Rate-induced biosphere collapse in the Daisyworld model

TL;DR

The paper addresses rate-induced tipping (R-tipping) in the Daisyworld model, showing that rapid changes in insolation, parameterized by $L(rt)$, can drive the biosphere to extinction even when slower changes would permit survival. It identifies two tipping pathways: bifurcation-induced tipping that occurs when $L$ crosses a dangerous bifurcation, and rate-induced tipping via basin instability where trajectories cross the stable manifold of a saddle $e_1(L)$. Using numerical continuation and nonautonomous forcing, the authors quantify the critical rate $r_c$ and characterize the basin boundary between surviving and collapsing attractors. The results generalize the relevance of R-tipping to a classic co-evolutionary framework, underscore its potential ubiquity in complex systems, and enhance Daisyworld as a teaching tool for nonautonomous dynamics and critical transitions.

Abstract

There is much interest in the phenomenon of rate-induced tipping, where a system changes abruptly when forcings change faster than some critical rate. Here, we demonstrate and analyse rate-induced tipping in the classic "Daisyworld" model. The Daisyworld model considers a hypothetical planet inhabited only by two species of daisies with different reflectivities, and is notable because the daisies lead to an emergent "regulation" of the planet's temperature. The model serves as a useful thought experiment regarding the co-evolution of life and the global environment, and has been widely used in the teaching of Earth system science. We show that sufficiently fast changes in insolation (i.e. incoming sunlight) can cause life on Daisyworld to go extinct, even if life could in principle survive at any fixed insolation value among those encountered. Mathematically, this occurs due to the fact that the solution of the forced (nonautonomous) system crosses the stable manifold of a saddle point for the frozen (autonomous) system. The new discovery of rate-induced tipping in such a classic, simple, and well-studied model provides further supporting evidence that rate-induced tipping -- and indeed, rate-induced collapse -- may be common in a wide range of systems.

Figures (6)

• Figure 1: (a) Equilibrium states in the Daisyworld model: $\alpha_b$ is the fraction of the planet covered by black daisies, $\alpha_w$ is the fraction of the planet covered by white daisies, and $L$ is the dimensionless luminosity (incoming sunlight). There are six physically relevant equilibrium states, which we label as $e_0$ through to $e_5$. $e_0$ is a dead planet, $e_1$ and $e_2$ are states in which there are only white daisies, $e_3$ and $e_4$ are states in which there are only black daisies, and $e_5$ is a "coexistence" state, in which there are both black and white daisies. Solid lines indicate stable equilibria, and dashed lines unstable equilibria. (b-g) Phase portraits for different values of the luminosity $L$. Filled circles are stable equilibria, and unfilled circles are unstable equilibria. Of particular interests are the stable manifolds of certain saddle points: these are highlighted in orange for easy future reference.
• Figure 2: Effective emission temperature $T_e$ as a function of varying luminosity $L$ for the different equilibrium states of the model. Solid lines are stable equilibria, dashed lines are unstable equilibria, and filled circles are bifurcations. Notably, when daisies are present, $T_e$ tends to be maintained at values that are much more conducive to the survival of daisies. For example, for larger values of $L$ (ca. 1.2-1.5) white daisies increasingly dominate, increasing the albedo of the planet and reducing overall temperature relative to a dead planet. Conversely, for smaller values of $L$ black daisies increasingly dominate, decreasing albedo and increasing overall temperature relative to a dead planet. This is the widely-discussed "self-regulation" of Daisyworld's biospherewatson83wood08.
• Figure 3: Bifurcation-induced tipping towards extinction in the Daisyworld model. In both (a) and (b) we initialise the system at a stable "living" equilibrium (black dot), and then gradually change the luminosity $L(rt)$. In panel (a), there is initially a large population of white daisies; however, eventually the equilibrium undergoes a saddle-node bifurcation (magenta dot), the only remaining stable equilibrium is that of a dead planet, and the biosphere collapses. In panel (b) we demonstrate the same phenomenon starting from the stable state in which there are only black daisies, and decreasing $L$. In each case, dashed lines denote unstable equilibrium states. Panels (c), (d) show time series of the luminosity $L(rt)$ with $r = 0.01$, corresponding to (a) and (b), respectively. Panels (e) and (f) show time series of $\alpha_w$ and $\alpha_b$ corresponding to (a) and (b), respectively.
• Figure 4: Rate-induced tipping towards extinction in the Daisyworld model. We initialise the system at the coexistence equilibrium $(e_5)$ with a value of $L=L_{\rm start}$. Then, we ramp $L$ towards some new value $L_{\rm end}$, at a speed determined by a dimensionless rate parameter $r$. Panel (b) shows time series of $L(rt)$, while panels (c) and (d) show time series of $\alpha_w$ and $\alpha_b$. When the change is slow enough (blue, $r = 0.5$), the biosphere survives; when it is too fast (red, $r = 1$), the biosphere collapses. This occurs because the system can enter the basin of attraction of the "dead planet" state, $e_0$, if the change in $L$ is fast enough. The threshold for entering this basin of attraction is the stable manifold of the saddle equilibrium $e_1$ (see also Figure \ref{['fig:BI_phase']} and panel f of Figure \ref{['fig:3Dbif_and_phase']}). In the space of $\alpha_w$,$\alpha_b$, and $L$, the threshold becomes a two dimensional surface $W^{s}(e_1(L))$: trajectories which cross it are precisely those which tip. This is shown in panel (a), in which the red curve crosses the surface and the blue curve does not.
• Figure 5: Rate-induced tipping in the Daisyworld model (e.g. as seen in Figure \ref{['fig:3Dbif_rtipping']}) is a consequence of basin instability. Panel (a) shows that, at $L=L_{\rm end}$ (the upper end of the perturbation in Figure \ref{['fig:3Dbif_rtipping']}) the phase space is partitioned into two basins of attraction, separated by the stable manifold of the saddle point $e_1(L_{\rm end})$. The stable coexistence equilibrium $e_5(L_{\rm start})$ lies within the basin of attraction of $e_0(L_{\rm end})$. Thus, if the system is initialised at $e_5(L_{\rm start})$ with $L=L_{\rm start}$, and then $L$ is instantaneously changed to $L_{\rm end}$, the system tips towards $e_0$ (the "dead planet" state). The equilibrium $e_5$ is thus basin unstableokeeffe20. Panel (b) shows the two basins at $L=L_{BI}$, which is where $e_5(L_{\rm start})$ intersects the boundary of the basins. We refer to this as the boundary of the basin instability region of equilibrium $e_5(L_{\rm start})$.