Dynamic Pool Models I:

7 Dynamic Pool Models I:

Interpreting the Past using Virtual Population Analysis

J.G. SHEPHERD AND J.G. POPE

Chapter 7

technique known as Virtual Population Analysis especially when the retrospective analysis and the (VPA), which is the subject of this chapter. The forecast are themselves integrated within a formal latter include both short-term catch forecast management procedure. These too have undoubt- methods, and long-term analyses such as that of

ed advantages, but when they are applied it can be- yield-per-recruit (introduced by Beverton and Holt come very difficult to figure out what is actually 1957), which are the subject of Chapter 8. There is going on, when surprising or controversial results no essential reason for this separation, and meth- are obtained. ods certainly exist in which both analyses are con- ducted together within a unified framework (for an up-to-date account of such methods see Quinn and

7.3 VIRTUAL POPULATION

Deriso 1999). There are advantages and disad-

ANALYSIS (VPA):

vantages in both approaches. However, it is a re-

THE BASICS

grettable but undeniable fact that fish stock assessments sometimes go wrong, or at least

7.3.1 The virtual population

become the subject of intense debate, and it is

and cohort analysis

not unusual for assessment scientists to spend The original concept of the virtual population – a considerable time trying to figure out what hap- pened, and why, either in reality, or in their analy- minimum estimate of the size of a year class at

any age, obtained by summing all the catches sis, or both. It is therefore usually helpful to have as clear a picture as possible of the actual (estimated) subsequent to that age – was introduced by Fry

(1949). The idea did not become popular, however, current state of the stock, and the information on which this estimate is based, as well as of its ex- until the 1970s, following the demonstration by

Gulland (1965) that one could also allow for natur- pected future evolution, and the principal factors determining this in both the short term and in al mortality to obtain a more realistic estimate.

Gulland’s working paper was not generally avail- the long term. Moreover, it is often necessary to include in the analysis some assumptions about able for many years, but is now included in the

compilation by Cushing (1983). changes, either those to be imposed as a result of

management action such as reductions of fishing Gulland’s method depended on the iterative so- lution of the conventional catch equation for each

mortality, or changes of the exploitation pattern given by the set of values of F on all ages in any year, age and year (equation 13.19 in Sparre and Hart,

Chapter 13, this volume), i.e. or to be expected from natural causes such as

changes of climate regimes. It is helpful if such

(7.1) critical evaluation and debate. Both of these pur-

assumptions are exposed as clearly as possible for C = ( FZ ) [ - 1 exp ( - ZP ) ]

poses, as well as the explanation of the workings of the analyses, are assisted to some extent by treat- where we have omitted the indexes y and a for clar- ing the retrospective analysis and the forecasting ity. This was a laborious and expensive calculation process separately. We have therefore adopted this when done on an electromechanical calculator, approach, which also reflects the structure of and it was not trivial even on an early digital com- much of the software available for these analyses, puter. The so-called ‘cohort analysis’ approxima- in the descriptions of these methods given here and tion introduced by Pope (1972) – also reprinted in in Chapter 8. However, there are also advantages in Cushing (1983) – was therefore a significant ad- using a unified approach to analysis and forecast- vance. It is in fact a remarkably good approxima- ing, and our approach is not intended to detract tion – good enough for all practical purposes. Now from these, except in so far as critical features of that computation is much cheaper, however, its the analysis may sometimes be more difficult to main virtue is that it exposes very clearly indeed discern in such unified procedures. This applies the essence of virtual population analysis (VPA)

Dynamic Pool Models I: Interpreting the Past

The fishing mortality values can of course be calculated immediately once the populations have 70 000

been estimated, since

Natural deaths

Adjusted catch

Zya ( , )= 1n [ PyaPy ( , ) ( + 1 , a + 1 ) ] (7.3)

Fya ( , )=( Zya , )-() Ma .

Clearly also, in order to estimate the total popu- Number of fish in the cohort

lations, we need to have and to use the total inter- 20 000

national catch numbers for C(y,a) – partial data from a few fleets is not enough, although the pos-

10 000 sibility of relaxing this restriction is discussed at the end of this chapter. Finally, it is clear that one

0 1 2 3 4 5 6 needs some estimate for terminal population size

Age

at the end of the final year for which a catch esti- mate is available, in order to start the calculation,

Fig. 7.1 Accumulation of the virtual population. which is carried out retrospectively, as implied by equation (7.2).

‘Exact’ VPA, based directly on the solution of equation (7.1) (with equations (7.3) and (7.4)) can of

as a technique, and for that reason we employ it course be carried out instead of the cohort approxi- throughout this chapter.

mation, and this is in fact the usual operational Pope’s formula

practice. A large number of algorithms for the iterative solution of this equation have been pro-

Pya ( , )= exp [ Ma () 2 ] Cya ( , ) posed, including the standard Newton–Raphson

(7.2) method (see for example Stephenson 1973). Given the terminal population, all the other can be derived, using a little ingenious algebra, as population-at-age estimates can thus be computed an approximation to (7.1). It is, however, also the immediately. And given these, all the fishing obvious result of assuming that all the catch is mortality values can then also be calculated. There taken instantaneously half-way through the year. are no fitting procedures involved, no residuals, no One is simply raising the number of survivors to lack of fit, and no degrees of freedom. VPA is allow for natural mortality over a full year, and the therefore in no real sense a statistical modelling number of fish caught to allow for natural mortal- procedure (G. Gudmundsson, personal communi- ity over only half the year. Clearly, multipliers cation). The catch data are accepted as though they other than exp (M/2) could also be applied if the were exact, and entered directly into the calcula- catches were concentrated at some other time of tion. VPA is therefore best thought of as a transfor- year. In any case, this formulation shows that VPA mation of the data. The catches are converted into is no more and no less than a method for estimating

+ exp [ MaPy () ] ( + 1 , a + 1 )

a set of equivalent population estimates, which the population size of a cohort at any age, by ac- may in turn be converted into the equivalent fish- cumulating the subsequent catches, with adjust- ing mortalities. Any errors in the original catch-at- ment for natural mortality. The process, which is age data feed through directly into the population normally applied sequentially for all age groups be- and fishing mortality estimates. longing to the same cohort, is illustrated in Fig. 7.1.

Thus understood, VPA is a very quick and con-

Chapter 7

venient procedure for turning indigestible catch- at-age data into things that are more easily under- stood, at least by those who have been suitably trained, namely fishing mortalities and estimates of year-class strength. Its virtues and vices will be discussed in more detail later on. Here it is neces- sary to stress one vital point: the transformation performed by VPA is not unique. For every choice of terminal population, and there are infinitely many such choices, there is a resultant and differ- ent transformation: a few of the possibilities are illustrated in Fig. 7.2. In addition, with no residuals and no lack of fit, there is no way to choose among them. All are equally good (exact) fits to the data. This is the central problem of VPA, that there are infinitely many possible ‘solutions’ for each and every cohort. To select among these the user must supply an estimate of the terminal population, or, equivalently, the terminal fishing mortality, and there is no objective way of doing this based on the catch-at-age data alone.

Additional information is therefore required. This may take the form of additional assumptions, or additional data. In the former case one is in ef- fect specifying a stronger and less highly parame- terized model. This is the approach adopted by separable VPA, discussed in Section 7.4. It is, however, only partially successful, since serious indeterminacy remains. The second approach, ad- ducing more data, is typified by the use of CPUE and survey abundance data for ‘tuning’ the VPA, and is discussed in Sections 7.5 and 7.6.

Before leaving the subject of ‘conventional’ (un- tuned) VPA, it is important to recognize some of its essential features. A first important point is that natural mortality, M, is taken to be known, and is not and cannot be estimated by the analysis. M is usually taken to be given as a constant or as a func- tion of age, but the analysis is easily generalized should values of M as a function of both year and age be available. VPA is also essentially a retro- spective technique, not only because that is the most convenient way in which to carry out the calculations, but because estimates of fishing mor- tality and population size for earlier years and younger ages become progressively less sensitive to the starting assumptions about terminal F. This

is usually referred to as convergence of the VPA, and is discussed in detail by Pope (1972). As a rule of thumb, estimates may be regarded as more or less converged if the cumulative fishing mortality, from the oldest age in the cohort back to that in question, is greater than one, but not otherwise.

The converse is also true, that running a VPA calculation forwards in time leads to divergent estimates for the older ages and years. In practice this means that estimates of terminal F or popula- tion obtained in this way are highly variable, and very sensitive to the initial estimates of year-class strength used to start the calculation. They may in fact easily become infeasible (zero or negative) for what are a first sight quite plausible initial as- sumptions. Given sufficient precision in the calcu- lation, forwards and backward VPA are of course exactly equivalent calculations, but in practice, because of the sensitivity to starting assumptions, the retrospective calculation is robust, whereas the prospective one is not, and it is useful only as a technical device in some special circumstances.