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Last month we introduced you to the more quantitative aspects of fisheries management. The example we used was Virtual Population Analysis (VPA) which uses a simple mathematical model to derive historic stock sizes based on the age distribution of past catches, an estimate of natural mortality, and an inspired guess of terminal fishing mortalities or terminal cohort sizes. Another commonly used method of stock assessment treats fish resources as a unit of biomass.
Reproduction and individual body growth rates continually tend to increase this biomass, while natural mortality and fishing continually decreases it. The biomass therefore experiences a net increase or decrease depending on its size which results in it achieving an average unexploited level known as the resource carrying capacity.
This method is simpler since it makes no assumptions about the size and/or age composition of the catch or of the broader population. This model of a renewable resource is known as a surplus production model and forms the main topic of this article.
Surplus production model analysis
The basic information used in surplus production models is catch-per-unit-effort (CPUE) data and records of landed catches. In this approach, consistent with most other stock assessment techniques, CPUE is regarded as an index of resource biomass. The problem is to estimate (1) the constant of proportionality linking CPUE to resource biomass, referred to as the catchability coefficient, and (2) to estimate the resource carrying capacity and the scale of the surplus production curve.
Our earlier articles presented a number of surplus production curves where the surplus production of the resource (the net annual increase in the resource biomass in the absence of fishing) was plotted against resource biomass. Since the surplus production and hence the resource biomass cannot increase indefinitely (resource biomass is assumed to achieves a maximum level known as the resource carrying capacity), surplus production is zero both at a resource biomass level of zero (when there is no biomass then it cannot produce any surplus production) and at a resource biomass level equal to the carrying capacity. Somewhere between these two extreme resource biomass levels (zero and the resource carrying capacity) the surplus production must reach a maximum value.
Most surplus production models assume that the relationship between surplus production and resource biomass is bell shaped, that is it is fairly symmetrical with a maximum about halfway between a resource biomass of zero and the carrying capacity (Fig. 1). In practice surplus production curves are seldom symmetrical and there are a variety of surplus production models which accommodate virtually all possible asymmetrical relationships that may be required for different situations. For example, in many finfish stocks the maximum is assumed to occur at a biomass smaller than the halfway point, whereas for whale stocks it is assumed to lie at a biomass larger than the halfway point.
In order to estimate the catchability coefficient, the resource carrying capacity and the scale of the surplus production curve, computer-based techniques have to be used. It is also necessary to provide a value for the resource biomass at the start of the fishery – this is normally assumed to be equal to the resource carrying capacity, but a different value may be used in certain circumstances. The computer-based techniques involve comparing the implied (or modelled) CPUE trend for the resource, which one can calculate based on an initial guess of all the model quantities, to the actual CPUE recorded. Subsequent ‘guesses’ of the catchability coefficient, the resource carrying capacity and the scale of the surplus production curve are then made to improve this comparison.
The ‘guessing’ process is actually reduced to a mathematical algorithm which makes intelligent guesses, and the whole process eventually converges to the best possible set of model quantities. The ideal is for the modelled CPUE and the recorded CPUE to agree exactly but because of statistical noise this will never be possible. Using "intelligent" computer code, the best estimates might be obtained after 20 to 100 guesses. In theory this can be done by hand, but the calculation of the outcome of a single "guess" might involve a few hours of work with a hand held calculator. Given that the surplus production model is one of the simplest population models, it is easy to understand why extensive use of modern computers is essential for any meaningful progress to be made.
Once the best estimates of the surplus production curve and the initial resource biomass level have been obtained, forward calculation through time allows one to deduce the present resource biomass level. The TAC that should then be recommended will follow from the harvesting strategy, or catch policy as it is often called. This policy might be quite straightforward, e.g. TAC should be 20% of the available resource biomass. Alternatively the policy might involve a more complicated calculation based on the estimated present resource biomass level.
VPA and surplus production models are based on a fairly simplistic view of fish population dynamics. In reality the processes governing fish population size are much more complex. Furthermore, these two methods suffer from specific peculiarities and limitations. VPA models assume that fish cohorts are well defined whereas in reality cohorts are mixed and difficult to distinguish. Closely related to this problem, VPA’s are heavily dependent on the ability to age fish, which is problematic.
VPAs also work best for long-lived species, and tend to be more reliable for estimating historic rather than recent population sizes. Surplus production models, on the other hand, are heavily dependent on the assumption of proportionality between CPUE and resource biomass. Although this assumption may be reasonable for non-shoaling species such as hake or cod, it is clearly invalid for small shoaling species like anchovy and pilchard caught by purse-seine gear. As a result both VPAs and surplus production models are unsuited to the management of small shoaling species.
Nowadays the availability of very powerful computers has made it feasible to explore the use of more sophisticated techniques which can combine the logic of the VPA technique with the surplus production modelling approach, and go even further to incorporate additional complexity. These are the set of age-structured production models and size-based modelling approaches. These models try to represent the full complexity of growth, death, fishing and reproductive processes in a population on the computer, and then use techniques to estimate the various important quantities governing the changes in population size over time.
The models can then be run forward in time to try to understand the implications of different future harvesting strategies for the resource. Our next articles will deal with some of the more advanced modelling techniques and also with analytical techniques which are used to eliminate some of the biases in the data used in such modelling approaches, especially those biases found in CPUE data.