THE APPLICATION OF MATHEMATICAL MODELS TO FISH 



POPULATIONS 



J. A. Gull AND 



Fisheries Laboratory, Lowestoft 



Marine fish populations are among the most intensively studied of natural 

 populations — because of course they sustain commercial fisheries of major 

 economic importance. In these studies a variety of mathematical models of 

 varying complexity have been apphed, with varying success. 



THE POPULATION TREATED AS A WHOLE 



In one class of model the entire fish population, or more strictly in most 

 apphcations that part of the population vulnerable to fishing, is treated as a 

 unit. The basis of all these models goes back to the classical theories of 

 population growth of authors such as Cause (1934) and Volterra (193 1), based 

 on populations of generally very simple organisms. An example of uses of 

 such a model is Graham's (1939) study of the North Sea and more recently 

 Schaefer (1954) has extended it to include the fishing fleet as the predator 

 element of a predator-prey system, and to study the expansions and decline 

 of the size of fleet. Such models assume that the gross rate of increase of a 

 population, i.e. the sum of the growth in weight of the individuals in the 

 population, plus the recruitment, less the natural deaths, is determined 

 directly by the size of the population; i.e. in mathematical terms dP/dt 

 = /(P). The general relation between population and increase will be a 

 peaked curve, the increase being zero at zero population and at the maximum 

 population, and having a maximum at some intermediate value. 



The effect of fishing may be readily included in the analysis, so that if 

 f{P) is the gross increase in population, the net increase is/(P) — C where C 

 is the catch (this net increase can of course be negative, indicating a decline 

 in the population). 



The population size will remain unchanged, and the fishery may be said 

 to be in a steady state if/(P) = C, i.e. the catch equals the gross increase. 

 The greatest steady yield will therefore be taken if the population is main- 

 tained at the size giving the maximum rate of increase. Such models have 

 one feature in common with observed natural populations of many kinds — 



