FISHERY BULLETIN; VOL. 73, NO. 4 



The model was employed in a number of 

 simulations for a variety of trophic webs. There 

 were not sufficient data at hand for all the species 

 of a real trophic web to permit simulation of such a 

 web in this way. The parameters and initial values 

 used in these simulations are, therefore, reason- 

 able illustrative values for fish in the natural en- 

 vironment, based mostly on the literature. 



Despite the scarcity of the real data that would 

 be required to use the model effectively for quan- 

 titative prediction of real systems at present, 

 models of this sort have considerable immediate 

 value. Representing and interrelating animal 

 functions analytically enforces a discipline in 

 thinking which tends to clarify perceptions of the 

 trophic relations. Formulation of a system in 

 mathematical functions makes clear the nature of 

 the data required, so that effort in gathering data 

 can be applied efficiently. Component functions 

 for a single species can be collected from a variety 

 of sources and fused. The trophic behavior of the 

 resulting model species can be studied, at least 

 qualitatively, to see if the model behaves as the 

 animal appears to behave. If the species model 

 appears to represent the animal reasonably well, 

 and if a trophic web is constructed from such 

 animals, some confidence may be placed in its 

 predictions of the behavior of the real system— a 

 system which may be much more difficult to 

 evaluate independently of the model. 



THE MODEL 



The basic model used for each species in every 

 trophic web was developed from an energy 

 balance of the individual and a formulation of the 

 population dynamics of the species. Table 1 con- 

 tains a glossary of symbolic notation used in the 

 model. 



(A) The Energy Balance 



The energy balance was written by equating 

 assimilated food intake, kC, to the sum of the three 

 physiological uses of the assimilated food: res- 

 piratory metabolism, Q, reproductive material 

 produced for spawning, S, and growth, G. 



kC = Q + S + G 



(1) 



(1971), and elsewhere. Mann (1965) has made one 

 of the very few attempts to include the S' term 

 quantitatively in the balance. Most workers (e.g., 

 Winberg 1956:209; Mann 1967, 1969) find that to a 

 very acceptable ecological approximation, most 

 fish under most circumstances assimilate a fairly 

 constant fraction, A-aO.8, of the food, C, consumed 

 (C, the feeding rate, is commonly called the ration 

 and will be so designated herein). Ten (1967) 

 should be consulted for a minority opinion on the 

 effective constancy of k. Kostitzin (1939:180) and 

 Beverton and Holt (1957:113) also deal with the 

 form of possible variation. 



All the above terms are time rates. In the 

 present simulations, the time unit used is the year. 

 Since G is dW/dt, the instantaneous value of body 

 weight, W, can be found by integration of G. Each 

 term in Equation (1) can be expressed as energy or 

 as the equivalent weight of body tissue, wet (live) 

 or dry. In these simulations, all terms for all 

 species are expressed in wet weight of tissue, 

 based on a standard conversion factor of 1 kcal/g 

 wet weight (Winberg 1956; Mann 1969). Recent 

 results (Davis 1968; Kausch 1968; Brett et al. 1969) 

 on changes in water content of fish tissues at 

 various nutritional states suggest that a dry 

 weight basis may be noticeably more accurate 

 where data are available. The use of different 

 conversion factors for different species or condi- 

 tions, when known, introduces no conceptual 

 problems. That Equation (1) can be balanced using 

 experimental values of A-, C, Q, and G determined 

 simultaneously in the laboratory for a group of fish 

 over a range of sizes, ambient temperatures, and 

 nutritional states, has been demonstrated by 

 Kausch (1968), using the carp, Ci/piinus carpio. 



The results of many investigations indicate that 

 respiratory metabolism can be expressed 

 approximately as a function of body weight, W, by 

 the relation 



Q = aW"^, 



(2) 



Similar expressions are found in Winberg 

 (1956:210, 1962), Ivlev (1961a), Warren and Davis 

 (1967), Mann (1967, 1969), Davis and Warren 



where Y is some fractional power. For most fish 

 species a value of Y = 0.8 appears to be sufficiently 

 reliable for ecological pui'poses (Winberg 1956:149, 

 1962; Mann 1965, 1969; Paloheimo and Dickie 1966). 

 Where a more accurate value of 7 is known for a 

 particular species, the model will accept it readily. 

 For the purpose of the present simulations, a 

 level of a for a constant (or long-term average) 

 temperature of 10°C is used. Based on a large 



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