The contribution of direct and indirect flows to the resilience of element cycles [An article from: Acta Oecologica]

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Description:
Element (e.g., nutrient) cycling is often modelled with linear, constant-coefficient differential equations. The transition rate matrix of such models shows only the direct interactions between ecosystem compartments. The indirect interactions here mean the element flows between compartments that are not connected by a direct flow and thus appear zero-valued in the transition rate matrices of continuous time models. This lack of explicitness does not allow their contribution to the system resilience to be evaluated from the elasticity analysis of the dominant eigenvalue of the transition rate matrix. I present a matrix exponential transformation of a continuous time model to result in an equivalent discrete time model, where a matrix of transition probabilities (or proportions) is explicit in both direct and indirect interactions. An elasticity analysis of a probability matrix then can be used quantifying the contributions of both direct and indirect interactions to the dynamics of the system. The method is given an example by a reanalysis of previously published data for the dynamic properties of nutrient cycling in tundra and temperate forest ecosystems. Tundra was shown to have a higher resilience (i.e., a short return time from perturbations) and more indirect flows than forest. Indirect flows generally contributed less to the dynamics of nutrient cycling than direct flows, but were more important in tundra than forest. An analysis of the internal nutrient cycling rate's effects showed that rapid nutrient cycling increases system's resilience and the importance of indirect flows.