APPLICATION OF MATHEMATICAL MODELS TO FISH POPULATIONS 205 



they are, in the sense used in mechanics, stable. That is, if they are disturbed 

 from the equihbrium state, e.g. by especially good conditions for the survival 

 of young fish in some year, they will return to the original state, while the 

 application of a continuing force, e.g. an established commercial fishery, will 

 alter the population, but only to a new equilibrium level. Though some 

 commercial stocks do show fluctuations, these are usually either short term, 

 due to variable strength in individual year-classes, or long term, due to 

 definite changes in the environment, such as the changes in the abundance 

 and distribution of the cod at Greenland. No fluctuations in amplitude or, 

 especially, regularity comparable to those of some Arctic mammals have 

 been noticed in marine fish. 



In most formulations, Volterra's relation between rate of increase and 

 population size has been used, i.e. increase is proportional to population size 

 multiplied by the difference between the present and maximum population 

 sizes i.e. dP/dt = aP (Pmax ~ ^)' This gives a parabola, with maximum rate 

 of increase at half the maximum population size. However, such observa- 

 tional data as there are, on both natural and artificial fish populations (e.g. 

 Silliman & Outsell, 1958) suggest that maximum rate of increase occurs at a 

 population size well below half the maximum size, perhaps at a third to a 

 quarter. It would obviously be possible to put forward mathematical equa- 

 tions having these properties, e.g./(P) = P{P — M){P + M) has a maximum 

 at JM and zero at O and M, but none has obvious theoretical qualifications. 



The model has other serious disadvantages. No distinction is made about 

 the composition of the stock; populations of a large number of small and 

 young fish, or a small number of old and large fish are considered as the 

 same, though their rates of increase in terms particularly of growth of 

 individuals are likely to be very different. It may be argued that in a steady 

 state only one population composition is possible for a given population 

 size, and that of the two populations with the structures suggested only one 

 at most could be maintained at the same level and with the same structure 

 indefinitely. This may be true, but conditions in the sea are rarely static. 

 The fishing effort is rarely uniform for more than a few years at a time — 

 usually less than the stocks would take to settle to a steady state, even if they 

 could. In most stocks there are considerable fluctuations in the strength of 

 year-classes which mean that the age structure is also fluctuating, the average 

 age being, for example, less than usual when a strong year-class is entering 

 the fishery. 



The rareness of any steady state, particularly in the amount of fishing, 

 leads to a further practical difficulty in applying the theory. A large contribu- 

 tion to the gross increase in any stock is due to the entry of young fish — 

 the recruits. These fish will have been born perhaps several years earlier. 



