Operationalization of Topology of Sustainable Management to Estimate Qualitatively Different Regions in State Space

Tim Kittel, Rebekka Koch, Jobst Heitzig, Guillaume Deffuant, Jean-Denis Mathias, Jürgen Kurths

Published 2017-06-08Version 1

In order to define a paradigm of sustainability for human development and our co-evolution with the earth system, Rockstr\"om et al. [58] introduced Planetary boundaries with superseding refinements by Steffen et al. [66]. They are a set of planetary-scale thresholds whose transgression may be catastrophic for the earth system. The question of their interaction with each other and their relations to Social Foundations [56], demanding normative human-oriented boundaries, have been under intensive investigation. The framework on Topology of Sustainable Management (tsm) developed by Heitzig et al. [30] shed a new light on the understanding of boundaries and how complex structures in state space corresponding to a hierarchy of safety levels may occur when taking the dynamics of the system and possible management options into account. In this paper, we present a variant definition of tsm based on viability theory (vt), a subfield of control theory. This enables us to use the Saint-Pierre algorithm in order to estimate the main partition of the tsm-framework, meaning we make the first step on the way of operationalization of tsm. Furthermore, we present an extension of the algorithm to compute implicitly defined capture basins, a notion from vt that is more elaborated in the article, as these come up in tsm. We use a low-complexity model coupling environmental and socio-economic dynamics to demonstrate the applicability of this approach. Furthermore, two common problems of estimations in viability theory are critical for this example: (i) the unbounded state space and (ii) the highly varying time scales. We solve both by introducing appropriate mappings and these solutions are applicable for general systems, too. An open source Python-library for viability computations called pyviability has been developed and is accessible under [link to be added].