## Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

Resolving the complexity of ecosystem models using network theory
Caner Kazanci

#### Abstract

Ecosystems are often modeled using directed graphs, representing flow of energy or nutrients among species. While ecosystems seem to be made up of flows among compartments, neither flows, nor compartments can function by themselves. Motivated by Flux balance analysis (Kauffman, 2003) and metabolic control analysis (MCA), we propose a new building block for ecosystems, called fluxes. A flux is a sub-network defined according to specific mathematical rules. In ecological terms, a flux represents the smallest process within the ecosystem that can theoretically sustain itself. This can be a material cycle within the ecosystem, or a simple food chain in a complex food-web. In mathematical terms, fluxes form a cycle basis of the ecological network assuming the environment is a node included in the digraph.

Fluxes have interesting properties that render them extremely useful for ecological studies. For example, any ecological network has a unique set of fluxes. And any ecosystem model can be expressed as a linear combination of its fluxes. Fluxes can be regarded as the "genes" of the ecosystem, as they indeed are the minimal functional group. Identifying important fluxes for an ecosystem model might be as important as identifying important compartments (e.g. keystone species) or important flows (e.g. betweenness measure, Freeman, 1977).

Fluxes provide a unique opportunity to study large and complex ecosystems. Since no connections are broken during this decomposition, system-wide properties of the full ecosystem, such as cycling, can be studied using individual fluxes. For example, the amount of material cycling that occurs within the whole ecosystem equals the sum of material cycling that occurs within each of its fluxes. This result holds regardless of the size or complexity of the model. Other similar system-wide ecosystem properties are conserved under this decomposition.

In this talk, we define what a flux is and describe its properties. We demonstrate the flux decomposition using the intertidal oyster reef and the Georgia salt marsh ecosystem models. We discuss how system-wide properties of an entire ecosystem can be studied using only fluxes, and demonstrate this using the cycling index. Finally, we show how to decompose a model into its fluxes using EcoNet (Kazanci, 2007), a free online software we have developed.

#### Keywords

network, graph, system, ecology