PAULY: ESTIMATING FOOD CONSUMPTION OF FISH POPULATIONS 



0.14 



Wo, estimated through 



Type I regression 



W m estimated through 



Type H regression 



■)£ Mean of y,x values 



W(a,) input 



from outside 



log in A B (TypeI)\. logotype I) 



10 



0.00 



K/ 



i X 



7 



2.0 



2.5 



3.0 



3.5 



4.0 



Body weight (g, log units) 



Figure 2.— Relationship between gross food conversion efficiency (KJ and body weight in 

 Epinephelus guttatus. Note that a Type I "predictive" regression leads to an overestimation 

 of W„ while a Type II "functional" regression leads to a value of W^ close to an estimate 

 of W^ based on growth data (see text). Based on data in Menzel (1960). 



model presented above can be made a part of a 

 model for estimation of food consumption per unit 

 biomass (Q/B), provided a set of growth parameters 

 is also used in which the value of W m or W {oo) is iden- 

 tical to that estimated from or used to interpret the 

 feeding experiments. 



In this case, inserting Equation (8) into Equation 

 (5) leads to 



K m = 1 - (1 - e -*«-<o))3P 



(15) 



where K l(t) is the food conversion efficiency of the 

 investigated fish as a function of their age t, and 

 K, t , and /? are as defined above. 

 Equation (1) is then rewritten as 



dq/dt = (dw/dt) IK 



i(«) 



(16) 



where r x = t - t . Equations (17) and (15) may be 

 substituted into Equation (16), which is a separable 

 differential equation and may be solved by direct in- 

 tegration. The cumulative food consumption of an 

 individual fish between the age at recruitment (t r ) 

 and the age at which it dies (t max ) is thus 



Q c = W„ 2,K 



'max 



/ 



(1 - exp(-Kr 1 )) 2  exp(-Kr 1 ) 



1 - (1 - exp(-Kr 1 )) 



3/3 



dt. 



(18) 



The food consumption of a population should de- 

 pend, on the other hand, on the age structure of that 

 population. The simplest way to impose an age struc- 

 ture on a population is to assume exponential decay 

 with instantaneous mortality Z, or 



N t = R e-W-U 



(19) 



where the "growth increment" is replaced by a 

 growth rate (dw/dt) and the "food ingested" is also 

 expressed as a rate (dq/dt ). The growth rate of the 

 fish is then expressd by the first derivative of the 

 VBGF (Equation (7)) or 



Q 



dw/dt = W^ 3K(1 - expi-KrJ) 2  exp(-ifr 1 ) (17) R 



where t r is the age at recruitment (i.e., the starting 

 age at which Z applies, assuming, if there is any 

 fishery, that t r = t c , the mean age at first capture), 

 R the number of recruits, and N t is the number of 

 fish in the population. As the model below assumes 

 a stationary population, the food consumption of the 

 population per unit time can be expressed on a per- 

 recruit basis or 



= W„3K 



f max 



/ 



(1 - exp(-Kr 1 )) 2 • exp(-(i!Lr 1 + Zr 2 )) 



1 - (1 - exp(-iTr 1 )) 3/? 



dt 



(20) 

 831 



