In Chapter 1, I deﬁned the research problem and methods to be used for com- pleting this project where I introduced the Deriso and Schnute model to lay down the foundation for modelling the growth of ﬁsh populations.

In Chapter 2, I reviewed basic probability theory, frequentist and Bayesian paradigms, dynamic Bayesian networks, non-parametric models and time series modelling for handling uncertainties in ﬁsh populations.

In Chapter 3, I developed and implemented methods for identifying the non- constant variance (heteroscedasticity) in the spawner-recruit relationship. I found

6.2. Thesis summary 155

heteroscedastic models tend to ﬁt the S-R model inputs better than constant vari-
ance models across the majority of stocks, and strong evidence for a negative co-
*eﬃcient of heteroscedasticity in seven cases (Table A.1), including exploited cod,*

*herring and whiting stocks in addition to olive ﬂounder and Peruvian anchoveta.*

I advocate that the non-constant variance parameter in these cases deserves to
be taken into account by managers. In contrast, only one stock was identiﬁed as
having a positive coeﬃcient of heteroscedasticity at the 95% conﬁdence levels.
*To determine whether I can reliably estimate the sign of η*1, I tested whether the

conﬁdence interval lies in a region showing a consistent sign with the coeﬃcient where I found that both frequentist and Bayesian methods led approximately to equivalent inference.

To reliably identify a negative coeﬃcient of heteroscedasticity, managers or ﬁsh-
eries scientists using the frequentist methods should check that their chosen con-
ﬁdence interval lies in the negative region; those using the Bayesian framework
*can consider the proposed priors (i.e. π*1 *or π*2) as a non-informative benchmark

prior and check whether their Bayesian credible interval lies in the negative re- gion. I note that Bayesian approaches may be particularly useful where priors can be speciﬁed based on information about similar stocks in other locations. To protect this work against false positives or negatives, I recommend ﬁsheries scientists to use both frequentist and Bayesian methods when assessing stocks for heteroscedasticity; if both methods agree then there would be strong support for our conclusion being correct; otherwise they should investigate the limitation of each method separately.

In Chapter 4, I extended the analysis of Chapter 3 by applying a Bayesian hier-
*archical model for assessing the sign of the coeﬃcient of heteroscedasticity η*1. I

found that the Bayesian hierarchical model applied to ﬁsh stocks living in a com-
munity has reduced the uncertainty in parameter estimates of the S-R relationship
compared to the case where the analysis is based on a single stock assessment.
I proposed four diﬀerent S-R relationships based on the Deriso-Schnute (Deriso,
1980; Schnute, 1985) model. These models (*M*1, *M*2, *M*3 and *M*4) are assessed

on ﬁve diﬀerent geographical areas: Celtic sea, Faroe Plateau, Georges Bank, North-East Arctic and North sea; and three other macro scale marine column zones: pelagic, demersal and all populations. The macro scale marine analysis is aimed to assess the inﬂuence of grouping ﬁsh species, across diﬀerent water depths, on the prediction accuracy of ﬁsh populations.

Results showed that the coeﬃcient of heteroscedasticity deserves consideration
in Faroe Plateau, North-East Arctic, pelagic and demersal communities because
it had a better predictive accuracy on the test set; in contrast to the constant
*variance model (η*1 = 0) that is found to have some signiﬁcance for the Celtic

Sea, the North Sea and all populations as it provided a higher prediction accu-
*racy. The reliability of the sign of η*1 is found to be approximately consistent in

two regions: Faroe Plateau and North-East Arctic with an approximate credible interval of 95% and 70% respectively.

*Because the Deriso-Schnute model presents a singularity at γ = 0, I partitioned*
*the search space into three disconnected zones (γ < 0, γ* *≈ 0, γ > 0) so as to over-*
come this limitation. The models are analysed along with diﬀerent constraints
*on γ; whereas the assessment on the test set helped us to identify the model with*
best prediction values. The model*M*1 *(with γ > 0 and γ≈ 0) is found to provide*

the best recruitment prediction for the Faroe Plateau, North-East Arctic, pelagic
and demersal water columns; *M*2 *(with γ < 0) provided best accuracy for the*

Georges Bank; *M*3 *(with γ < 0) found to provide the best prediction for the*

Celtic sea; and the model *M*4 *(with γ > 0) found to provide the best prediction*

for the North sea and all populations. I checked whether the prediction accuracy of ﬁsh recruitment is aﬀected by pooling multiple populations across the water column zones; my results showed that estimating ﬁsh recruitment by restricting the analysis to species within their own community (i.e. pelagic or demersal) has a higher accuracy than pooling all ﬁsh populations together.

In this work, I provide to ﬁshery managers (or policy makers) a new way of assess- ing the size of ﬁsh recruitment: I advocate the importance of collecting as many ﬁsh stock assessments within the same community as they can, then apply them to the four (or other) possible models so as to check the one that has the lowest RMSE value in recruitment prediction. A more compelling reason to conduct stock assessment selection based on water column is because I found an increase in accuracy compared to the case of pooling pelagic and demersal populations together.

On the ecosystem level I conclude that the sea surface is to some extent dis- connected from the sea bed in the sense that the water column mixing does not aﬀect enormously nutrient supplies of these two habitats: I found that ﬁsh stock assessments are best analysed on their own community.

6.2. Thesis summary 157

In Chapter 5, I aimed to provide a statistical model capable of analysing ma- rine ecosystem response to environmental perturbations. The model consisted of analysing marine ecosystem data from several sources, with diﬀerent temporal resolutions, bounded by the Northwest Coast of Scotland and Northern Ireland (Division VIa). This geographical area can have temperate oceanic plankton taxa as well as colder water plankton as the shelf-edge current can bring warmer water species around the top of Scotland and into the North Sea.

I collected 13 diﬀerent random variables for both biotic and abiotic factors and
ﬁve additional ICES ﬁsh populations from which I derived two data sets: the
ﬁrst is based on a listwise deletion for the ﬁsh populations, and the second is
based on analysing the biotic and abiotic variables only. Since there is no general
theory on how to combine processes and organisms that operate on diﬀerent time
and space scales together into a model, especially when one goes up the trophic
levels, I applied a set of ﬁrst order autoregressive models from which I built a
revised ecological model (REMO) that outperformed the other models. I used
REMO to simulate future responses with a set of diﬀerent enquiries (E1-to-E8)
to understand the impact of weather change on the marine ecosystem; the model
showed serious risk values for ﬁsh larvae, ﬁsh eggs, krill and dinoﬂagellates when
the sea surface temperature increases by 2*◦*C.

The analysis shows that the number of ﬁsh eggs appear to be increasing under
the eﬀect of temperatures, I explain this behaviour as the fecundity of female ﬁsh
increases with water temperatures; the abundance of diatoms and large copepods
are found to increase under warmer water temperatures; but the abundance of
copepods are found to decrease with temperature. This is consistent with the the-
*ory that Calanus ﬁnmarchicus and Calanus helgolandicus copepods would occupy*
a niche (and grow) in cold and warm water temperatures respectively (Bonnet
et al., 2008); these two species can co-occur in the same geographical region such
that the former species becomes more abundant in cooler temperatures earlier
in the year and the latter species becomes more abundant in warmer tempera-
tures later in the year. The results showed that an increase of 20% in wind speed
could destabilise the ecosystem by reducing the number of diatoms and copepods.
However, the most serious risk remains that of the sea surface temperature where
an increase by 2*◦*C could jeopardize the ﬁsh larvae in the short run; and hence
deplete the ﬁsh stock size in the long run.