Equivalence of the realized input and output oriented indirect effects metrics in Ecological Network Analysis

Abstract

A new understanding of the consequences of how ecosystem elements are interconnected is emerging from the development and application of Ecological Network Analysis. The relative importance of indirect effects is central to this understanding, and the ratio of indirect flow to direct flow (I/D) is one indicator of their importance. Two methods have been proposed for calculating this indicator. The unit approach shows what would happen if each system member had a unit input or output, while the realized technique determines the ratio using the observed system inputs or outputs. When using the unit method, the input oriented and output oriented ratios can be different, potentially leading to conflicting results. However, we show that the input and output oriented I/D ratios are identical using the realized method when the system is at steady state. This work is a step in the maturation of Ecological Network Analysis that will let it be more readily testable empirically and ultimately more useful for environmental assessment and management.

Static ecological system analysis

“…Moreover, the indirect flow contributions were defined as the flow contributions carried by subsequent steps after their first entrance into a compartment. Therefore, the higher-powered terms (m > 1) in the series expansions are considered as indirect contributions in these formulations [37,39,22,48,5,32]. Even the direct transactions between compartments after the first step (m = 1) are counted as indirect.…”

Dynamic Ecological System Measures

“…Therefore, it is anticipated that the proportion of indirect flows varies seasonally, contributing more in the spring and less in the winter. There are attempts in the literature to demonstrate the seasonality in indirect effects that failed possibly due to the authors' indirect effect formulations as discussed in Section 2.2 [41,43,6,4,35]. The graphs of the composite indirect effect efficiencies presented in Fig.…”

Static ecological system measures

“…Therefore, it is anticipated that the proportion of indirect flows varies seasonally, contributing more in the spring and less in the winter. There are attempts in the literature to demonstrate the seasonality in indirect effects that failed, possibly due to the authors' indirect effect formulations as discussed in Sections 2.2 and 2.6 [39,41,6,4,33]. The proposed indirect effect indices can clearly capture the temporal system behavior as anticipated (see Fig.…”

“…Moreover, the indirect flow contributions were defined as the flow contributions carried by subsequent steps after their first entrance into a compartment, as discussed in Section 2.6. Therefore, the higher-powered terms (m > 1) in the series expansions are considered as indirect contributions in these formulations [39,41,23,54,4,33]. Even the direct transactions between compartments after the first step (m = 1) are counted as indirect.…”

“…The indirect effects are considered to be flow contributions carried by subsequent steps after the first entrance into a compartment in static ecological network analyses. Even the direct transactions between compartments, after the first step, are considered as indirect contributions in various formulations proposed in the literature, as discussed in Section 2.2 [39,41,23,54,4,33]. These indirect effect indices are formulated without actually defining the indirect flow between two system compartments.…”