FISHERY BULLETIN: VOL. 70, NO. 2 



12 



(16) 



(17) 



This linear expression is used to approximate 

 the following modification of Ivlev's exponential 

 relation: 



-ID 



R = H{1 — e ~^)' ^^^) 



where D and H are defined as before. 



The implementation of Equation (14) in the 

 graph of Figure 2 (a) is seen by summing inputs 

 at vertex 812: 



S12 = ( + l)Z>i2 + { — l)Hx2 = D12 — H 



and at vertex R12: 



R12 = ttviHio + bi2 812 . 



Although lacking in mathematical rigor, this 

 linear approximation can be made to give rea- 

 sonably accurate results over a limited range 

 of prey density, and it is considered to be an 

 improvement over the simple density-indepen- 

 dent assumption. Figure 3 provides a sample 

 comparison of an Ivlev exponential relationship 

 according to Equation (15) with the linear ap- 

 proximation of Equation (14). The coefficients 

 a and b should, of course, be chosen in any real 

 case to approximate either a desired analytic 

 function of known utility or a set of data on 

 feeding observations. 



The "network analysis" techniques described 

 previously were applied to the graph of Figure 

 2(a) to produce the simplified graphs shown as 

 Figure 2(b) and Figure 2(c). The derivations 



§ 04 



Z 



I 



s 



OS 10 



ACCESSIBLE PHEY 

 MAXIMUM RATION 



Figure 3. — Linear and exponential approximations of 

 feeding behavior. 



used do not require Pi or mi. Furthermore, Mi, 

 M2, and U2 cannot give "inputs" at any vertex 

 since they are all directed outward. Therefore, 

 the above parameters were eliminated with no 

 effect on the solutions. Vertices P2 and Q2 were 

 absorbed using graph theory network techniques 

 to produce Figure 2(b). The same figure shows 

 the similar absorption of vertices Dri, 812, and 

 Hv2. Parallel inputs to a vertex can be combined. 

 In this case, the two self-loops at vertex B2 were 

 combined, and the two edges from B2 to R12 were 

 combined. The resulting simplified graph, Fig- 

 ure 2(c) , is the most basic graph that expresses 

 the assumed relationships. 



The above formulation describes a two-species 

 predation model where Species 2 preys on Spe- 

 cies 1. At this trophic level, and for the form- 

 ulations to be used, the term "predation" is 

 applied in its broadest sense. Since the formula- 

 tion does not make use of production, mortality 

 or any other vital property of Species 1, Species 

 1 is really just a resource. It could be vegetation, 

 or with some reinterpretation of coefficients, 

 even living space. Clearly, the above graph is 

 a building block from which a variety of more 

 complex food webs can be constructed. Only 

 limited applications of this concept are made in 

 the following material, and its validity awaits 

 the test of further applications. 



SOME MODELS AND THEIR 

 INTERPRETATION 



Since relatively little observational informa- 

 tion is available concerning the important prob- 

 lem of community interactions, it was believed 

 that a model study such as this might assist 

 in a further understanding when additional ob- 

 servational data are taken. 



Competitive and predatory interactions, with 

 and without exploitation, were examined using 

 trophic graphs made from the building block 

 developed previously. Figure 4 shows Species 2 

 and Species 3 competing in their utilization of 

 resource Bu A relation was derived for the ra- 

 tio of the biomass of each of the competitors 

 to that of the resource: B2/B1 and B3/B1. In 

 either case this was done by writing the very 

 simple linear equations for each of two vertices 



388 



