Figure 1 system than of length 1 (Table 1 ). For example, there are two length 2 paths 

 from the demersal carnivore compartment (4) back to itself, and there are four such 

 paths from benthic residents (6) to benthos (9). These are (Figure 1) 4—4—4 and 

 4—6—4 in the first instance, and 6—1—9, 6—5—9, 6—6—9 and 6—9—9 in the second. 



The middle matrix of Table 2 shows the cumulative strengths of the length 2 

 influence paths in terms of daily fractional carbon flows (values all denoted "> 0" 

 range < a,'j< 0.0005). The value from compartment 4 back to itself is 0.903, which is 

 95% of the self influence over the length 1 path (0.950, Table 1). The value from 

 compartment 6 to 9 (Table 2) is 0.01 1, which is greater than the direct path value of 

 0.006. The total effect over length 1 and 2 paths is obtained by summing their 

 individual effects (Table 2, lower matrix). These values can now exceed I because 

 each parallel path between two compartments has a value < 1, and the individual 

 path values are summed. 



The total effect of compartment 4 on itself over length 1 and 2 paths is 1.853(51% 

 direct and 49% indirect), and for compartment 6 on 9 the value is 0.017 (35% direct 

 and 65% indirect). All the diagonal entries in Table 2 are smaller than corresponding 

 values in Table 1, so the direct paths are more significant in this category. The 

 nondiagonal values are all larger, however, so in every intercompartmental 

 interaction the length 2 paths are more important than those of length 1. 



Third Order 



The same holds for paths of length 3, which are derived as product matrices (Table 

 3) of the respective Tables I and 2 matrices. The number of length 3 paths (Table 3, 

 upper matrix) is greater than the number of paths of length 2. There are. for example, 

 seven such paths from compartment 4 to itself: 4—4—4—4. 4—4—6—4. 4—5—9—4. 

 4—6—1—4. 4—6—4—4. 4—6—6—4 and 4—6—9—4. And there are fourteen length 3 

 paths from compartment 6 to 9: 6—1 — 1—9. 6—1—5—9. 6—1—9—9, 6—4—5—9. 

 6-4-6-9. 6-5-5-9. 6-5-9-9. 6-6-1-9. 6-6-5-9, 6-6-6-9. 6-6-9-9. 

 6-9-5-9, 6-9-6-9. and 6-9-9-9. 



The cumulative intTuence. as fractional daily carbon flow, generated over these 

 paths is 0.858 from 4 to 4. and 0.017 from 6 to 9 (Table 3, middle matrix). As in the 

 case of the length 2 paths, the diagonal entries are smaller in value than the 

 corresponding entries in Table 1 representing direct effects, but the nondiagonal 

 values are greater indicating greater indirect inOuence over paths of length 3 than 

 direct between these compartments. 



Total influences overall paths of lengths 1 through 3 are given in the lower Table 3 

 matrix. The total effect of compartment 4 on itself over paths through length 3 is 

 2.710 (35% direct and 65% indirect), and for compartment 6 on 9 the value is 0.034 

 (18% direct and 82% indirect). Thus, at the level of length 3 paths, indirect effect^ 

 already are becoming more important than direct ones. 



The total effect of one compartment on another within a system is given by the 

 cumulative influence propagated over all paths of all lengths connecting the two 

 compartments. Thus, the matrix multiplication process continues, to form an infinite 

 series which converges in the limit to a final, transitive closure matrix'"^-'^ in which all 

 the influence over all paths of all lengths is fully accounted for. Of course, as path 

 lengths increase the influence over any one of them becomes small, but the 

 combinatorial increase in the number of paths may be dramatic enough that their 

 cumulative influence is significant (Notes, i). 



Tenth Order 



Table 4 gives for the Figure 1 system the number of paths of length 10 (upper 

 matrix), and the daily fractional carbon flow over these paths (middle matrix) and 

 over all paths of all lengths through 10 (lower matrix). The combinatorial increase in 

 number of paths is evident. For example, there are 40,619 paths of length 10 from 

 compartment 4 to itself, and 80,937 from compartment 6 to 9. Numbers like these and 



97 



