460 RILEY [CHAP. 20 



remainder of the analysis will continue as before, with the assumption of a 

 single group of forage animals. 



There is no indication in the equations that have been presented of an easy 

 way to evaluate the biomass of potential tuna and its predation coefficient. 

 However, several earlier equations have demonstrated that, if either one can be 

 evaluated, the other follows. It is doubtful whether the biomass can be directly 

 determined with any degree of confidence. Alternative methods would require 

 feeding experiments on tuna, which are not likely to be as simple as in the case 

 of small fish. Nevertheless, it seems desirable to examine the matter theoretically 

 to determine what kind of information is needed. 



We can write an equation for the rate of change of potential tuna, which has 

 the same general form as (26). It is 



(IT 



— = T[h{cp r F-y')-Ml (37* 



where fi' , y and h' have the same connotation as in the analysis of forage fish 

 but refer specifically to physiological properties of tuna, and M is a coefficient 

 of total mortality. In the steady state. 



cTfi' h'F = T (h'y' + M). (38) 



T has been retained because expressions for cT can be obtained from equations 

 (27) and (33). These lead to 



a/3 S/S' 8'Fh = T(8'y' + M) (39) 



and 0.001 9/( T)P' 8'F = T(8'y' + 31). (40) 



If M can be evaluated by the usual fisheries technique of analyzing the age 

 structure of the stock, solutions of T and c will follow without difficulty. It is 

 also apparent that M . which is stated in general terms, can be elaborated into 

 a treatment of both natural and fishing mortality. 



It will be noted in equation (39) that potential tuna vary in proportion to 

 the product of zooplankton and forage animals. This will lead to a non-linear 

 relation between zooplankton and tuna, if M is constant and if a positive 

 correlation exists between zooplankton and forage animals. The latter is almost 

 certainly the case, judging from data published by King. Austin and Doty 

 (1957). 



Steele (in lift.) has pointed out a possible variant of this relationship. If 

 there is a significant amount of fishing mortality in certain areas, it may be 

 supposed that M will be proportional to T because the fishermen will tend to 

 congregate in areas of greatest tuna abundance. It then follows from equation 

 (39) that variations in T will more nearly approximate the square root of Fh. 

 Thus the more intensive the fishing becomes, the more nearly will linearity 

 between tuna and zooplankton be realized. 



The application of equations (38) to (40) is not expected to be entirely 

 straightforward. Some of the physiological coefficients probably are not con- 



