FISHERY BULLETIN: VOL. 81. NO. 2 



linearly with increasing c (Fig. 1, Bl). The energy ex- 

 penditure to exogenous nitrogen excretion (E f K ) also 

 increases linearly, whereas E h N and respiration (T r K 

 and T f K ) are constant (Fig. 1, B2). The curves rep- 

 resenting energy intake and expenditure both in- 

 crease linearly with increasing values of c (Fig. 1,B3), 

 and thus growth (G K ) increases linearly and gross 

 growth efficiency increases asymptotically (Fig. 1, 

 B4). 



InFigure 1, Cl-C4,/i increases, whiles andc remain 

 constant. Here, also, the energy intake (R andpR) in- 

 creases linearly with increasing h (Fig. 1,C1). The en- 

 ergy expenditure to endogenous nitrogen excretion 

 (E b K ) remains constant, while exogenous nitrogen 

 excretion (E fJi ) and the respiration during feeding (T /A ) 

 increase linearly, and the routine respiration (T r K ) 

 declines linearly (Fig. 1, C2). The curves describing 

 the energy intake and expenditure increase linearly 

 with increasing values of h, (Fig. 1, C3), and again we 

 find that growth (G K ) increases linearly, and gross 

 growth efficiency (K" 1-A -) increases asymptotically 

 (Fig. 1, C4) 



These examples demonstrate that in order for an 

 Atlantic menhaden, which forages ats cm/s for/? h/d, 

 to obtain a maintenance ration, the concentration of 

 food must equal a minimum threshold value, c min 

 (i.e., 0.0021 kcal/1 in Fig. 1, B3-B4). Similarly, a 

 menhaden foraging ats cm/s when the plankton con- 

 centration = c kcal/1, must feed for some minimum 

 period h mm (in Fig. 1, C3 and C4; 6.2 h/d) in order 

 to obtain a maintenance ration. There will also be a 

 minimum foraging speed, s min , required to obtain a 

 maintenance ration for each combination of c and h 

 (in Fig. 1, A3 and A4; 7.0 cm/s). If growth is to occur, s, 

 c, and h must exceed s mm , c^ n , and h min . The general 

 rule is that for any swimming speed (s), the more 

 abundant the food, the smaller the maintenance ra- 

 tion, and the shorter the feeding time required to ob- 

 tain the ration (Fig. 2, A, B). If an Atlantic menhaden 

 forages at 41.3 cm/s, for example, the lowest concen- 

 tration of Ditylum at which it could obtain a main- 

 tenance ration would be about 0.0018 kcal/1, 

 assuming that it fed for 24 h/d. The maintenance ra- 

 tion would be about 0.143 kcal/g dry weight per d. 

 With an increase in plankton concentration, the re- 

 quired feeding time and the maintenance ration 

 decline very rapidly, reaching 4 h/d and 0.05 1 kcal/g 

 dry weight per d at c = 0.0039 kcal/1, and declining 

 more slowly thereafter to 1 .3 h/d and 0.038 kcal/g dry 

 weight per d at c — 0.009 kcal/1. 



An interesting feature of the energy budget is that 

 for any combination of c and/?, there is a single forag- 

 ing speed which will maximize the growth rate(s (; 0PT ) 

 (Fig. 1, A4). Similarly, growth efficiency reaches its 



186 



UJ j 12 



< ^ 04 



s 



s = Observed 

 s = 4i3 cm/sec 



_i i I I i_ 



s : Observed 

 s= 41 3 cm/sec 



00006 00018 00030 00042 00054 00066 00078 00090 



PLANKTON CONCENTRATION (c.kcol//) 



FIGURE 2. — A, relationship between the concentration of plankton 

 and the maintenance ration of Atlantic menhaden which are 

 assumed to swim at a constant speed of 41.3 cm/s (Model I) and at 

 their actual speeds in response to plankton concentration (Model II). 

 B, the foraging time required for the Atlantic menhaden to obtain a 

 maintenance ration at different concentrations of plankton, assum- 

 ing that they swim at 41.3 cm/s (Model I) or at the actual speed which 

 has been observed in the laboratory (Model II). 



maximum value at a unique foraging speed (s A0PT ), 

 which is always less thans (; 0PT . s G 0PT increases cur- 

 vilinearly with increasing food concentration (Fig. 3), 

 but is independent of the duration of feeding (Fig. 4, 

 Equation (36)). In contrast, s AOPT declines as the 

 duration of feeding increases (Fig. 4), but is indepen- 

 dent of food concentration (Fig. 3, Equation (45)). It 

 should be remembered however that the values of G. 



and K UK when the fish swim at s G 0PT and s, 



'K 

 'A, OPT are 



determined by both c and /?. For example, if c — 

 0.0030 kcal/1, a fish will maximize its growth rate if it 

 swims at 33.9 cm/s although the actual rate of growth 



0030 00060 0090 



PLANKTON CONCENTRATION (c. kcal/Z) 



FIGURE 3. — The relationship between plankton concentration and 

 the foraging speed which maximizes the Atlantic menhaden's 

 growth rate (s L; opt'- s G OPT ' s independent of foraging time (h). 



