The formulation of each term of the biomass 

 budget equation will now be presented in detail. 



The Model 

 Biomass Production 



Production (P) for a cohort of animals over 1 yr is 

 defined as 



1 J 



F = f N, — (w,) dt 

 dt 



and mean annual biomass (B) for the cohort is de- 

 fined as 



B = J NtWfdt 



where iV, is the number of animals and w, the mean 

 individual weight at time t. 



Allen (1971) investigated the production to bio- 

 mass (PIB) ratio for a cohort over a range of mortal- 

 ity and growth functions. For a number of growth 

 and mortality functions, including negative exponen- 

 tial mortality and von Bertalanffy growth, the ratio 

 of annual production to mean biomass for a cohort is 

 the annual instantaneous total mortality (Z,). For a 

 species-group which consists of n cohorts or species, 

 with instantaneous annual total mortality (Z,) for 

 cohort or species i, where mortality is determined by 

 a negative exponential function and growth by a von 

 Bertalanffy growth function, the total species-group 

 production (P) is the sum of the cohort production 

 (P,) and can be expresed as 



P= I P,= 1 Z,B, 



(2) 



i:=i 



( = 1 



Under the assumption that the Z's are all equal to 

 say Z, then total species-group production can be ex- 

 pressed as 



P = Z  B 



where B is the mean annual species-group biomass. 

 Allen (1971) has also shown that when growth in 

 weight is linear, the PIB ratio is equal to the recipro- 

 cal of the mean age for a range of mortality func- 

 tions. For a number of other growth and mortality 

 functions the ratio of cohort PIB can be the recipro- 



cal of the mean lifespan. Thus, for a range of growth 

 and mortality functions, total species-group produc- 

 tion can be expressed as 



P = C  B 



where B is the mean annual species-group biomass, 

 and C is a parameter. 



In an application of ECOPATH to an ecosystem of 

 French Frigate Shoals where there was very little 

 fishing mortality, the PIB ratio for fishes and crusta- 

 ceans was taken as the annual instantaneous natural 

 mortality (M); whereas, for primary and secondary 

 producers whose growth is more likely to be linear 

 than the von Bertalanffy, the PIB ratio was esti- 

 mated as the reciprocal of the mean age (Polovina 

 1984). 



Predation Mortality 



The predation mortality is the fraction of the 

 biomass of a species-group which is consumed by all 

 predators excluding fishing mortality. Two types of 

 information are needed. First the food web or 

 predator-prey relationships must be defined. A diet 

 composition matrix DC,, must be specified where an 

 entry DC,j from this matrix refers to the proportion 

 (by weight) of prey j in the diet of predator (. The 

 primary source of this information is the analysis of 

 stomach contents data. At least in one study it has 

 been shown that there is a high correlation between 

 diet indices based on weight, volume, and percentage 

 of occurrence for stomach content data, and thus 

 either index may be used to generate the DC matrix 

 (Macdonald and Green 1983). The second type of in- 

 formation needed to ascertain predation mortality is 

 the food requirements of the predator. The 

 ECOPATH model requires the user to specify FR„ 

 the ratio of annual consumption to mean annual 

 biomass. The annual food required by the predator is 

 the product of FP, and P,. 



Some values of daily food required as a fraction of 

 body weight range from 0.005 to 0.02 (Laevastu and 

 Larkins 1981). Based on these daily estimates a 

 range of annual food required as a fraction of mean 

 biomass (FPj is 1.8 to 7.3. 



Nonpredation Mortality 



All mortality attributable to causes other than 

 predation and fishing is termed nonpredatory mor- 

 tality. The ECOPATH model defines ecotrophic effi- 

 ciency e, as the fraction of total production which is 

 removed by fishing and predation mortality. This 



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