FISHERY BULLETIN: VOL. 70, NO. 2 



from the network is presented as a matrix (Fo 

 of Figure 8) of frequencies wliich was normal- 

 ized to form a probability matrix (M of Fig- 

 ure 8). This probability matrix M is one in 

 which the i, j entry gives the proportion of the 

 samples from community Vi which resembled 

 community Vj during the sampling period. An 

 important theorem concerning probability mat- 

 rices states that if B and C are probability mat- 

 rices, so is their product BC. A corollary to this 

 theorem states that if M is a probability matrix, 

 then so is every power M", for any positive in- 

 teger n. If the assumption is made that the 

 probability matrix M remains constant over time, 

 then if one knows the initial frequency matrix 

 Fo and the probability matrix M, it is possible 

 to find the sample frequency distribution at a 

 subsequent time tn by finding the nth power of 

 M and then forming the product PoM" where 

 Po is the vector of row sums equal to the initial 

 vector of frequencies by community type. In 

 our case the frequency in year 2 is A = PoM". 

 This matrix-vector multiplication is illustrated 

 in the lower part of Figure 8. It is suggested 

 that this derived frequency might be useful as 

 the expected value basis for comparison with the 

 sample observations made during subsequent 

 years, if it is assumed the probability matrix 

 remains constant over time, 



CONCLUSIONS 



The examples provided in this study are given 

 primarily to illustrate the wide range of possi- 

 bilities for the use of graph theory in studying 

 the stability of interacting competitive and pred- 

 atory relationships. The tentative results of this 

 model study suggest that a nonselective exploi- 

 tation strategy, which includes both predator and 

 prey organisms, may be "best" from the point 

 of view of maintaining community stability in 

 complex ecosystems. The high desirability of 

 obtaining experimental values for certain coef- 

 ficients was also pointed out. 



The limitations of a linear, steady-state model 

 are many and obvious. However, if such a model 

 can sometimes be utilized to provide approximate 

 results suitable for use in practical management 

 at the early stages of marine ecosystem manage- 



ment, then the model is a worthwhile tool, and 

 the method utilized has some merit. If the meth- 

 od (graph theory) can be used not only to obtain 

 some basic insight into system behavior but can 

 also be used as an empirical tool, then it seems 

 particularly worthwhile. Both these possibilities 

 seem to await the results of future imaginative 

 development. 



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