FISHERY BULLETIN: VOL. 81, NO. 2 



E K = 0.1023 R K + 0.002423 (kcal/g 

 dry weight per d). 



(28) 



where K 



B (log,, 10)7) 



(35) 



Since the total daily ration is given by R K — 

 0.079574 s c h (kcal/g dry weight per d), we can sub- 

 stitute and obtain an expression for the total energy 

 lost per day through nitrogen excretion, as a function 

 of the foraging speed (s, cm/s) of the Atlantic men- 

 haden, the concentration of food (c, kcal/1) and the 

 foraging time (/?, h/d): 



E K = 0.008140 sc h + 0.002423 (kcal/g 



dry weight per d). (29) 



Growth Rate, G K and 



Gross Growth Efficiency, K x K 



Equations (12), (19), and (29) may be combined to 

 provide an estimate of the daily growth rate, G K 

 (kcal/g dry weight per d), as a function of menhaden 

 foraging speed (s, cm/s), the concentration of plank- 

 ton in the water (c, kcal/1), and the foraging time (h, h/ 

 d), since 



G K = pR K — T K - E K (kcal/g dry weight per d) 



G K = h [0.06311 s c - 0.00958 (io 002948 s " 1 - 5342 ) 

 + 0.000974] - 0.025803 (kcal/g dry 

 weight per d). (30) 



The gross growth efficiency, K l is calculated ac- 

 cording to 



K - G 



Thus K l in calories is equal to 



Equation (30) 

 Equation (8) 



K\.K ~ 



(31) 



(32) 



From Equation (30) we can also determine the 

 foraging speed which maximizes growth rate (s G 0PT ), 

 for any given values of c and h. First restating Equa- 

 tion (30) in a more general form, replacing the con- 

 stants by A,B, C,D,E,J, andM, 



G K = h\Asc- B(10ids-e>) +J]-M (kcal/g 



dry weight per d). (33) 



We then differentiate Equation (30) with respect to 



s, i.e., set —j—= 0, and we find 

 ds 



\og l0 K + E 



>6\OPT 



+ l ~ log 10 c 



(34) 



In the present study where D. brightwelli is the 

 food, 



s g ,opt= H9.4433 + 33.9213 log 10 c. 



(36) 



To determine the equation for the swimming speed 

 which maximizes gross growth efficiency (s KO pt)> i-e., 



when = 0, we use the following general equa- 

 ls 



tion: 



Ki - 



Equation (30) 



Equation (8) 



_ /? \Asc -fl(10'»>-^) +J] 

 P s c h 



- M 



(37) 



where P is the constant in Equation (8), i.e., in the 

 present example, P — 0.079574. We next define the 

 new constants 



A' = 



(38) 

 (39) 



(40) 

 (41) 



(42) 

 (43) 



ds 



^ -J'= (B"s -B') 10<*-fi> 



(44) 



This identity must be solved iteratively for s KOPT by 

 using a given value of h and trial values of s. 



In the present example using D. brightwelli, we 

 find 



0.32426 



0.01224 = (0.0081722 s - 0.12039) 



X 10<°-02948.s-1.5342) _ 



(45) 



Each term in the energy budget has now been 

 defined in the same three variables: The foraging 

 speed (s), the food concentration (c), and the foraging 



182 



