully used78
to calibrate the model. An ideal calibration approach would enable time series data to be more79
fully utilized combined with a formal statistical framework for uncertainty.80
It is important to report uncertainty associated with model derived research findings when81
used for advising policy makers and environmental managers (Harwood and Stokes 2003). Un-82
certainty can be separated into four main types: parameter uncertainty, structural or model83
uncertainty, residual variation and data uncertainty. Parameter uncertainty comes from un-84
certain knowledge about parameters (Li and Wu 2006); structural uncertainty is uncertainty85
associated with the model itself caused by simplifications, uncertain processes or even numerical86
approximations; residual variation is the uncertainty caused by demographic and environmental87
stochasticity (Kennedy and O’Hagan 2001; Vernon et al 2010) and data uncertainty is often88
referred to as measurement or observation error. This can be transferred to the parameters but89
can be propagated through to the model output and can be caused by sampling biases or errors90
in data collection (Harwood and Stokes 2003).91
Parameter uncertainty has not been formally explored in dynamic size spectrum models92
4
although some work has been done with a length-based multi-species model (Thorpe et al93
2015) and an age-based model (Tsehaye et al 2014). To improve the utility of multi-species94
size spectrum models for supporting fisheries management, the parameter, model and data95
uncertainty need to be quantified. Here we further investigate the model of Blanchard et al96
(2014) (see the Supplementary Material for the model description) using a Bayesian framework,97
a more realistic error model and an improved estimation strategy to assess uncertainty from98
parameters and the data and demonstrate how this uncertainty can be included in evaluating99
multi-species effects of fisheries management scenarios.100
2 Methods101
In this section we describe the model parameters, their prior distributions and how the model102
outputs can be related to the observed data in a probabilistic way. We then describe the steps103
used to sample from the posterior distributions using a Markov Chain Monte Carlo (MCMC)104
algorithm (Gelman et al 2013) (see the Supplementary Material for details).105
Uncertain parameters106
In the multi-species model there are a number of uncertain parameters to estimate. For the107
inputs Rmax.i, where irepresents the species as described in Table 1, we specify priors in terms108
of ψi= log Rmax.i for i= 1 . . . 12 and ψ0= log κtaking ψi∼U(·|αi, βi), where αi< βi. So the109
prior densities for Rmax.i and κare p(Rmax.i|αi, βi) and p(κ|α0, β0) where110
p(x|α, β) =
exp(x)
β−αif exp(α)≤x≤exp(β)
0 otherwise.
For the present analysis, we represent identical priors for each species, by αi= 0 and βi= 50111
for i= 0 . . . 12 which means that they are not very constraining.112
The dynamic model requires a spin-up period, where the fishing mortality, Fi, is fixed,113
so that the model reaches a steady state before the fishing mortality is varied and output is114
collected in 1967, the first year of the empirical time series. It is not obvious what the fishing115
mortality should be whilst the model is in the “spin-up” period so we have added the spin-up116
fishing mortality as an additional parameter to estimate for each of the 12 species, [Fi]12
i=1. The117
spin-up period is used to run the model into the best fitting stationary states before the fishing118
5
mortality is varied. It does not make sense for Fito be negative so we decided on119
Fi∼Half-normal · |0,(1.824)2
for i= 1 . . . 12.120
We used the same fishing mortalities as Blanchard et al (2014) based on stock assessments121
(www.ices.dk) for the 12 species from 1967 to 2010. According to these inputs, the fishing122
mortality for Norway Pout in 2005 was 0. This is inconsistent with the fact that there were123
landings in that year. In order to estimate this we have added the fishing mortality of Norway124
Pout in 2005 as another parameter, ρ. We assumed that the zero value was likely due to a125
rounding error for Norway Pout so we used an informative prior on ρsuch that126
ρ∼Exp ·
1
0.34.
We elicited (see e.g. O’Hagan et al 2006) these values using expert knowledge from JLB by127
examining the 50th percentiles of the distributions and then confirming the priors graphically.128
Table 2 summarizes the uncertain parameters and their prior distributions.129
Likelihood130
The model was fit to landings data, Y(in tonnes), from stock assessments (www.ices.dk) for the131
years shown in Table 1 using a Bayesian framework. For an introduction to Bayesian Statistics132
see McCarthy (2007); for a more detailed review of the area see Gelman et al (2013). If the133
modeled landings, assumed to be the same as the catches (i.e. discards are ignored), were134
expressed as M(θ) where the unknown parameters are defined as θand the other inputs are135
implicit in M(·) then we assumed136
log Y= log M(θ)+Σ1
2
where Σ’s off diagonal elements are 0 and the diagonal elements are σ2
i(i= 1 . . . 12) and is a137
vector of standard normals (Nielsen and Berg 2014; Tsehaye et al 2014).138
All the variance parameters, σ2
i, had independent inverse-gamma prior distributions defined139
as:140
σ2
i∼Inv-Gamma(·|0.0001,0.0001)
for i= 1,...,12.141
The simulation model is a solution of partial differential equations (PDEs) which is in-142
tractable and is approximated by discretizing both time and size (Hartvig et al 2011). The year143
6
is divided into intervals of length δt and the PDEs are estimated at these points. Initially we144
experimented with δt = 1, the same value used by Blanchard et al (2014), i.e. the PDEs were145
estimated every year, and we found that the likelihood surface was very unstable and that often146
made a large difference to the model output. As δt decreases the numerical estimation becomes147
more accurate. Changing δt we found that the estimate stabilized at around δt =1
4. However148
as we decreased δt, the model took longer to run so we has the classic problem of efficiency149
versus accuracy.150
Exploration of the parameter space151
The model output, M(θ), from the 26 dimensional input space is not smooth, even with a152
low value of δt. It contains many local minima that an MCMC chain would get stuck in and153
the quality of the fit, as measured by likelihood or posterior density, varies by many orders of154
magnitude. Thus a standard MCMC algorithm would be unable to fully explore the parameter155
space in any reasonable time. To overcome this, our strategy involved firstly carrying out an156
extensive search of the space, followed by local optimisation and then a parallel tempering157
algorithm (Swendsen and Wang 1986).158
Our initial search could have been carried out by selecting completely random points in the159
parameter space. However, in view of the computational costs, we instead used a more efficient160
design for the selection of the points, Latin hypercube sampling (LHS) (McKay et al 1979). For161
efficiency, we also carried out these exploratory runs with δt =1
2.162
In the first round we used LHS to sample 50,000 parameter sets and evaluated the model163
at each of these, setting all of the σ2’s to 1, which is effectively using the sum of squared errors164
as a measure of how good a parameter set was.165
We then performed a second round of LHS around each of the ten best points found in round166
1. For each top-ten point (θ1, . . . , θ26), we applied LHS on the Cartesian product, j= 1,...,26,167
of the parameter intervals168
P−1
j(max {Pj(θj)−, 0}), P −1
j(min {Pj(θj) + , 1})
where Pj(·) is the prior cumulative distribution function of parameter j, and we took = 0.025.169
From the best 49 points from the second round, plus the point representing the parameters170
that Blanchard et al (2014) found, we then optimized using a Nelder-Mead algorithm (Nelder171
and Mead 1965) to find 50 high local maxima, capping the number of model runs in order to172
keep the computational effort down. This gave us 50 candidate points, fitting the data much173
7
better than randomly selected starting points, and we applied the Metropolis-within-Gibbs174
algorithm described in the Supplementary Material, running 50 chains starting from these local175
maxima, to explore their neighborhoods in the parameter space, using δt =1
4for accuracy and176
allowing σ2
1:12 to vary.177
We took the best 5 points and performed parallel tempering starting from these points (see178
the Supplementary Material for details). From the parallel tempering we found that two of these179
Metropolis-within-Gibbs runs identified a region fitting so much better than any others that180
effectively all of the posterior probability was associated with these two runs. The quality of181
the fit, and the posterior probability, associated with each of the other regions of the parameter182
space was so low in comparison that they had essentially no effect on the parameter estimates183
or uncertainties. The fit is also, of course, very much better than would be found by a na¨ıve184
random search; some further detail is given in the Discussion.185
To explore the consequences of alternative management strategies, we sampled 2500 param-186
eter sets from the posterior distribution, and for each set we ran the model until 2010 and then187
projected the model to 2050 under two contrasting scenarios: 1) a status-quo scenario where188
each species fishing mortality is held at 2010 levels, F2010, and 2) a single-species FM SY scenario189
suggested by ICES using the values shown in Table 1. To evaluate the uncertainty associated190
with population we estimated191
BFscenario
BF0
,
where Bis the total spawner biomass with the fishing mortality set to either FMSY or F2010
192
divided by the SSB at the baseline, F0, where the fishing mortality is 0 for the whole of the193
simulation (including the spin-up period). We also estimated the large fish indicator (LFI), the194
proportion of biomass of demeral fish that are >40cm in length, for each of the three fishing195
scenarios and the slope of the community size spectrum for demersal fish as described in the196
Supplementary Material.197
3 Results198
The results in this section are based on running the final MCMC chain from the previous section199
for 60 000 iterations and discarding the first 10 000 as burn-in.200
8
Posterior distributions201
We found that the marginal posteriors for the recruitment parameters are unimodal; summaries202
are shown in Figure 1 using violin plots (Hintze and Nelson 1998) and in Table 3.203
Many of the posterior distributions of the the fishing mortality parameters, F1:12, were not204
too dissimilar to their respective prior distributions, others were more concentrated (see the205
Supplementary Material).206
The variance parameters describe the estimated distribution of the error around the observed207
landings. These were close to zero (Figure 1), suggesting that the modeled landings captured the208
observed landings reasonably well on average. This was particularly the case for sole, whiting,209
plaice and saithe. The model was particularly poor at estimating gurnard landings; the error210
parameter for gurnard is omitted from Figure 1 because it is too big to plot on the same scale.211
The posterior mean fishing effort for Norway pout in 2005 was about 0.019, confirming our212
suspicion that there may have been a rounding error in either the landings or fishing mortality213
for that species.214
Time-series model output215
A comparison of the observed time series of the landings to the model output (Figure 2) showed216
that the model does a reasonable job of fitting the dynamics of the data. We more formally217
assessed how well the model fit the dynamics of the landings by calculating the values of σ2
i
218
relative to the variabilities of their respective landings. Figure 3 shows the posterior distribution219
of σ2
i/ιi, with ιibeing the unbiassed estimate of the variance of the landings,220
1
n−1
2010
X
t=1967
(Y(t)
i−¯
Yi)2,
where ¯
Yiis the mean landings for species i. We found lowest values of relative variance (meaning221
best fit) for sprat, Norway pout and plaice. Higher values of relative variances were for gurnard222
and dab, implying poorer fits. Figure 4 shows the model output for SSB for 9 of the species and223
compared it to single species stock assessments (www.ices.dk). This comparison is not intended224
to evaluate goodness of fit but rather to examine differences between our model predictions225
with the single-species model outputs. We found lower SSB for most of the species, except226
for sandeel, Norway pout and herring compared with single-species assessments. The temporal227
trends in SSB were broadly similar.228
9
Scenarios229
We simulated the model forward to 2050 under the two scenarios described in the Methods but230
the model was almost in a steady state by 2020. The results of these forecasts are shown in231
Figure 5.232
Under the status-quo scenario sprat, sandeel and cod were the most depleted with the233
spawner biomass of sandeel and sprat ranging from 0.368-0.394 and 0.307-0.317 of their respec-234
tive unexploited spawner biomasses. Cod is the most depleted, ranging from 0.133-0.094 with235
a 0.02 probability of being less than 0.1 of its unexploited biomass, which has been used as236
a threshold for collapse. Several species have a high chance of being higher than unexploited237
biomass, due to the much lowered biomass of cod resulting in prey release. Under the single-238
species FM SY scenario these species have higher probability of being closer to their unexploited239
values. Plaice, saithe and cod were the most depleted ranging from 0.324-0.496, 0.438-0.476240
and 0.486-0.514 of their respective unexploited values.241
The uncertainty is higher for some species, such as haddock under the status-quo (standard242
error is 0.155) and plaice under FMSY (standard error is 0.028), than others, such as sandeel243
under the status-quo (standard error is 0.002) and cod under FM SY (standard error is 0.004).244
Consistent with the findings of Blanchard et al (2014), the LFI did not differ under the two245
fishing scenarios (the median is 0.385 and 0.380 under the status-quo and FMSY respectively)246
whereas the FMSY scenario gave a much shallower size spectrum slope (the median is about247
-2.12) than the status-quo (the median is about -2.35) for all parameter sets.248
4 Discussion249
An ecosystem approach to fisheries management requires tools that can evaluate the risks of250
fisheries management actions on both target and non-target species. Although extensive work251
on model uncertainty has been carried out through simulation approaches such as management252
strategy evaluation, a wide range of ecosystem and multi-species models being used to support253
ecosystem advice rely on projections from single best-fitting parameter sets, ignoring parameter254
uncertainty, and are considered to be strategic or “big picture” rather than of tactical use to255
support management decision (Plag´anyi et al 2014). Robust estimates of uncertainty in model256
parameters are also important for reporting results of management scenarios to policy makers257
(Harwood and Stokes 2003). Few attempts have been made to explicitly address parameter258
10
uncertainty in more complex models (Thorpe et al 2015) and this study is the first to develop259
such a framework for multispecies size spectrum models. Multispecies size spectrum models are260
still in their infancy in fisheries and fall into the strategic category. Our methods demonstrate261
how this class of models can be developed further using a Bayesian framework. The key ad-262
vantage, as illustrated here through two simple fisheries scenarios, is that it is possible to make263
probabilistic statements of scenario outcomes which enable more informed assessments of risk.264
Fisheries landings data are often assumed to not contain error but in reality contain high265
uncertainty due to misreporting and discarding. Here, we treated the landings data as uncertain,266
assuming the model and data uncertainty result in independent Gaussian errors on the log scale.267
In addition to quantifying the uncertainty around the modelled landings, we also estimated268
variance parameters of the Gaussian errors for each species in the model. These parameters269
take into account the data uncertainty and the residual variability and can be interpreted as270
how well, on average, the model does at recreating the observations. A small value of σ2
imeans271
that, on average, the model recreates the landings of species iwell. If all of these parameters are272
the same then the likelihood of the observations is related in a simple way to the sum-of-squares273
metric used by Blanchard et al (2014). If the variance parameters are not equal, the appropriate274
metric becomes the weighted sum of squares, with a lower value of σ2
iimplying that observations275
on species ishould be more highly weighted. We found these variances clearly do differ across276
species and may be a possible reason why the points found by Blanchard et al (2014) are not in277
the posterior distribution (see Figure 1). Other reasons may be that Blanchard et al fitted their278
model with time-averaged SSB and landings data, whereas we account for temporal dynamics279
over the 1967-2011 period for landings only; we also fitted the model using dt = 1/4 instead280
of dt = 1. Blanchard et al’s choice of dt = 1 succeeded in fitting the equilibrium behaviour of281
the model, which is largely unaffected by dt, to the time-averaged data; their fitted parameter282
values were shown to capture time-averaged size distributions and growth rates from survey data283
well. However, for fitting to time-varying data or using time-varying [fishing mortality] inputs284
to predict time-series, it is necessary to describing the dynamics of the model in more detail,285
and hence to use a smaller value of dt. This does affect the likelihood surface and potentially286
the parameter estimates. Our experiments with fitting to SSB as well as landings (Spence 2015)287
did not give parameters close to Blanchard et al’s (despite starting one of the MCMC chains288
there). Blanchard et al also used a penalty function for species that went extinct when fishing289
mortality was zero; however, since all the species coexist with both fitted parameter sets, this290
11
is unlikely to have a substantial effect on the precise parameter estimates. Overall, it seems291
that the differences in species weighting, (i.e value of σ2) and the choice of dt are most likely to292
account for the parameter differences.293
In spite of these differences, the consequences of the two fisheries scenarios explored resulted294
in similar qualitative outcomes. Under the status-quo scenario both models showed agreement295
in cod being the most heavily depleted with the same smaller-bodied species (herring, whiting,296
plaice, haddock) having biomasses higher than their unexploited biomass. The latter is a feature297
that would not emerge from single-species models that ignore food web interactions. Under the298
FMSY scenario both models also show that cod biomass and the community size spectrum299
slope return closer to their unexploited levels, whereas the large fish indicator does not. The300
major advantage of the approach shown here is that the scenarios account for the range of likely301
parameters (as opposed to a single parameters set), enabling a probability distribution of the302
model outcomes formally linked to the parameter uncertainty.303
Thorpe et al (2015) use a multispecies length-structured model and show a stronger correla-304
tion between the response of the size spectrum slope and the large fish indicator than reported305
here. There are a few reasons that could explain this difference. First the Thorpe et al (2015)306
model differs from ours in terms of the dynamics. The model used here contains more complex307
dynamical feedbacks; the growth process is food-dependent and the dynamics are governed by308
a system of partial differential equations whereas growth is non-dynamic and with discrete time309
dynamics in the models used in Thorpe et al (2015). It is worth noting that Thorpe et al (2015)310
reported higher power of the size spectrum slope to detect a change over a 5 or 15 year fishing311
scenario compared to the LFI. Second, the species composition between models and inclusion312
in the calculation of the community metrics differed. Here, demersal species only were used to313
calculate community metrics (in keeping with empirical analyses, Fung et al (2012)) and, from314
further experiments, we found that the LFI is more sensitive to species subsetting than the315
slope of the community size spectrum.316
We are not limited to forecasting the SSB, LFI and size spectrum but can make forecasts,317
with robust measures of uncertainty, of any indicator that the model is able to predict. In318
Figure 4 we compared the model output and the SSB from single species stock assessments.319
Stock assessments use landings and survey data to estimate fishing mortalities and predict SSBs320
for each species separately, with different underlying assumptions across models. We used fishing321
mortalities from stock assessments as inputs to the multispecies model and fitted it to landings322
12
data. Because of the fundamental differences between single and multispecies models we a priori323
expected SSBs predictions to differ from single-species SSB estimates. The multispecies model324
predicts an overall higher SSB for sandeel than the single-species model, reflecting the need to325
meet predation requirements of larger fish in the model. With the exception of herring, lower326
SSBs were evident for several species which is a result of the higher and explicit dynamically327
changing predation mortality present in the multispecies model.328
In reality the North Sea was not in a steady state in 1967 which could be a reason why we329
do not fit the dynamics of the landings well for all of the species (as indicated by larger values330
of σ2
i/ιi). We could, instead of restricting the spin-up period to the set of steady states, look at331
all possible states of the model before the dynamical fishing mortality was added to the model.332
This may be difficult to do in practice. Another possible reason for some of the poorer fits is333
that we are assuming that landings and catches are equivalent. For some species there is likely334
to be a systematic difference between these two due to discards e.g. gurnard.335
The trend of the model simulations is the same for the most of the possible parameter336
values that make up the posterior distribution, i.e throughout the posterior we overestimate the337
landings at one time and always underestimate the landings at another. Further experiments338
(for details see Spence 2015) show that this is a feature of the model and is not sensitive to339
the parameter estimates. However, rather than assuming that the errors are independent and340
identically distributed, we could re-model the error structure so that the errors are correlated341
through time, possibly using an autoregressive model of order 1 (AR1; see for example Brockwell342
and Davis (2002)). We believe this would improve the representation of the errors.343
Figure 1 and Table 3 show no systematic pattern between the estimated maximum recruit-344
ment and asymptotic size as suggested in Andersen and Pedersen (2009) and Andersen and345
Beyer (2015). It is believed that Rmax changes over time, possibly due to changes in habitat346
and temperature that have occurred in the North Sea (Bigg et al 2008). We could include347
dynamic changes in Rmax by including it as the hidden state in a state space model (see e.g.348
Rabiner 1989). This approach could also be used to estimate other useful parameters and even349
the model inputs (such as the fishing mortality) for each year.350
We have used a carefully designed strategy, involving Latin hypercube sampling, numerical351
optimisation and parallel tempering methods, to explore a complex likelihood surface over a352
large parameter space as thoroughly and efficiently as possible. The high dimension of the space353
means that na¨ıve methods would perform very poorly, or be completely infeasible. For example,354
13
a simple systematic search with all combinations of two levels of each parameter would require355
226 or 67 108 864 runs of the model, and numerical integration over the parameter space even356
more. Numerical optimisation, with or without derivative information, and MCMC applied357
in isolation, would be hampered by the many local maxima, though it is worth noting that358
our MCMC algorithm performs well locally, and so there is little to be gained by varying the359
details of the sampler. One way of improving the posterior distribution would be to use more360
informative priors. This could be done by eliciting the parameters (O’Hagan et al 2006) or361
using simpler, more tractable models in order to produce priors (e.g. the single species model362
of Andersen and Beyer (2015)).363
As it stands, our overall strategy gives an enormous improvement over the results of even a364
relatively efficient single-stage Latin hypercube search. The best point out of the 50 000 sampled365
in the first round of our search had a log likelihood of -13 790.19 and in the MCMC round, the366
best point from the sampled posterior had a log likelihood of -322.08. Thus the likelihood itself367
is higher by a factor greater than 105000. As an informal interpretation, this means that the368
latter point represents a model that, using a simple model selection criterion such as the AIC,369
would be preferred statistically even if it involved thousands of extra parameters (whereas in370
fact it uses none). This leads us to believe that the method described here gives a good estimate371
of the posterior distribution, and certainly much better parameter estimates and uncertainties372
than in previous work (Blanchard et al 2014) or would be obtained with standard methods.373
Our analysis allows for parameter uncertainty and for observation error. As it stands, it374
does not allow for the effects of structural uncertainty due to imperfections or limitations of the375
model itself. That could be handled by adding a discrepancy term, δ(·), (Kennedy and O’Hagan376
2001) to the formulation under “Likelihood”377
log Y= log M(θ) + δ(θ)+Σ1
2.
Note that this is likely to have a similar effect to allowing for autocorrelation in the observation378
errors, as outlined above. The discrepancy term is used to allow for structural uncertainties.379
Such uncertainties are often caused by simplifications in the model, e.g. the dynamic model380
fitted here did not model discards.381
Another source of uncertainty in predictions is stochasticity in the model, not addressed382
here since the model we use is deterministic. With a stochastic model such as that of Andersen383
and Pedersen (2009), the principles of our approach would remain the same, but the details384
would differ. Instead of MCMC, we would need to use Approximate Bayesian Computation385
14
(Tavar´e et al 1997; Beaumont 2010); the inclusion of observation errors means that so-called386
exact ABC (Wilkinson 2013) or likelihood free MCMC (Wilkinson 2010) could be used. This387
approach would retain the key advantages of the analysis described here: proper allowance388
for parameter and observation and uncertainty, and its propagation through to predictions.389
More generally, this Bayesian predictive framework can be applied to a wide variety of models390
and ecosystems. The range of computational tools to permit this in practice is constantly391
increasing; Spence (2015) gives some recent examples. As an alternative to formalizing the392
discrepancy within a single model, a promising approach is to consider a number of distinct393
models collectively, forming a multi-model ensemble. This can improve understanding of the394
strengths and weaknesses of individual models, and potentially give better predictions and395
assessments of uncertainty overall. We are at present working on an ensemble that includes the396
current model as one of its members by considering discrepancy shared between models and397
specific to each model as used in climate modelling (for example Chandler 2013).398
Further work on model uncertainty with size-spectrum and other ecosystem models will en-399
able multi-species forecasts to be reported to decision makers in a manner that is comparable to400
single-species decision tables. This would help further develop the use of formal risk assessment401
in ecosystem approaches to fisheries management, which has been fairly limited to date but is402
a burgeoning area of research (Plag´anyi et al 2014).403
5 Acknowledgements404
This work was supported by the Engineering and Physical Sciences Research Council [grant405
EP/I000917/1, National Centre for Statistical Ecology] and by the Natural Environment Re-406
search Council and Department for Environment, Food and Rural Affairs [grant number NE/L003279/1,407
Marine Ecosystems Research Programme]. We would also like to thank Toni Ingolf Gossmann,408
Christopher Griffiths, Abigail Marshall, Beth Mindel, Philipp Neubauer and an anonymous409
reviewer for their useful comments on earlier version of the manuscript.410
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Tables522
iSpecies Name Landings F2010 FMSY
1Sprattus sprattus Sprat 1967-2010 0.31 0.2
2Ammodytes marinus Sandeel 1983-2010 0.36 0.2
3Trisopterus esmarkii Norway Pout 1983-2010 0.42 0.2
4Limanda limanda Dab 1967-2010 0.14 0.2
5Clupea harengus Herring 1967-2010 0.12 0.25
6Eutrigla gurnardus Gurnard 1967-2010 0.10 0.2
7Solea solea Sole 1967-2010 0.34 0.22
8Merlangius merlangus Whiting 1990-2010 0.27 0.2
9Pleuronectes platessa Plaice 1967-2010 0.24 0.25
10 Melanogrammus aeglefinus Haddock 1967-2010 0.23 0.3
11 Pollachius virens Saithe 1967-2010 0.38 0.19
12 Gadus morhua Cod 1967-2010 0.68 0.3
Table 1: The species used in the model and their data sets used to estimate the parameters as
well as the fishing mortality in 2010, F2010, and at the maximum sustainable yield, FMS Y , as
shown in Blanchard et al (2014).
20
Parameters Also Units Prior Notes
ψ1:12 log Rmax log(m−3g−1yr−1)U(·|0,50) Log of the maximum recruitment
for each species.
ψ0log κlog(gλ−1vol−1)U(·|0,50) Log of carrying capacity of
resource spectrum
F1:12 yr−1Half-normal(·|0,(1.824)2) The fishing mortality during the
spin-up period for each species.
ρyr−1Exp(·|1/0.34) The fishing mortality for Norway
pout in 2005.
σ2
1:12 Unitless Inv-Gamma(·|0.0001,0.0001) The standard deviation of the
error on the log landing.
Table 2: The uncertain parameters. ψ0:12 ,F1:12 and ρare needed to run the model and σ2is
the error between the model output and the observed landings.
21
ψ σ2
Species Mean SE Mean SE
Sprat 26.659 0.124 0.236 0.070
Sandeel 26.008 0.091 0.208 0.062
Norway Pout 30.684 0.326 0.212 0.085
Dab 23.108 0.126 0.304 0.071
Herring 26.556 0.145 0.355 0.084
Gurnard 25.381 0.422 3.049 0.861
Sole 22.948 0.087 0.059 0.14
Whiting 26.034 0.158 0.099 0.035
Plaice 30.562 0.307 0.046 0.011
Haddock 28.375 0.252 0.269 0.067
Saithe 26.920 0.177 0.078 0.019
Cod 22.767 0.125 0.268 0.065
Background Resource 25.210 0.056
Table 3: The means and standard errors of the marginal posterior distributions of ψ0:12 and
σ2
1:12 rounded to 3 decimal places.
22
Figures523
1 (a) gives the marginal posterior distribution for ψ0:12 with the estimates from524
Blanchard et al (2014) being the points. The units for ψ1:12 are m−3g−1yr−1and525
gλ−1vol−1for ψ0. (b) gives the estimates of the error parameters for all of the526
parameters except Gurnard which is very uncertain and has a mean of 3.05 and527
variance of 0.75. The order of the species is that of their asymptotic size. . . . . 24528
2 Runs of the model with parameters sampled from the posterior distribution. The529
grey line shows the median model output, the dotted lines are the 5th and 95th530
percentiles for the model output and the thick black line is the observed landings. 25531
3 The posterior distribution for σ2
i/ιiwhere ιiis an unbiassed estimate of the532
variance of the observed log landings for species i. ................. 26533
4 log SSB of the model with parameters sampled from the posterior distribution.534
The grey line shows the median output, the dotted lines are the 5th and 95th535
percentiles for the model output and the think black line is the log SSB estimates536
fromstockassessments. ................................ 27537
5 The forecast for 2020. (a) shows the spawning stock biomass with the fishing538
mortality at that of 2010 (grey) and FMS Y (black) divided by the spawning539
stock biomass when the fishing mortality is 0 for the whole of the simulation. (b)540
shows the large fish indicator (LFI) for the fishing mortality equal to 0 (white),541
FMSY (black) and that of 2010 (grey). (c) is the same but for the community542
sizespectrumslope. .................................. 28543
23
log(units)
24 28 32
(a)
σ2
0.0 0.5 1.0 1.5 2.0
Sprat
Sandeel
N. pout
Dab
Herring
Gurnard
Sole
Whiting
Plaice
Haddock
Saithe
Cod
Background resource
(b)
Figure 1: (a) gives the marginal posterior distribution for ψ0:12 with the estimates from Blan-
chard et al (2014) being the points. The units for ψ1:12 are m−3g−1yr−1and gλ−1vol−1for ψ0.
(b) gives the estimates of the error parameters for all of the parameters except Gurnard which
is very uncertain and has a mean of 3.05 and variance of 0.75. The order of the species is that
of their asymptotic size.
24
1970 1980 1990 2000 2010
11.0 12.0 13.0
Sprat
11.0 12.0 13.0
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
12.0 13.0 14.0
Sandeel
12.0 13.0 14.0
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
8910 12
N. pout
8910 12
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
8.0 8.5 9.0 9.5
Dab
8.0 8.5 9.0 9.5
1970 1980 1990 2000 2010
1970 1980 1990 2000 2010
10 11 12 13
Herring
10 11 12 13
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
246810
Gurnard
246810
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
9.6 10.0 10.4
Sole
9.6 10.0 10.4
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
9.6 10.0 10.4 10.8
Whiting
9.6 10.0 10.4 10.8
1970 1980 1990 2000 2010
1970 1980 1990 2000 2010
11.4 11.8 12.2 12.6
Plaice
11.4 11.8 12.2 12.6
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
10.5 11.5 12.5 13.5
Haddock
10.5 11.5 12.5 13.5
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
11.4 11.8 12.2 12.6
Saithe
11.4 11.8 12.2 12.6
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
11.0 12.0
Cod
11.0 12.0
1970 1980 1990 2000 2010
year
log Landings
Figure 2: Runs of the model with parameters sampled from the posterior distribution. The grey line shows the median model output, the dotted
lines are the 5th and 95th percentiles for the model output and the thick black line is the observed landings.
25
Sprat
Density
0.1 0.3 0.5 0.7
0123456
Sandeel
Density
0.5 1.5
0.0 0.5 1.0 1.5
N. pout
Density
0.0 0.4 0.8 1.2
0246
Dab
Density
12345
0.0 0.4 0.8
Herring
Density
0.2 0.6 1.0
01234
Gurnard
Density
0.5 1.5
0.0 1.0 2.0
Sole
Density
0.5 1.5
0.0 1.0 2.0
Whiting
Density
0.5 1.5
0.0 1.0 2.0
Plaice
Density
0.2 0.6 1.0
01234
Haddock
Density
0.2 0.6 1.0
0123
Saithe
Density
0.2 0.6 1.0 1.4
0.0 1.0 2.0 3.0
Cod
Density
0.4 0.8 1.2 1.6
0.0 1.0 2.0
Figure 3: The posterior distribution for σ2
i/ιiwhere ιiis an unbiassed estimate of the variance
of the observed log landings for species i.
26
1970 1980 1990 2000 2010
11 12 13 14
Sandeel
11 12 13 14
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
10.5 11.5 12.5
N. pout
10.5 11.5 12.5
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
11 12 13 14
Herring
11 12 13 14
1970 1980 1990 2000 2010
1970 1980 1990 2000 2010
10.0 10.5 11.0
Sole
10.0 10.5 11.0
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
10.5 11.5 12.5
Whiting
10.5 11.5 12.5
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
10.5 11.5 12.5
Plaice
10.5 11.5 12.5
1970 1980 1990 2000 2010
1970 1980 1990 2000 2010
10.5 11.5 12.5 13.5
Haddock
10.5 11.5 12.5 13.5
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
10.0 11.0 12.0 13.0
Saithe
10.0 11.0 12.0 13.0
1970 1980 1990 2000 2010 1970 1980 1990 2000 2010
10.5 11.5 12.5
Cod
10.5 11.5 12.5
1970 1980 1990 2000 2010
year
log tonnes
Figure 4: log SSB of the model with parameters sampled from the posterior distribution. The grey line shows the median output, the dotted lines
are the 5th and 95th percentiles for the model output and the think black line is the log SSB estimates from stock assessments.
27
0.1 0.25 0.5 124
BFscenarioBF0
Sprat
Sandeel
N. pout
Dab
Herring
Gurnard
Sole
Whiting
Plaice
Haddock
Saithe
Cod
(a)
F0FMSYF2010
0.36 0.40 0.44
LFI
(b)
F0FMSYF2010
-2.40 -2.30 -2.20 -2.10
Slope
(c)
Figure 5: The forecast for 2020. (a) shows the spawning stock biomass with the fishing mortality
at that of 2010 (grey) and FMSY (black) divided by the spawning stock biomass when the fishing
mortality is 0 for the whole of the simulation. (b) shows the large fish indicator (LFI) for the
fishing mortality equal to 0 (white), FMS Y (black) and that of 2010 (grey). (c) is the same but
for the community size spectrum slope.
28
