FISHERY BULLETIN: VOL. 70, NO. 2 



1969). One of the generalizations which 

 appeared to emerge from the s^-mposium was 

 that ecosystems characterized by high species 

 diversity tended to be relatively stable. Elton 

 (1958) has shown that if a predator has several 

 alternate prey species to utilize, it will persist 

 even if one of the prey becomes very scarce. 

 Thus, it seems as if there is some correlation be- 

 tween diversity and stability. 



In the case of marine invertebrates, there is 

 some observational evidence (Paine, 1963) to 

 indicate that marine predators at high levels in 

 the food chain feed on more species of prey than 

 do those on lower levels. Observations, such as 

 the above mentioned, have led some ecologists to 

 suggest that high level predators might contrib- 

 ute more to community stability than the lower 

 level predators. 



Removal of predators from rocky shore inver- 

 tebrate communities (Paine, 1966) resulted in 

 a reduction of the species diversity of the an- 

 imal community. In addition, removal of graz- 

 ing herbivores from rocky shores has resulted 

 in the rapid growth of some of the formerly eaten 

 plant species and a change in community com- 

 position toward lower species diversity (Jones, 

 1948; Southward, 1964; Paine and Vadas, 

 1969). The observations and experiments of 

 Paine ( 1966) also indicated that diversity among 

 competing species of marine invertebrates could 

 be decreased by removal of predators in some in- 

 stances. A theoretical dynamic analysis (Par- 

 rish and Saila, 1970) of a trophic subweb using 

 Lotka-Volterra type interactions offered some 

 support to Paine's conclusions. 



With the exception of some pioneering con- 

 ceptual work by Larkin (1963, 1966) in describ- 

 ing models for interspecific competition and ex- 

 ploitation applied to natural fisheries, very little 

 seems to have been done in an effort to predict 

 the effects of man's activities on aquatic com- 

 munities consisting of several interacting spe- 

 cies. One of the reasons for this appears to lie 

 in the degree of complexity required to establish 

 and express all the basic interrelationships in 

 such a system (Mann, 1969). Recently, Men- 

 shutkin (1969) suggested graph theory as a use- 

 ful tool for minimizing some of the difficulties 

 of constructing models of interacting systems 



if certain simplifying assumptions, such as lin- 

 earity and steady-state conditions, could be 

 tolerated. Recognizing that any set of mathe- 

 matical equations represents at best a crude ap- 

 proximation of the actual behavior of complex 

 ecosystems and that empirical values of coeffi- 

 cients for complex models are largely unavail- 

 able, we have proceeded under the assumption 

 that the simplest models should first be explored 

 and carefully evaluated before proceeding to 

 more elaborate formulations. In addition, it is 

 believed that the simplicity of the methods de- 

 scribed herein may enhance their utility, espe- 

 cially when considering the initial effects of ex- 

 ploitation or environmental modification on in- 

 teracting ecosystems. 



The objectives of this work were to: (a) in- 

 troduce a subset of graph theory as used in net- 

 work analysis; (b) describe a graph theoretic 

 formulation of a basic ecological trophic unit, and 

 to demonstrate some effects of predation and ex- 

 ploitation on model ecosystems consisting of 

 these units; and (c) demonstrate some other 

 uses of graph theory with a view toward stim- 

 ulating further interest in its applications. 



BACKGROUND 

 AND DEVELOPMENT 



By definition, a graph is a set of vertices 

 (nodes) connected by a set of edges (branches) . 

 If the graph has polarity or direction, the edges 

 have arrows, and the graph is said to be directed. 

 In this report we are concerned only with direct- 

 ed graphs. Two very simple directed graphs are 

 illustrated in Figure 1. 



The ecological graphs utilized herein are based 

 largely on graph theoretical techniques of net- 

 work analysis, for which the theory has been 

 clearly and concisely presented by Mason and 

 Zimmermann (1960). To analyze a network, 

 each edge connecting two vertices is given a co- 

 efficient, a "transfer function" or "branch trans- 

 mission." The "transmission" from one vertex 

 A to a distant one C can then be expressed as a 

 combination of these individual coefficients. The 

 important principle is that the value of any ver- 

 tex is the sum of the directed inputs, regardless 

 of the outputs. In the very simple case of Fig- 



384 



