MULTISPECIES VIRTUAL POPULATION ANALYSIS

7.7 MULTISPECIES VIRTUAL POPULATION ANALYSIS

We saw in section 7.3 that single-species virtual population analysis requires the assumption that natural mortality rate is a known value, often thought to be both constant with year and with age. But in practice we might well expect that natural mortality rate would vary by age, because smaller fish are more vulnerable to predation than larger fish. We might also expect levels of preda- tion mortality to vary from year to year because numbers of fish predators are known to vary from year to year. For example, if the numbers of cod in the North Sea doubled then we might expect as a first approximation that the amount of food they ate would double. We might also expect the preda- tion mortality to vary if the total amount of prey available to the predators varied from year to year. For example in the North Sea, saithe eat mostly Norway pout and herring, hence if in a particular year there was more herring we might expect predation mortality on Norway pout to decrease. Clearly, such changes in natural mortality would influence the results of VPA and so might have pro- found effects on our perception of the numbers of fish in the sea. It is also likely that realizing that natural mortality could vary would also influence our perception of how the fish stock should be managed.

Such considerations persuaded Andersen and Ursin (1977) to make a model of the North Sea fisheries, which included a variable predation mortality. Their model worked using a forward simulation of fish stocks based upon the solution of simultaneous differential equations. Their most important innovation was to partition natural mortality rate (M) into two components, M2, due

to predation mortality, and M1, due to other forms of natural mortality.

M = M1 + M2 (7.37) This idea was carried into multispecies versions of

virtual population analysis (Helgason and Gisla- son 1979; Gislason and Helgason 1985) and cohort analysis (Pope 1979; Pope and Knights 1982). The strength of these two, essentially similar, ap- proaches, were that they showed how the existing catch-at-age data from a number of predator and prey species could be combined to estimate preda- tion mortality rate. However, it was clear that in addition to the catch-at-age data, additional data would be needed to characterize the feeding of each age of each predator. Thus these early papers led to the establishment of a working group of ICES, which undertook to make a comprehensive study of the feeding of the major predatory fish of the North Sea during one year, to provide suitable inputs to these models.

This study was first conducted in 1981 and was known unofficially as ‘the International Year of the Stomach’. The predator species considered in the study were cod, whiting, haddock, saithe and mackerel. Stomach samples from size-stratified ranges of these species were taken from all parts of the North Sea for all quarters of 1981 because, as shown for example by Daan (1973), the stomach contents of species such as cod were known to vary by size by area and by season. Analysis of the con- tents of the stomachs of the five predators were undertaken by five designated laboratories to ensure consistency of their interpretation. The contents of the stomachs of each predator were analysed to provide, where possible, estimates of how much of each species was eaten; not an easy task with partially digested remains. In particular, their feeding on species for which catch-at-age data were available were carefully reconstructed, as in- gested weight by prey size range. There were nine species of particular interest as prey: these were the five predators themselves and the prey species herring, sprat, Norway pout and sandeel, for which catch-at-age data were also available. Thus the data on these predators and prey species were in a

Dynamic Pool Models I: Interpreting the Past

157 form that could be converted to estimate how Fyas ( ,, )= M 2 ( yasCyasDyas ,,* )( ,, )( ,,. )

(7.40) many fish, of each age, of each prey species, were eaten by each age of each predator species. These Equation (7.38) shows how we would include pre- data became available in 1983 (Daan 1983; Daan dation, if predators provided catch statistics in the 1989) and the ICES multispecies assessment work- same way that fishermen do. If this were the case ing group was then established to analyse the data the multispecies cohort analysis equation would using Multispecies Virtual Population Analysis suffice to solve the problem. However, since preda- (MSVPA). The early history of this working group tors do not provide catch statistics, we must find is summarized in Pope (1991) while more recent other ways of estimating D(y,a,s). We can do this approaches can be seen in ICES (1997).

by considering the quantity d(y,A,S,s,a), the num- The actual analysis made by the working bers of prey species (s) of age (a) eaten by a particu- group was based on the Helgason and Gislason lar predator species (S) of age (A) in year (y). Thus: MSVPA model, but since the Pope multispecies

cohort analysis model is easier to understand, Dyas ( ,, )= Â Â dyASas ( ,,,,. )

(7.41) it is used here to explain the approach. However,

all A all S

the formulation of feeding relationships used by Helgason and Gislason (1979) is more robust Of course we do not know what d(y,A,S,s,a) is than that of Pope (1979) and so is adopted in the either, but we can write an equation for it in terms following.

of the ration R(y,A,S) of all the predators of age A The single-species cohort equation (7.2) can be and species S in the year y, and the diet proportion modified quite simply to include a term due to DP (y,A,S,a,s) that the predator obtains from prey feeding, by all quantified predators on the prey of age a and species s. Thus: species age in question. We may call this term

D (y,a,s), the numbers of prey species (s) of age dyASas ( ,,,, )=( R y A S DP y A S a s ,,* ) ( ,,,, )

(7.42) perhaps in some other portion of the year such as a quarter).

a eaten by all quantified predators in year y (or

Wt y a s ( ,, )

where Wt(y,a,s) is the average weight of the prey in the predator’s stomach. Note that this may differ

Pyas ( ,, )= exp [ Mas 1 ( , ) 2 ] [ Cyas ( ,, )+( Dyas ,, ) ]

from the weight of the prey in the sea, but can be

+ exp [ MasPy 1 ( , ) ] ( + 1 , a + 1 ,. s )

(7.38) obtained from stomach samples. Ration may also be written as the average popu-

lation sizes of the predator, times the average Note that not only does this equation contain the annual (or quarterly) diet r(y,A,S). Average values amount of the prey species eaten by all predators in the equations are denoted by an over-bar:

included on the analysis, but it also uses M1 in- stead of M. In effect it treats predation deaths as

RyAS ( ,, )=( PyASryAS ,, )( ,,. ) (7.43) just another form of catch. Notice also that once

we have successfully performed a multispecies co- Average population size is estimated from the hort analysis and estimated population sizes then

results of VPA or MSVPA, while average ration we can estimate the predation mortality rate (M2) is provided by the results of feeding experiments.

and the fishing mortality rate (F) by solving: If necessary, ration may be modified from year

to year in response to the size and growth of ln [ Py ( + 1 , a + 1 , sPyas )( ,, ) ]

predators.

=( Zyas ,, ) Diet proportion DP(y,A,S,a,s) may be written in

(7.39) terms of the populations of the prey species and of the predator, and a factor called the ‘suitability’ and

=( Fyas ,, )+ Mas 1 ( , )+ M 2 ( yas ,, )

Suit (A,S,a,s). Suitability expresses how much a

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predator of a certain age would eat of a certain prey, compared with the proportions that were found relative to the total prey biomass available, of all in stomach samples for that year. This allows possible ages a¢ of all prey species s¢. Note that in the estimates of suitability to be corrected. The this formulation it is considered to be constant for corrected estimates of suitability may then be all years, though it may be allowed to vary between used to estimate new values of d(y,A,S,a,s), and quarters, or other divisions of the year, to reflect the process is repeated until the estimated values changing diet preferences/availabilities with of suitability converge to a stable value. An season:

exactly similar procedure can be adopted using the ‘exact’ multispecies virtual population analy-

DP y A S a s ( ,,,, )

sis if required.

In pseudo-code we can write this as an iterative P y a s Wt y a s Suit A S a s ( ,,* ) ( ,,* ) ( ,,, ) =

loop, as follows:

  Pyas ( ,,* ¢¢ ) Wt y a s ( ,,* ¢¢ ) Suit A S a s ( ,,, ¢¢ all a ) ¢ all s ¢

Pseudo-code for multispecies VPA

(7.44) Choose initial values of Suit(A,S,a,s). For all species for which catch-at-age data are

available . . .

This equation can be solved, if we know the num- bers of predators and of prey, and also the suitabil-

carry out single-species cohort analysis to obtain estimates of P(y,a,s).

ity of each prey for each predator. Population numbers will of course ultimately be obtained Begin iterative loop

estimate d(y,A,S,a,s)

from the multispecies cohort analysis equation. Suitabilities, however, need to be estimated using

perform a multispecies cohort analysis calculate estimated feeding EF(y¢,A,S,a,s) of

additional data. For prey species whose numbers are not known, Helgason and Gislason (1979)

predators compare with actual feeding AF(y¢,A,S,a,s)

adopted the approach of including an additional group of ‘other prey’ whose joint biomass was con-

obtained from stomach samples recalculate Suit(A,S,a,s,new) = Suit(A,S,a,s,

sidered, in the absence of any information, to be constant through time. This turned out to be a very old )*AF(y¢,A,S,a,s)/EF(y¢,A,S,a,s).

Repeat until values of population and suitability stabilizing assumption, and moreover the biomass of these prey could actually be set at any arbitrary converge.

level without affecting the model. This is because When this procedure was first used concern was the model effectively estimates the product of the raised that even if the process converged, the re- biomass and suitability for these prey and hence an sults might not be unique. Studies by Magnus and assumption of a higher biomass would be balanced Magnusson (1983) indicated mathematically suffi- by an estimation of a lower suitability.

cient conditions for uniqueness which appeared to Suitabilities are estimated by first guessing

be robust enough to ensure convergence in most their values for each predator age and prey age com- cases. However, these sufficient, but not neces- bination, and using these guesses in combination sary, conditions excluded the possibility that a with population sizes of predator and prey age predator could eat conspecifics of the same age. groups obtained from singles-species cohort analy- Since this phenomenon is observed from time sis. This gives initial values of d(y,A,S,a,s). A mul- to time, for example small 1-year-old cod are eaten tispecies cohort analysis run with these values by large 1-year-old cod, the possibility of non- provides new estimates of population size for each uniqueness of the above procedure cannot be predator and prey species. The proportion that the entirely ruled out. However, it has yet to be ob- various prey items that the MSVPA predicts to served in practice. form of the diet of each age of each predator species

In principle it is possible to tune MSVPA to in a particular year, in our case 1981, can then be trends in CPUE or survey data, in a similar fashion

as described in single-species assessments, and such results could be used for setting quotas. How- ever, the ICES multispecies assessments working group found that for the predator species, at least, there was little to be gained from calculating multispecies-based quotas rather than those based upon single-species analysis. This is because the predation mortality on the larger species occurs mostly on prerecruitment ages. Thus, while this affects the long-term yield, it has much less impact on short-term predictions. This is because these are typically made using estimates of the numbers of prerecruitment ages obtained from the results of surveys, often using results from ages older than those on which the majority of the predation has occurred. Multispecies effects might well af- fect quotas set for prey species, since predation af- fects all ages of the species, but in the case of sandeel and of Norway pout in the North Sea these were usually set on an average basis. In other areas, however, multispecies effects are considered in the setting of quotas for prey species such as capelin, and in these cases tuning of some sort be- comes important. For more details of the behav- iour and ecology of age-specific predation, see Chapters 11, 12, 13, Volume I, and Chapter 16, this volume).

In the North Sea the more interesting features of the analysis relate to the changes observed in nat- ural mortality rate with age and with time, and the consequent effects this has on recruitment esti- mates. This is illustrated in Fig. 7.14, which shows the cumulative predation mortality of North Sea cod at age, estimated by MSVPA. The fig- ure clearly shows that predation mortality rate can vary with both year and age and that thus the as- sumption of constant natural mortality rate typi- cally used in single-species models is violated. The figure indicates that the cumulative predation mortality to age 3 varied by nearly 1.0 between a high value in 1980 and a low value in 1992. This is equivalent to a change in survival and hence of apparent recruitment over these ages by a factor of 2.7.

Inevitably, multispecies virtual population analysis, and multispecies cohort analysis, are heavily over-parameterized, since not only do they

estimate the population and the fishing mortality terms of the single species VPA models, but they also estimate predation mortality and suitability. As with the single-species models there is no good- ness-of-fit measure. This is because all the degrees of freedom in the data are absorbed by the model. At the time the models were developed, more restrictively parameterized, least-square-fitted models would not have been practical, due to limi- tations in computer power. It was, however, pos- sible to fit more restrictive models to the results of MSVPA using simple statistical packages. This was performed for both estimates of M2(y¢,A,S,

a ,s)/[P(y¢,A,S)*Wt(y¢,A,S)] and of Suit(A,S,a,s). Quite reasonable fits of both outputs can be obtained using predator–prey species interaction terms int(S,s) and an Ursin (1973) log-normal predator weight/prey weight food-preference model, N{ln[Wt(A,S)/Wt(a,s)], m, s}.

(7.45) These statistical models can absorb up to half of

the variation in estimates of suitability or of M2. The remaining variation e is probably what one

Suit A S a s

( [ ) { ] } ms + e

Dynamic Pool Models I: Interpreting the Past

1984 1988 1992 Year class

Age 3 2 1

Fig. 7.14 North Sea cod predation mortality rate by age, for selected years, from MSVPA. M2 (1980)–M2 (1992) = 0.99, equivalent to a exp(0.99) = 2.7 change in recruitment.

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should expect from what must be rather variable data on stomach content data broken down by age of both predator and prey.

Clearly, simple MSVPA is dependent on the assumption that suitability is constant through time. If this assumption were not made it would not be possible to base it upon one year’s stomach content data. Historical estimates might still be obtained if stomach content data were collected each year and suitability estimated on an annual basis. However, since in the North Sea the stomach content data collected in 1981 required a large sampling effort and the dedicated work of five stomach-content analysis teams for two years, this would scarcely be practical. Moreover, if suitabil- ity changed from year to year it might prove dif- ficult to make forward predictions. Thus, it was considered important to test the hypothesis of con- stant suitability. This was attempted by collecting partial stomach samples in 1985, 1986 and 1987. Analyses of these are given in Rice et al. (1991). However the partial nature of these samplings made the tests incomplete and it was decided to repeat the full sampling in 1991. This allowed equations of the form of (7.45) to be fitted jointly to estimates of suitability based upon the 1981 data and the 1991 data.

For most predator species the joint model cap- tured about 40% of the variation, and a smaller but significant fraction of the variation (between 6% and 29%) could be captured by allowing suitability to change with year. Thus, from a strictly statisti- cal point of view the null hypothesis of no-year effect has been disproved. This suggests that as- sumption of constant suitability is also disproved, though one might note that an alternative explana- tion might be a drift in stomach-sampling proce- dures over a 10-year period. It is also worth bearing in mind the point made by ICES (1994) that rejec- tion of constant suitability implies a more com- plex model, but not a return to single-species dynamics. While the differences found were statis- tically significant, the high number of degrees of freedom of the fits meant that in many cases the differences were not large, and not necessarily of practical significance, as may be judged from Fig.

7.15. This shows the differences of the fitted suit-

ability of cod (as prey) for cod (as predator) by ln(size ratio), estimated from the 1981 and 1991 data respectively. Analyses of these data may be found in the 1993 and 1997 reports of the ICES Multispecies Assessment Working Group (ICES 1994, 1997).

In the North Sea the comparison between 1981 and 1991 was quite testing because the herring, which is an important prey, had gone from low abundance in 1981 to high abundance in 1991, and other species abundances had also shifted. Given the problems of sampling the North Sea, it is diffi- cult to carry investigations of the constancy of suitability further there. Clearly, one might expect suitability to be a function of size ratio, of species overlap and of the behaviours of predator and prey. One might also speculate that the diet proportion function described by equation (7.44) might be in higher powers (positive switching) of prey biomass or lower powers (negative switching) rather than in unit powers.

It is rather easier to investigate changes in suit- ability and thus the applicability of MSVPA in areas where there are fewer species. Work in the Baltic (ICES 1999) and at Iceland and the Barents Sea (Stefansson and Pálsson 1997; Tjelmeland and Bogstad 1998) has this benefit and the challenge has been taken up. The latter areas have also seen

10 1 Predator wt/prey wt

Fig. 7.15 Suitability of cod as prey for cod as a function of predator weight/prey weight.

developments of forward simulation models that fit to data using more statistical principles. Again this is more simply achieved in areas with fewer species to consider.