We chose the log model instead of the logistic model because they perform equally well, but the log model has fewer parameters and thus is more convenient to apply. Finally, we contrasted the model MS3, including interspecific spatial correlations, with a sub-model, MS4, for which there is no assumed species correlation i.
Although multispecies models combine species, spatial and size correlations separately for each year, the exact same type of models are considered, where species and year switch roles. These models are called multi-year models, where for each species, the correlation between year, space and size is modelled. An overview of the models considered and tested is given in Table 1 , where single, multispecies and multiyear models are covered. In addition, the hierarchical structure of the model testing is indicated. The hierarchical order of the models is as follows:.
This states that the and models are sub-models of the model. Finally, we illustrate the potential of an extended correlation structure by predicting the abundance distribution of a target species, using only indirect data, i. For the cases of cod and whiting in and , the possible correlation across species and year are further investigated. We illustrate the potential of the correlation model to make spatial abundance predictions, first using species correlation models and secondly using time correlation models.
No humans, primates or laboratory animals were involved in the study. There was no sampling from private land, and the field studies did not involve endangered or protected species. Only fish sampled in public sea areas have been used. The sampling and handling of fish strictly followed all ICES guidelines, procedures, legislative rules, and permissions from national governments for sampling and handling of fish in fisheries research surveys.
The sampling was conducted by national government-owned research vessels following Danish national legislation, permissions, and ethics for handling of wild caught fish. The sampling was performed under repeated international standardised surveying where the research vessels had full permission to sample from all relevant national public authorities governments in the Baltic waters.
This was not necessary because the sampling and handling of fish strictly followed all ICES guidelines, procedures, legislative rules, and permissions from national governments for sampling and handling of fish in fisheries research surveys. The sampling was conducted by national government-owned research vessels following Danish national legislation, permissions, and ethics for handling of wild-caught fish. The different size correlation structures tested are shown separately in the 4 panels of each figure, covering an hierarchical testing procedure see Fig.
The main purpose of the tests was to investigate for each species and year which of the size correlations models that could be rejected or not Hypothesis H01 and the models SS2, SS3 or SS4.
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It was especially important to establish whether the free unconstrained model could be reduced to a structured model so that abundance predictions could be made using a model with few parameters. All models were parameterised from the same correlation function Eq. The spatial correlation parameter was first tested for independence of size and time for each species and both years, and it was found that it was possible to distinguish between the space and size-time correlation.
The Chi-Square likelihood ratio tests of the different models SS1. SS2, SS3, and SS4 Table 2 and Table S1 in File S1 did not have sufficient power to reject any of the correlation structures for both cod and whiting in and because of the very high number of degrees of freedom in the main model. This indicates that there is no significant difference between SS1 and SS2, SS3, and SS4 with respect to the description of the correlation between size groups for cod any of the years Table 2 , and consequently we cannot reject a structured correlation model hypothesis H Higher power of the tests can be obtained by re-binning the data to 2-cm size groups and re-fitting the models SS1 — SS4 , as this will very much reduce the degrees of freedom of model SS1.
When using 2-cm size groups, the test results for cod in both and not shown came out in favour of the log- and logistic transformations parameterising a structured size correlation model, as the identity transform was rejected. Thus, the identity transformation was excluded from the analysis.
There is a trade-off between precision in the model by using 2-cm size groups compared with the high resolution in the rate of change in distribution when using 1-cm groups. In the present approach, we use the 1-cm groups to retain as much information as possible in the distribution dynamics of the fish. On average, a juvenile Baltic cod grows 1 cm in 6 weeks, and when using 2-cm size groups, the time resolution of 1. Consequently, a size-structured model cannot be rejected for any of the species for both years.
In the multi-species and -year model extensions, the log transformation parameterisation was chosen because it is simpler than the logistic transformation in the sense that it does not contain any further parameters. An alternative criterion for model selection is parameter consistency over time. In this case, the question regarding the four independent analysis Figs. In other words, are the images Figure 2c and Figure 3c or 2d or 3d for cod significantly different?
Parameter estimates related to the logistic d and log c transforms display equal consistency over time, and here we have reported the results of the log-transform c Table 3. All correlation parameters Eq. It is remarkable that the parameter describing the angular velocity of the oscillating part Eq. This result basically indicates that small and large cod occur in the same spatial regions, whereas this is not the case for whiting. This pattern is also visible from the initial analyses with animations of cod and whiting abundance patterns across size-groups from the surveys as described in Supporting Information Appendix A, i.
Consequently, there is a rather consistent structure in the size correlation within the species over time. The next step is to use the model to assess the possible correlation between species to test hypothesis H This is accomplished by comparing the four multispecies correlation models MS1—MS4 and testing whether the free, unconstrained species-size model MS1 can be reduced to the sub-models M2—MS4.
The combined analyses with the multi-species extension of the model considering species correlation involved runs under the log-transformed size for both species Figs. The results are presented in Figure 5 for and Figure S4 for and are covered in detail in the panel a that presents an image of the combined correlation of cod the large square block and whiting the smaller square block using the previous separate analysis for each species combined with the assumption of unconstrained species-size correlation between the two species.
The next panel b is visually very similar to panel a and represents the model reduction where cod and whiting are assumed to have a common correlation function. Despite the visual similarity between panel a and b , the likelihood ratio test strongly rejects this reduction Table 2. Panel c is an image of the separable model reduction, and this model does not fit very well.
Finally, panel d shows the model fit, assuming species independence. The likelihood ratio tests for c and d indicate that it is not possible to distinguish between b , c and d. In summary, none of the model reductions are valid because of the rejection of model b because of the combined nested structure of the test procedure.
Consequently, a will have to be chosen as final model in this case, i. The conclu are the same for cod and whiting in Fig. S4 , except that here the separable model visually appears to perform better than for , and the species correlation is significantly non-zero. Below, we return to the overall conclusions on species correlation. MS1, unconstrained species correlation with separate parametric size correlation for each species a ; MS2, unconstrained species correlation with common parametric size correlation for both species b ; MS3, separable species-size correlation c ; MS4, no species correlation d.
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In MY1—MY4, the cod-whiting species correlation has been replaced by the corresponding — year correlation to test hypothesis H For both species cod and in Fig. S5 , the common correlation function hypothesis in b cannot be rejected as and , for cod and whiting, respectively Table 2 and Table S1 in File S1.
Furthermore, the separable model c cannot be rejected, but the independence test d is rejected for both species. Consequently, for both species, one will select model c , i. Unconstrained year correlation with separate parametric size correlation for each year MY1, a , Unconstrained year correlation with common parametric size correlation for both years MY2, b , separable year-size correlation MY3, c , No year correlation independence MY4, c.
Three main conclusions can be drawn. For the multiyear correlation models, it was possible to obtain a parameterised year correlation model, the separable model MY3, which was not rejected, and this indicates that Hypothesis H03 can be accepted. In contrast, we cannot reject that species correlation occurs hypothesis H02 , but this correlation is very complex and resisted the parameterisation described here. The current models appear to perform better when describing time correlation than species correlation. A free, unconstrained species-size correlation model with many parameters is therefore still needed for describing the correlation between species.
The potential of the correlation model to make abundance surface predictions was analysed, first using unconstrained species correlation models Figs.
An extended species or time correlation structure is useful to predict abundance surfaces of a target species when using only indirect data, i. The visual inspection reveals that of the performance of all predictions, the species-based predictions are perhaps the most accurate.
CHAPTER 7 - ESTIMATION OF PARAMETERS
These findings also support that we cannot reject hypothesis H02 when using a much more complex model in this comparison, i. The three row panels indicate three whiting size groups in cm. The three row panels indicate three cod size groups in cm. A length-based stochastic model of single-species stock dynamics including densities  was applied to the Baltic cod species based exclusively on survey data; however, this model was not spatially explicit.
In the present study, an extension of the statistical LGCP model  is applied to the standard and extended BITS data for Baltic cod and whiting to investigate not only intraspecific size correlations, including spatial and temporal distribution patterns, but also potential interspecific correlation between species in relative density according to space, size, and time. The motivation for developing size-based density models including species, time and spatial correlation is based on the apparent visual relationship between species from sequential abundance maps as for instance presented in Figures 7 , 8 , S6 , S7 , first column, illustrating the spatial distribution by species by size group as well as described in Supporting Information Appendix A.
Quantification and modelling of the covariance functions is performed either for the same species at different time periods or for different species at the same time. The aim is to empower spatial predictions of relative density of fish within and across species after constructing spatial abundance models that support hypotheses testing regarding alternative model specifications. We formulated natural model reduction hypotheses based on a size transformation that results in fish being able to be compared on a size scale. The separate single species analyses assumed that the spatial and the size correlations in density are independent among species.
The analyses further revealed that the idea that fish can be compared on a size scale by transformations of the natural size scale cannot be rejected for any of the species for both years, and there is a rather consistent structure in the size correlation within species over time. The log and logistic size transformations, but not the natural no transformation for cod, were not rejected when changing the bins to 2-cm length groups.
On this basis, hypothesis H01 cannot be rejected. As such, the model enables prediction, interpolation and animation of unobserved relative distribution and density patterns at any location and season of the year in the area for, e. Regarding model complexity, the following conclusions have been drawn. For the multispecies relationship, the simple structured models developed are not adequate to describe the correlation pattern, and thus, we were not able to decide if a significant species correlation exists. In general, the strength in the approach lies in the detailed description and testing of the combination of species-size and time-size correlations.
For the single species models, area and time spatial variations in log fish density, , of a point in the space, , for a given size are assumed random by nature. However, if some structure in relative fish densities according to animal behaviour exists, we expect to observe fish of similar sizes occupying the same spatial areas. For the two competing species, cod and whiting, we expect that fish of similar sizes of occupy the same spatial areas sharing the same habitats, food sources, etc. Both cod and whiting have, for the North Sea, been demonstrated to be competing species for the same habitats and to predate on each other e.
Interspecific relationships may play a role in the distribution patterns of WBC cod and whiting, but this phenomenon is not well understood  ,  ,  , .
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There is spatial and temporal variation in biological interactions due to predation by cod in the Baltic Sea, where also cannibalism has been documented as an impacting factor in certain periods  ,  ,  ,  , . The levels of cannibalism are dependent on the abundance of juveniles and larger cod predators, their overlap in distribution, and the availability of alternative prey items for larger cod, such as sprat and herring  ,  ,  , .
In the WBS, there are also abundant competing gadoid predators in the form of whiting . The basic single species model used Eq. The large-scale variation is chosen as generally as possible, i. The small-scale variation and correlation are assumed to be the same for each point in the space and depends only on the size correlation.
This model has the advantage that paths crossing land can be avoided.