Application of input to state stability to reservoir models

Abstract

Reservoir models play an important role in representing fluxes of matter and energy in ecological systems and are the basis of most models in biogeochemistry. These models are commonly used to study the effects of environmental change on the cycling of biogeochemical elements and to predict potential transitions of ecosystems to alternative states. To study critical regime changes of time-dependent, externally forced biogeochemical systems, we analyze the behavior of reservoir models typical for element cycling (e.g., terrestrial carbon cycle) with respect to time-varying signals by applying the mathematical concept of input to state stability (ISS). In particular, we discuss ISS as a generalization of preceding stability notions for non-autonomous, non-linear reservoir models represented by systems of ordinary differential equations explicitly dependent on time and a time-varying input signal. We also show how ISS enhances existing stability concepts, previously only available for linear time variant (LTV) systems, to a tool applicable also in the non-linear case.