FISHERY BULLETIN: VOL. 70, NO. 3 





(7) 



where p is the number of steps of the series 

 eliminated. 



Other types of formulations and summations 

 are, of course, possible. For example, differen- 

 ces in the time for living-to-living and recov- 

 erable-to-living steps may be incorporated by 

 multiplying the constant appropriate to the slow- 

 er step by another constant representing the in- 

 verse ratio of the times of the two respective 

 processes. 



As another example, the average content of 

 each point in time in the matrix can be derived 

 by dividing each term of the series by the num- 

 ber of points along the particular diagonal. 



Other manipulations of this matrix will be 

 suggested later. 



CALCULATION OF THE COEFFICIENTS 

 OF THE PREDACEOUS FOOD WEB 



For the purpose of examining Young's results 

 on cesium, we will proceed with equations (4), 

 (5), and (7) and test the hypothesis that the 

 carnivorous fishes of the Gulf of California are 

 members of an unstructured food web. That is 

 to say that they randomly draw from the mean 

 composition of the food web above the plant 

 level. 



Since, from the Salton Sea results, the Cs/K 

 ratio increases by a factor of about 3 for each 

 living step, we can say that 



Kic ^ SKif, for organic food material (8) 



and since the invertebrates in the Salton Sea 

 which feed partially on detritus show no sub- 

 stantially diflferent increase in Cs/K ratio, I will 

 assume that a double step from living to recov- 

 erable to living material also results in an in- 

 crease in the Cs/K ratio of only about 3, and, 

 hence, that 



K, 



3c 



Ksf 



(9) 



where subscript c denotes the coefficients Ki and 

 Ki applicable to the Cs/K ratio, and subscript / 

 those applicable to food. 



For the conditions in the Gulf of California, 

 it is clear that, with the exception of the mullet, 

 the fish do not consume plant material directly, 

 and we can eliminate the first step in the series, 

 and thus, for non-herbivorous omnivores, p — I, 

 and equation (7) becomes 



M. 



Mo 



K,iKr + Ks) 



(10) 



'" 1—iKr+K,) 

 Let us also define the Cs/K ratio in the plant 



material as: 



then 



Mpc 

 Mpf 



Mac 



^' l-iKrc+Ksc) 



= 16^0 (11) 



l—(Ku + Ksf) 



where the coefficient 16 is approximately the 

 mean increase in the Cs/K ratio from algae to 

 predaceous fish in the Gulf (Table 1), and sub- 

 scripts are as in equations (8) and (9). 



Substituting (8) and (9), equation (11) for 

 foo(f becomes 



S9Ki' + Kxi52Ks—7) —lBKz(l—Ks) = 0.(12) 



Assumption of a range of reasonable values of 

 Ka results in the following: 



Values of ratios MZ/Mo (living material to in- 

 put) , Mt'VMo (recoverable to input) and Mt/Mt" 

 (living to recoverable) are derived, but it is not 

 known what values of these ratios might exist 

 in nature. 



The values of Ki, however, are within the 

 range of values of conversion commonly found in 

 laboratory experiments and culture operations 

 involving growing animal populations. 



We can thus conclude that the assumption of 



K 



In the remainder of this paper, K^ ^ K^f, and K^^ = 



3/- 



1056 



