Indirect effects in ecological interaction networks I. The chain rule approach

A mathematical method for evaluating indirect effects propagated through ecosystems consisting of multiple species is developed. The time-backward expansion of the sensitivity matrix of a system at steady state represents the tracking back of the total effects received by species. Aggregating those portions of the total effect between two species that travel through a common path with various schedules gives the path partitioning of the total effect. From this path partitioning, a chain rule is derived that expresses the indirect effect transmitted through an individual path as the products of direct effects associated with the links constituting the path. The evaluation of indirect effects by this chain rule is applied to example systems to reveal the entire structure of influence propagation through the systems. The results of this application suggest three basic mechanisms through which indirect effects contribute to the complexity and contingency of species interactions: (i) the globalization of influence by bundles of long indirect paths, (ii) the amplification (or reduction) of effects by positive (or negative) cycles, and (iii) the alteration in sign of interactions between a pair of species due to the change in dominance among the effects carried by parallel paths connecting the species.