Indirect effects in ecological interaction networks. II. The conjugate variable approach

A new method called the conjugate variable approach for the analysis of indirect effects propagating through an ecosystem network is developed. For a given ecosystem of n species, the 2n variables representing the inflows and abundances of the n species are related in the n equations that define the system's steady states. There are 2n alternative ways of designating, for each species, either an inflow or an abundance variable as an independent variable and the other (i.e., its conjugate variable) as a dependent variable. Each of these alternative ways defines a unique configuration of constraints in the influence propagation through the ecosystem network. In particular, designating as independent variables the inflow variables of all but two focal species leaves their abundance variables free to change, thus leading to the evaluation of the total effect between the two focal species. On the other hand, choosing their abundance variables as independent variables to fix prohibits influence to pass through any of these intermediate species, thus selectively evaluating the direct effect between the two focal species. Any other way between these two extreme cases sets partial constraints on influence propagation, thus evaluating a partial effect between the two species. Four categories of effects between two species--inflow to abundance, abundance to abundance, inflow to inflow, and abundance to inflow--can be distinguished and explicitly evaluated in terms of community matrix elements.