From lakes and glades to viability algorithms: automatic classification of system states according to the topology of sustainable management

Abstract

The framework Topology of Sustainable Management by Heitzig et al. (Earth Syst Dyn 7:21. https://doi.org/10.5194/esd-7-21-2016, 2016) distinguishes qualitatively different regions in state space of dynamical models representing manageable systems with default dynamics. In this paper, we connect the framework to viability theory by defining its main components based on viability kernels and capture basins. This enables us to use the Saint-Pierre algorithm to visualize the shape and calculate the volume of the main partition of the Topology of Sustainable Management. We present an extension of the algorithm to compute implicitly defined capture basins. To demonstrate the applicability of our approach, we introduce a low-complexity model coupling environmental and socioeconomic dynamics. With this example, we also address two common estimation problems: an unbounded state space and highly varying time scales. We show that appropriate coordinate transformations can solve these problems. It is thus demonstrated how algorithmic approaches from viability theory can be used to get a better understanding of the state space of manageable dynamical systems.

Appendices

Appendix

A Existence of eddies

Definition of eddy-like sets: We call a pair of sets eddy-like if and only if they fulfill the following two conditions: (i) and (ii) . Note the inverted order of the signs in the last term.

Union of two eddy-like pair sets are also eddy-like: For two eddy-like pairs of sets and , the union pair is eddy-like, too.

Proof: The first condition is trivially fulfilled and the second one follows straight away from for two state sets . Hence, the union of all eddy-like pairs of sets is maximal and eddies exist. $\square$

B Working definition of TSM bifurcations

For the analysis in Sect. 5.8, we use the following working definition of tsm bifurcations. Let $\psi =(\psi _1,\ldots ,\psi _k)$ be a vector of parameters of the control system, let be the various tsm regions in dependence on parameter values, and let be the function that maps parameter values to the tuple of these regions. Now, we say that at a certain point $\psi ^*$ in parameter space a tsm bifurcation happens if and only if $f(\psi )$ is discontinuous at $\psi ^*$ with respect to the Hausdorff metric between subsets of . In particular, we have a tsm bifurcation if the volume of some of the regions changes discontinuously at $\psi ^*$.

C Parameter estimation

To get a roughly realistic setting, we estimated the parameters of the model using several publicly available data sources.

$A_0$ was taken from [86] and slightly rounded. $\tau _A$ and $\beta$ were taken from [45]. $\phi$ was based on the ton oil equivalent of various fossil fuels and a typical mass share of 90% carbon in fossil fuels, as described in [76].

Assuming that two degrees warming correspond to a carbon concentration of around 450 ppm [87] and thus to a carbon stock of 950 GtC (both being 1.6 times their pre-industrial value), we require that the total growth rate $\beta - \theta A_1$ becomes zero for $A_1 \approx 950\,$GtC${} - A_0 = 350\,$GtC, hence $\theta$ was taken to be $\beta / A_1$ $\approx$ $8.57\cdot 10^{-5}/{}$(GtC a).

$\epsilon$ was estimated from the World Bank’s primary energy intensity data [88].

For $\tau _S$, the characteristic depreciation time of renewable energy knowledge, no reliable source was found, so we made a very coarse guess by setting it roughly to the length of an average working life of 50 a.

The break-even knowledge level $\sigma$ was also estimated very coarsely. According to past cumulative world consumption of renewable energy is $\approx 2\cdot 10^{18}\,$Btu $\approx 2\cdot 10^{12}\,$GJ or roughly 20 years of world energy consumption. To be on the conservative side and avoid overestimating the potential of renewables, we took $\sigma$ to be two times that value.

$\rho$ was set as follows. We assume fossil and renewable energy production costs of $C_F\propto F^{1+\gamma }$ and $C_R\propto R^{1+\gamma } / S^\lambda$, where $\gamma > 0$ is a convexity parameter and $\lambda > 0$ is a learning exponent. Then, energy prices are $\pi _F\propto \partial C_F/\partial F$ $\propto F^\gamma$ and $\pi _R\propto \partial C_R/\partial R$ $\propto R^\gamma / S^\lambda$. In the price equilibrium, $\pi _F = \pi _R$, hence $R / F \propto S^{\lambda / \gamma }$, and thus $\rho = \lambda / \gamma$. According to [89], the learning rate $LR = 1 - 2^{-\lambda }$ of several renewables is around 1/8, hence $\lambda \approx \log _2 (8/7) \approx 0.2$. Assuming a mild convexity of $\gamma \approx 0.1$, we get $\rho \approx 2$.