SAILA and PARRISH: EXPLOITATION EFFECTS 



(a) 



ab 



D 



(b) 



Figure 1. — An illustration of two simple directed graphs. 

 A "self-loop" is shown in part (b) of the figure. 



ure 1(a), the value of B is equal to the input 

 from A plus the input from C: 



B = aA -^ cC. 



(1) 



Similarly, the value of C is equal to the only- 

 input: 



C = bB. 



(2) 



Substituting B from Equation (1) into Equation 

 (2) and solving for C gives: 



C = 



abA 

 1—bc 



(3) 



Thus the ratio of the value of C to the value of 

 A is: 



C  



ab 



(4) 



1 — be ' 



which is the transmission from A to- C. This 

 type of term is used later as a biomass ratio or 

 "trophic efficiency." 



It can easily be seen that the graph in Figure 1 

 simply represents a set of linear algebraic equa- 

 tions. Such sets of equations can, of course, be 

 solved classically. However, solution by inspec- 

 tion of some graphs or parts of graphs is pos- 

 sible. For example, in the graph of Figure 1 (a) , 

 observe that: 



C = {a X b) X A + {c X b) X C. (5) 



The graph can be simplified to that shown in 

 Figure 1 (b) . A "self-loop" has been created 

 that has the effect of making the value of C to be 



_ , , L — ., __ — _ times what it 



1 — loop transmission coefficient 



would have been without the loop. The situ- 

 ation becomes only slightly more complex when 

 the transmission from A to D is considered. 

 The value of D can be obtained from the value 

 of C in Equation (3) as: 



D 



eC = e 



abA 



be 



(6) 



Or, making use of the known effect of a self- 

 loop, it is possible to simply see by inspection of 

 Figure 1(b) that: 



D = {abe)A 



abA 



1 — bc 



1 — bc 



(7) 



These simple principles and techniques are 

 considered adequate for formulating some use- 

 ful trophic graphs. 



GRAPH OF A TROPHIC UNIT 



Graph theory has been applied to the analysis 

 of a variety of problems in engineering, oper- 

 ations research, and the social sciences (Berge, 

 1958; Busacker and Saaty, 1965; Kaufmann, 

 1967; Harary, 1969). Its use in biological sci- 

 ences has been much more limited. However, 

 Benzer (1959) and Maruyama and Yasuda 

 (1970) have applied these concepts to genetics, 

 and Landau (1955) and Trucco (1957) have 

 used graph theory in describing animal behavior- 

 al problems. Menshutkin (1969) appears to 

 have been the first to apply graph theory to the 

 study of communities of aquatic organisms. He 

 used graph theory to derive expressions to illus- 

 trate the relationship of the biomass of harvested 

 organisms (fish) to primary production under 



385 



