Ecosystems are commonly modeled as networks of matter or energy stocks connected by flows. Network environ analysis (NEA) is a set of mathematical methods for using powers of matrices to trace energy and material flows through such models. NEA has revealed several interesting properties of flow–storage networks, including dominance of indirect effects and the tendency for networks to create mutually positive interactions between species.

There are several limitations on the applicability of present NEA methods. The methodology can only be applied to models at constant steady states. This greatly limits the range of applicability because not all models reach constant steady states, and those that do may also have significant, but unanalyzable, transient behavior.

Dynamic environ approximation (DEA) is a computational approach to dynamic environ analysis. It uses products instead of powers of matrices, which allows it to be applied to systems that are changing in time.

We analyzed energy flow in two implementations of a model of energy flow through an oyster reef ecosystem. The linear model incorporates donor-dependent flows, while in the nonlinear implementation, internal flows are proportional to the product of donor and recipient abundances; only flows to the detritus compartment are exclusively donor-controlled.

Results/Conclusions

In the linear model with donor-dependent flows, the fraction of energy flow between two compartments that travels over walks of length 2 or greater remained constant as state variables changed. In the nonlinear model, the indirect fraction of flows varied with time. Both models showed significant, often dominant, contributions by indirect flows.

The dynamic environ approximation approach described here has a broad range of potential applications. It should be possible to investigate energy cycling and system properties such as dominance of indirect effects in models that have not previously been subjected to such analysis. One promising area of application for dynamic environ approximation is the analysis of bioenergetic food-web models; individual-based models, including those incorporating evolution, could also be investigated. Such applications could provide a much-needed link between conventional and systems ecology.