daily to predator group i, i.e., f,, = a|"xj, is obtained as follows. One-step transition 

 probabilities p-, = P[f'(t-1) = X||f'(t) = xj for the reverse Markov chain jf'(t)}. 

 modeling the history of carbon flows, can be formulated as p[, = a[, T,for i =/ j, and 

 Pi'i = 1-T, (a,,, + a,,) for i = j, where denotes environment. If P[f (t) = x,^] = Ui^ a 



constant, k = l n, where I^ji, u^ = 1, then {f "(t)} can be constructed from {^t)} in 



the following manner: Since p,', = P[f'(t-1) = xjf'(t) - x,] = P[nt-» = Xj nf'(t) = 

 X,] P[r(t) = X,], then p; = P[f'(t-1) = X,] P[f'(t-1) = X,] = (P[f'(t-1) = x, H f'(t) = 

 X,] P[f'(t-1) = xJ)(P[f'(t-l) = x^] P[nt) = X,]). if P[f'(t-1) = X,] = P[^'(t) = .Xj] = u^, 

 then p,', (P[f '(t-1) = xJ P[^'{t-1) = X,) reduces by definition to P[f '(t) = x,|f '(t-1) = 

 Xj]. This is a one step transition probability p,"for a forward Markow chain 

 {f "(t)). whose relationship to the Table 1 daily fractional transfers a"is p,"= a,'Tj, 

 where T^ is turnover time of the prey compartment j. 



g. it is the principle that indirect causality or influence in an interactive network it is 

 important which is to be demonstrated. "Influence" may be manifested in many 

 different ways in a real system, involving objective and subjective, quantitative 

 and qualitative, processes. In the Figure 1 model carbon is taken as a surrogate 

 for general causality. It is assumed that influence can be modeled and quantified 

 in a manner analogous to Table 1. Then, the properties to be developed from 

 Table 1 are general and not especially restricted to carbon flow, which serves in 

 this instance merely as a concrete example. 



h. M. Craig Barber performed the calculations for Tables 2-5. 



i. The proliferation of paths also is pertinent to ecosystem diversity and stability 

 considerations. R. H. MacArthur-'Uouched off a long standing controversy in 

 ecology when he suggested in a network (food web) context that species diversity 

 confers community stability; "Where there is a small number of species (e.g., in 

 arctic regions) the stability condition is hard or impossible to achieve. . . Where 

 there is a large number of species (e.g., in tropical regions) the required stability 

 can be achieved. . ." Resolution of the controversy has been inconclusive, bogged 

 down on the finer points of exact definitions and measures of both diversity and 

 stability, and other technical problems. MacArthur wrote about food webs that, 

 "Stability increases as the number of links increases." This might now be 

 extended to read, stability increases as the number of paths increases, where a 

 path may be direct (a "link") or indirect between any two species or compart- 

 ments. All the paths of all lengths between two compartments represent 

 alternative routes; they are parallel in the network no matter how tortuous. J. 

 Hill and S. L. Durham-' have recently suggested an alternative mechanism to 

 feedback control in ecosystem stability, labeled "congeneric homotaxis." Hill-- 

 writes of this concept; "Congeneric homotaxis is a prototype concept, a new 

 hypothesis of control *** The term congeneric homotaxis identifies a control 

 mechanism resulting from many functionally similar, related or congeneric 

 components. These exist in a similar or homotaxial position in the system 

 structure but each has differing responses to noise or system inputs *** The 

 preferred connotation of the term homotaxis is that of an abstract control 

 mechanism. . .[which] results from the parallel connection of components that 

 are functionally identical with respect to one input but only functionally similar 

 with respect to another. . . For example, a community of phytoplankton, 

 consisting of species with differing optimal temperatures of nutrient uptake 

 rates, exhibits an insensitivity (controlled response) of total biomass to 

 temperature variation, nutrient fluctuations, and even species extension. . .that 

 results from congeneric homotaxis." Hill and I discussed mechanisms of network 

 control in the context of consumer regulation of ecosystems several years before 

 he identified the specific mechanism described above. The focus then was on 

 locating keystone positions in the network to become occupied by evolutionarily 

 "expensive," and therefore highly reliable, species (i.e., the top consumers) which 



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