446 RILEY |CHAP. 20 



diffusion and the plankton gradient are slight at z', so that the error is negligible. 

 It may also be noted in equation (14) that phytoplankton appears in both the 

 numerator and denominator. Technically this does not cancel the effect of 

 phytoplankton biomass on the zooplankton population, but in practice it very 

 nearly does so. Thus the size of the equilibrium population of herbivores largely 

 depends on the ratio of the production coefficient to the grazing coefficient. 

 This is in no way contradictory to the sense of (2) and (10), in which the growth 

 rate of predators was postulated to depend on the quantity of prey. In the one 

 case an increase in herbivore biomass simply requires an abundance of food at 

 the moment, while in the other case the maintenance of the animal population 

 requires sufficient productivity to ensure renewal of the food supply. 



The equation for the rate of change of herbivores may be restated for present 

 purposes 



— = h(gp-r h -fc), (15) 



where/ is the feeding rate of the carnivores, c. Then in the steady state 



o = (gp-~ h )lf. (16) 



Certain boundary conditions must be met in order to obtain a steady-state 

 solution, but neither these nor the derivation of the physiological coefficients 

 need be discussed in detail. Any further work of this type would require 

 revision of the coefficients in order to make use of new physiological findings. 



The equations can be solved simultaneously because they are all related 

 either through physiological coefficients or through shared biomasses and 

 chemical quantities at successive depth intervals. Solution of the equations 

 leads to relative vertical distributions of phytoplankton and phosphate and 

 relative quantities of animals. One absolute value has been postulated, namely 

 phosphate at the lowest depth interval, and this serves to convert all quantities 

 to absolute terms. 



An example of the kind of results obtained is shown in Fig. 3. Most of the 

 vertical curves were not highly realistic, but total populations were predicted 

 with an average error of not more than 25%. 



Steele (1956) made a theoretical study of plant production on the Fladen 

 Ground. An approximate integration of equation (11) was obtained, since a 

 steady-state solution was inadequate for his seasonal study. Regarding g and v 

 as unknowns, he first applied the equations to two depth intervals and obtained 

 a simultaneous solution of grazing and sinking rates. They were then used in 

 an examination of the form of the vertical profiles. The equation was 



$J>-*>* + t' 



+^r, L„( J ',-2' 1 )<k+=lr iPi-lPi+Pilfa), (") 



