FISHERY BULLETIN: VOL. 73, NO. 4 



(C) The Trophic Web 



All the fish species in a trophic web can be 

 modeled more or less as described above. Many 

 invertebrates that serve as fish food can be 

 modeled in much the same way, with some 

 appropriate changes in individual component 

 functions and by use of the proper parameter 

 values (see Winberg 1962, and Mann 1969 for dis- 

 cussion relative to invertebrates). Since this model 

 uses feeding functions based on prey abundance, 

 an operational limitation is imposed that the ul- 

 timate resource base-the lowest item in the food 

 chain— cannot be modeled fully in this way. In 

 terms of total ecosystems, this is natural enough. 

 Although the ultimate autotroph might be thought 

 to "prey" upon inorganic nutrients, and models for 

 plant growth as a function of nutrient abundance 

 exist, the present model is obviously not 

 appropriate for autotrophs. 



Thus, any trophic web modeled in this way must 

 have at its base an arbitrarily defined species or 

 group of species. The purpose of this first exercise 

 with the model is to explore trophic interactions 

 among fishes. Therefore, the cause of clarity seems 

 best served by modeling all the species of interest 

 as fishes. The level(s) below the lowest fish 

 species-the food base for the fish community-is 

 then given only the simplest representation. 



Two types of food base have been used in these 

 simulations: 1) the "constant input", and 2) the 

 "exponential growth." Properties of the constant 

 input base are that biomass, B^ , enters the system 

 at a constant rate, P^ , and is reduced only through 

 predation by the higher level, fully modeled 

 species. Thus, the rate of change of food base 

 biomass is 



at 



(19) 



where the summation is over all predators with 

 their individual rations, Cp , and populations, Np . 

 Ecologically, this system might correspond to a 

 fish community whose base prey enters the com- 

 munity feeding area at a constant rate; e.g., as 

 brought in by water circulation or by migration as 

 prey individuals continuously reach a particular 

 life stage. Because of its extreme simplicity, this 

 type food base model is preferred for studying the 

 trophic relationships of fishes higher in the web. 



Properties of the exponential growth food base 

 are that biomass is produced at a rate directly 



proportional to the current standing crop of 

 biomass and is reduced only through predation by 

 the higher level species. Thus 



(IB I 



= c,B, 



^CpNp, 



(20) 



where (\ is a numerical parameter corresponding 

 to the "instantaneous coefl^cient of natural 

 increase" of classical population growth theory. 

 Again the summation is over all predator species 

 preying on the food base. Without predation, B^ 

 would of course increase exponentially and 

 indefinitely. This makes stability of such a system 

 precarious, a fact borne out by experimentation 

 with the model. 



For these simulations, numerical values of P^ 

 and c'l were selected arbitrarily to be compatible 

 with the standard equilibrium state of the trophic 

 webs constructed. 



SIMULATION TECHNIQUE 



Combination of the previously described func- 

 tions produces the basic species model. A single 

 such model species, with one of the food 

 base models described above as prey, was exercised 

 over a range of conditions and with some variety 

 in certain component functions, in an effort to 

 become familiar with some of the dynamic 

 properties of the basic species model. Groups of 

 such model species were then interconnected in 

 various ways to explore the behavior of various 

 trophic webs. Interactions between species occur 

 through Equations (7) and (15). Where a predator 

 feeds on more than one prey species, P in Equation 

 (7) for that predator is the total abundance of all 

 the )t species. For each of the u prey species, the 

 predation mortality imposed by that predator is 

 given by Equation (15) in which the Coupon the fth 

 prey is 



^Pj ~ ^p 



TOTAL „ 



2v,A^, 



(21) 



where A^, is the current population of the iih 

 species, and /', is a coeflRcient expressing predator 

 preference and availability (vulnerability) of the 

 prey. The two (or more) elements contained in v 

 can be separately expressed by making r a product 

 of separate coeflficients. By Equation (21), the 

 predator tends to adjust the makeup of its diet 



702 



