478 SCHAEFER AND BEVERTON [CHAP. 21 



He had data to estimate AUtjAt, Ut and Xt for each of 20 years. Taking the 

 sums in (23) for the first ten years and the second ten years separately, and the 

 sums in (24) for the whole series, he then had three simultaneous linear 

 equations to solve for the three parameters, l/c, a and M . The solution was made 

 by a method of iteration, but we may note that the same result may be more 

 easily obtained by a straightforward simultaneous solution of the three 

 equations. 



It was found that the estimate of c by this means, for the tuna fishery, had a 

 very low precision but, on the other hand, that there was little difference in 

 the relationships among the variables for a rather wide range of values of c. so 

 that even an approximate estimate of this constant gives as good an estimate 

 of the steady-state relationships among X, U and Y as the data can provide. 



Equation (20) predicts that the equilibrium relationship between effort and 

 catch-per-unit-effort should be linear, and in the same paper the author tested 

 this on the yellowfin tuna data from 1934 to 1955. No significant deviation 

 from linearity could be detected, so that over the observed range of population 

 size, which included sizes near to the maximum (unfished) level down to about 

 half this magnitude, the simple linear interpretation of the function f(P) was 

 a sufficient approximation. This is a valuable confirmation of the Schaefer 

 approach in at least one fishery within the limits of the data, but it is, of course, 

 possible that a more complex function of f(P) might be required to fit a wider 

 range of data. 



In this connection it has to be remembered that the simple law relating rate 

 of population increase to population size formulated by (14) implies certain 

 assumptions about the dynamics of fish populations which are. to a greater or 

 lesser degree, unrealistic, and which, if the degree of departure from reality 

 be large, can limit its usefulness. For some fisheries these will not constitute 

 serious sources of error, but for others they may ; some aspects of these matters 

 have been discussed by Watt (1956), Beverton and Holt (1957) and Richer 

 (1958). 



There are, in fact, two assumptions about the fish population and its applica- 

 tion to data implied in (14) : 



(1) That the rate of natural increase responds immediately to changes in 

 population density. That is, delayed effects of changes in population density 

 on rate of natural increase, such as the effects of the time-lag between spawning 

 and recruitment of resulting progeny, are ignored. 



(2) That the rate of natural increase at a given weight of population is 

 independent of any deviation of the age-composition of the population from 

 the steady-state age-composition at that population weight. 



Neither of these conditions is exactly fulfilled by populations of fishes or 

 other multicellular organisms. The effects of intraspecies competition on in- 

 dividuals between the egg and the age of entry into the catchable stock involve 

 some time lag. Some of the effects of time lags on models similar to those used 

 here have recently been examined by Wangersky and Cunningham (1957). 



