matrix). Similarly, the total inniicnce of compartment 6 on 9 propogated over the 

 80,937 length 10 paths is 0.048 (I able 4. middle matri.x), represent mgonl\ 5.93 .\ iO"' 

 per path. The cumulative effect, however, is eight times the influence of the direct 

 linkage from 6 to 9(0.006. Table I. lower matri.x). The diagonal entries in the Table 4 

 middle matri.x are smaller, and the nondiagonal values larger, than in anv of the 

 previously illustrated corresponding matrices. Thus, while self influences (diagonal 

 elements) due to paths of increasing lengths are decreasing steadily, at the level of 

 length 10 paths, intercompartmental influences (nondiagonal entries) are still 

 increasing with path length. Eventually, this trend must reverse for the series of 

 partial influence matrices to converge to a final matri.x of total influences. 



The lower matrix of Table 4 shows that the total effect of compartment 4 on itself 

 over paths through length 10 is 7.632 (I2.4'"f direct and 87.6*^7 indirect), and for 

 compartment 6 on 9 the total influence is 0.282 (2. 1 '7 direct and 97.9*^7 indirect). 

 Thus, at the level of length 10 paths, higher order effects are already \ery 

 predominant over direct ones. 



Infinite Order 



The final convergent matrix for the Figure I model is shown in the upper matrix of 

 Table 5, which represents the total influence as fractional daily carbon flow 

 propagated over all paths of all lengths in the system. Comparison with the lower 

 matrix of Table 4 indicates that paths through length 10 hardly begin to account for 

 all the influence in this model. For example, paths of lengths 1 through 10 account for 

 37.69f (7.632 20.298) of the carbon flow from compartment 4 to itself, but lor only 

 1.8% (0.282 15.302) of the flow from compartment 6 to 9. Comparing the lower 

 Table I matrix with the upper matrix of Table 5, the direct influence of compartment 

 4 on itself (0.950) represents only 4. 7*^7 of the total (20.298). and that of compartment 

 6 on 9 (0.006) only 0.04'7 of the total (15.302). Indirect effects in the system are 

 summarized in the middle matrix of Table 5, which represents total influence(upper 

 matrix) less direct effects (Table I, lower matrix). The preponderance of causality 

 propagated as carbon flow in the Figure I model is obviousU indirect, not direct. 



CONCLUSION 



The numbers generated in this simple exercise are impressive. Natural ecosystems 

 must be even more impressive. Real ecosystems have hundreds or thousands of 

 species; the number of causal paths connecting each pair of them must be truly 

 astronomical in most cases. What we have is a situation where influence is 

 propagated so broadly and diffusely in ecosystem networks that its origins for all 

 practical purposes cannot be traced. Add dynamics to the network model, and the 

 situation becomes even more complex. Only direct causes are experienced instan- 

 taneously; as path length increases so does the time from source to destination. 

 System components that have long since gone out of existence could still be exerting 

 significant influence at any given locus. 



Science is not going to deal easily with these realities, which manifest the core of 

 holistic philosophy. The predominance of indirect causality in ecological networks is 

 going to challenge biology right down to its roots. For example, a central 

 consequence of organism-environment separatism is the paradigm of adaptation, 

 strongly rooted in Darwinism. But how may species adapt, much less develop 

 adaptive strategies (Notes, c), in ecosystem networks where there is little relationship 

 between the immediate signals (direct causes) upon which adaptation is based and the 

 total causality emanating from a source? This might be possible if a constant 

 relationship existed between the direct and indirect causes, so that adjustment to the 

 first might automatically provide or imply adaptation to the second. The lower 

 matrix of Table 5, giving indirect 'direct influence ratios for the Figure I model, 

 dispels this possibility immediately. Not counting the =» values denoting division by 

 zero, there are one to three orders of magnitude variation in these ratios for the 



100 



