INDIRECT EFFECTS IN ECOLOGICAL INTERACTION NETWORKS .2. THE CONJUGATEVARIABLE APPROACH

A new method called the conjugate variable approach for the analysis o f indirect effects propagating through an ecosystem network is develop ed. For a given ecosystem of n species, the 2n variables representing the inflows and abundances of the n species are related in the n equat ions that define the system's steady states. There are 2 '' alternativ e ways of designating, for each species, either an inflow or an abunda nce variable as an independent variable and the other (i.e., its conju gate variable) as a dependent variable. Each of these alternative ways defines a unique configuration of constraints in the influence propag ation through the ecosystem network. In particular, designating as ind ependent variables the inflow variables of all but two focal species l eaves their abundance variables free to change, thus leading to the ev aluation of the total effect between the two focal species. On the oth er hand, choosing their abundance variables as independent variables t o fix prohibits influence to pass through any of these intermediate sp ecies, thus selectively evaluating the direct effect between the two f ocal species. Any other way between these two extreme cases sets parti al constraints on influence propagation, thus evaluating a partial eff ect between the two species. Four categories of effects between two sp ecies-inflow to abundance, abundance to abundance, inflow to inflow, a nd abundance to inflow-can be distinguished and explicitly evaluated i n terms of community matrix elements.