was 0.95 in the French Frigate Shoals model. The 

 nonpredator mortality rate is (1 - e,) • Z|, and the 

 amount of production which goes to nonpredation 

 mortality is 



(1 - e,) P, = (1 - e,) C, B,. 



For n species-groups the biomass budget Equation 

 (1) becomes a system of n simultaneous equations as 

 follows: 



C,5i - 1 {FR,)B,DC„ - (1 - e^)C,B, = catchj, 



k= 1 



C,B, - I^ (FRk) B^ DC,, - (1 - e,) C,B, = catch. 



Schoals in the Northwestern Hawaiian Islands pro- 

 vided the estimates for many of the input parameters 

 required by the ECOPATH model as well as some 

 estimates of biomass and production to serve to 

 evaluate the estimates produced by the model. The 

 estimates of biomass and production generated by 

 the application of ECOPATH to French Frigate 

 Shoals are given in Figure 1 . In general the model's 

 estimates of biomass and production are in good 

 agreement with the available field data (Polovina 

 1984). In the application of the French Frigate 

 Shoals, the biomasses of the top level carnivores are 

 treated as fixed inputs thus a particularly appropri- 

 ate validation of the model is the comparison of the 

 estimate of net benthic primary production with an 

 independent estimate from field data. The model 

 estimated net benthic primary production, adjusted 

 to the total 1,200 km^ habitat of French Frigate 

 Shoals, at 2.3 x 10*' kg • km-^ • yr-i while the esti- 

 mate based on field data was 2.5 x lO'^kg • km"-  

 yr-i (Griggetal. 1984). 



C„B„ - I iFR,)B,DC,„ - (1 - e,;)C„B„ = catch„. 



*■=! 



With input estimates for parameters C„ FR„ DC,j, 

 and e, for all i and j, and catches (catch,) if there is 

 fishing, this system of equations is a system of n 

 simultaneous equations linear in the unknown B-s. 

 This system of equations can be expressed in matrix 

 form as AB = C, where A is an n x n matrix of 

 coefficients, B is an n-dimensional vector of mean an- 

 nual species group biomass, and C is the vector of 

 fishery catch where the ith element is the total catch 

 of the ith species-group. 



If the matrix A is of full rank and if there are some 

 fishery catches for some species so the vector C is not 

 null, then there typically exists a unique nontrivial 

 solution vector of biomass B. If there are no fishery 

 catches then it is necessary to provide an estimate of 

 at least one of the mean species group biomass 5, 

 before there exists a unique nontrivial biomass vec- 

 tor B which solves the budget equation. In the appli- 

 cation of ECOPATH to an ecosystem at French 

 Frigate Schoals where there was no fishing mortali- 

 ty, the biomasses of three apex predators were esti- 

 mated from field censuses and treated as known in- 

 puts. In this application the ith element of C vector 

 was the annual predation by the three apex 

 predators on the ith species- group. 



Five years of field work targeting most of the com- 

 ponents of the marine ecosystem at French Frigate 



The Computer Program 



The ECOPATH model has been implemented via 

 two BASIC language programs. The "dialect" of the 

 language used is BASIC-80, version 5.21, by Micro- 

 soft (CP/M version). These programs are designed to 

 be used interactively on a terminal or a hard-copy 

 printer. The first program is the input parameter 

 program which accepts the input parameters and 

 formats them into a BASIC sequential file. The sec- 

 ond program is the ECOPATH model itself. 



Literature Cited 



Allen, K. R. 



1971. Relation between production and biomass. J. Fish. 

 Res. Board Can. 28:1573-1581. 

 Anojersen, K. p., and E. Ursin. 



1977. A multispecies extension to the Beverton and Holt 

 theory of fishing, with accounts of phosphorus circulation and 

 primary production. Medd. Dan. Fisk. Havunders., New 

 Sen, 7:319-435. 

 Grigg, R. W., J. J. Polovina, and M. J. Atkinson. 



1984. Model of a coral reef ecosystem: Part III Resource 

 Limitation, Community Regulation, Fishery Yield, and 

 Resource Management. Coral Reefs 3:23-27. 

 Laevastu T., and H. a. Larkins. 



1981. Marine fisheries ecosystem: its quantitative evaluation 

 and management. Fishing News Books, Farnham, Surrey, 

 Engl., 162 p. 



Macdonald, J. S., and R. H. Green. 



1983. Redundancy of variables used to describe importance of 



prey species in fish diets. Can. J. Fish. Aquat. Sci. 40:635- 



637. 

 Pauly, D. 



1982. Notes on tropical multispecies fisheries, with a short 



459 



