In studying the state of the fish stocks and the effect of fishing on them, the fishery biologist should carry out his analysis in precise quantitative terms. To do this he must use mathematics, and to use mathematics the complexities of the real situation must be replaced by more or less simplified and abstract mathematical models. Such models may be used to represent both quantities of interest (abundance of the population, size of the individual fish), and the relation between these quantities.
In their simplest form, mathematical models are regularly used by biologists; for instance, it is commonplace to represent the size of a fish by the number of centimetres between the tip of its jaw and the end of its caudal fin. This model conceals many factors about the actual fish - whether it is fat or thin, or whether it is a cod or a tuna - but enables many analyses to be carried out - e.g. the construction of a length-frequency distribution of a sample of the fish population.
The value of a model may be judged by its simplicity and the closeness with which events or values predicted by the model fit the actual observation. A model cannot be considered as being either right or wrong, but as giving a satisfactory fit to the facts over a wide or narrow range of situations. A good model is one that is simple, but gives a good fit over a wide range.
The best test of a model is its usefulness in prediction; in this sense prediction covers not only the prediction of future events but also any values or events not considered in proposing the model. Thus a model describing the growth of cod in length may be proposed from the analysis of data of mean length at age; it will be a more useful model if, subject to the fitting of a minimum number of constants, it can be used to predict (estimate) the mean length at age of haddock, or any other species.
All the models described in this manual have proved to a greater or lesser extent to be useful models in that they have provided useful quantitative descriptions of events in various fisheries. Most have also been successful in making predictions, in the sense used above. A good example of the testing of a model by a successful prediction is the analysis of the Antarctic whale stocks. At the 1963 meeting of the International Whaling Commission there was considerable discussion on the quota, in terms of numbers of whales caught, to be set for the 1963/64 season. The Commission's Committee of Three Scientists had devised a model for the population of Antarctic fin whales, taking into account the probable rates of mortality and recruitment, etc. On the basis of this model the Committee had recommended that in order to rebuild the depleted stocks, not more than 5 000 fin whales should be killed, but that if whaling activities were continued at the 1962/63 rate about 14 000 fin whales would be caught, equivalent, including catches of other whales, to some 8 500 Blue Whale Units (BWU). Members of the Commission were unwilling to make so large a reduction in the quota from that for the 1962/63 season (15 000 BWU), and therefore agreed on a quota of 10 000 BWU for 1963/64. This gave no effective restriction on whaling activities, and the catches were very close to the predicted values (13 870 fin whales, and a total of 8 429 BWU).
Such close agreement between observed and predicted values was partly a matter of luck, that is, certain assumptions made in constructing the model were quite closely fulfilled. In particular the model assumed that some factors influencing the catch, such as the weather and the skill of the gunners were, in the 1963/64 season, close to the average of previous seasons. These assumptions were nearly fulfilled, but a slight difference in weather or in the skill of a few gunners could have increased the difference between observed and predicted catches to perhaps three or four hundred animals. The closeness of the agreement did not prove that the model is correct, or that a slightly different model might not give a rather closer fit, but does prove that the model used can predict within useful limits the results of one pattern of whaling activity. Presumably the model could also predict the results of other patterns of activity, and in particular the result of a severe restriction in catches for a period long enough to allow the stocks to rebuild. Therefore the model serves as a usable basis for managing the whale resources.
As in the whale example, most models include simplifying assumptions about the factors which are not of immediate interest (e.g. weather), usually stated, when stated explicitly, in the form "assuming the conditions are constant." This should not be taken to mean that the validity of the model depends on the constancy of these conditions, and that the model should not be used if conditions vary. Rather it means that for the purposes for which the model has been constructed, the variations caused by these conditions are not of primary interest, and will be ignored.
This manual is mainly concerned with studying the effect of fishing on the stocks and on the catches; it is, therefore, essentially concerned with the long-term effects. Also when predicting the catch from a given pattern of fishing the important question is often not the absolute amount of catch but the catch relative to that which would have been taken with some other pattern of fishing. Many fluctuations and variations are therefore irrelevant to the purposes of this manual. For instance, very often the average catch per trawl haul is taken as a measure of the abundance of the stock; the actual catch taken in one haul will depend on a large number of factors, e.g. the size of the trawler, the skill of the skipper, the precise ground, the season, the weather and the time of day, some of which may have a considerably greater influence on the catch of that particular haul than the overall abundance of the stock. Fish population dynamics is not however concerned with studying or predicting the catch of one haul, but the average catch over a period, say a year. Most of these factors will have average annual values which are effectively constant from year to year, and thus can be ignored; only if these annual values vary, and particularly if they show a trend (e.g. an increase in the average size of trawler) may they have to be studied in detail.
When comparing the catches from two different patterns of fishing, e.g. trawling with large or small meshes in the cod-end, even year-to-year fluctuations can often be ignored. For instance, variations in recruitment may make big differences to the catch in any one year, but will not in general affect the conclusion that say with the present amount of fishing, the use of meshes of 120 mm in the cod-end will give sustained catches 5 percent larger than those taken when meshes of 110 mm are used.
Any model will eventually have to be replaced or modified, possibly merely by adding a little complexity to take into account further factors (e.g. weather conditions in the whale example above) to attain rather greater precision, or perhaps being replaced by a different model. In any case, the process of construction of a model, testing it, and modifying or replacing it is an essential part of the study and eventual understanding of the dynamics of fish populations, or indeed any subject of scientific study. The process of model construction is the complement of the collection of data, and in fact it is only by constructing and using models that it is possible to decide what data should be collected.
The basic model used in nearly the whole of this manual is one in which the stock considered is taken as being self contained, and the various factors causing the stock to change are considered separately. These factors are as follows:
Recruitment (see section 7) Growth of individuals (see section 3)
Deaths due to fishing; these will be determined by the fishing effort (see sections 4 and 5.3)
Deaths due to other causes (see section 5)
The changes with age in these factors and in the total weight of a year-class are illustrated in Figure 1.1. This shows the growth in weight of the individual; the decrease in the numbers of individuals (the full line shows the decrease in the absence of fishing, the broken line the decrease if fishing occurs after some age, tc); and the total weight of the stock. The last has a maximum at some intermediate age, the "critical" age of Ricker.
In the early sections each of these factors are considered separately, and the models are developed to describe each in quantitative terms. In the final sections the factors are combined to provide a quantitative model for the changes in total weight of the stock, and in catch, for various patterns of fishery. The modifications to the model which are necessary to take into account interactions between the various factors, e.g. that increased fishing will reduce the stock and therefore may affect the recruitment, are discussed briefly in section 11.
FIGURE 1.1. - Changes in numbers and weight of a year-class of fish during its life
