Significance The great environmental disruptions of the geologic past remain enigmatic. Each one results in a temporary change in the oceans’ store of carbon. Although the causes remain controversial, these changes are typically interpreted as a proportionate response to an external input of carbon. This paper suggests instead that the magnitude of many disruptions is determined not by the strength of external stressors but rather by the carbon cycle’s intrinsic dynamics. Theory and observations indicate that characteristic disruptions are excited by carbon fluxes into the oceans that exceed a threshold. Similar excitations follow influxes that are either intense and brief or weak and long-lived, as long as they exceed the threshold. Mass extinction events are associated with influxes well above the threshold.

Abstract The history of the carbon cycle is punctuated by enigmatic transient changes in the ocean’s store of carbon. Mass extinction is always accompanied by such a disruption, but most disruptions are relatively benign. The less calamitous group exhibits a characteristic rate of change whereas greater surges accompany mass extinctions. To better understand these observations, I formulate and analyze a mathematical model that suggests that disruptions are initiated by perturbation of a permanently stable steady state beyond a threshold. The ensuing excitation exhibits the characteristic surge of real disruptions. In this view, the magnitude and timescale of the disruption are properties of the carbon cycle itself rather than its perturbation. Surges associated with mass extinction, however, require additional inputs from external sources such as massive volcanism. Surges are excited when CO 2 enters the oceans at a flux that exceeds a threshold. The threshold depends on the duration of the injection. For injections lasting a time t i ≳ 10,000 y in the modern carbon cycle, the threshold flux is constant; for smaller t i , the threshold scales like t i − 1 . Consequently the unusually strong but geologically brief duration of modern anthropogenic oceanic CO 2 uptake is roughly equivalent, in terms of its potential to excite a major disruption, to relatively weak but longer-lived perturbations associated with massive volcanism in the geologic past.

Introduction Earth’s carbon cycle is a loop between photosynthesis, which converts carbon dioxide ( CO 2 ) to organic carbon, and respiration, which converts organic carbon back to CO 2 (1⇓⇓–4). The cycle has undergone many disruptions throughout Earth’s history (5⇓–7). These events are expressed in the geologic record as relatively abrupt and large changes in the isotopic composition of sedimentary carbon (8) compared with background fluctuations. Fig. 1 shows 2 examples in one 800-ky time series. Such events are typically attributed to changes in the fluxes and concentrations of carbon (8⇓–10). Thus, Fig. 1 could reflect the transient addition of 2 pulses of isotopically light carbon originating, for example, from the respiration of a previously unreactive reservoir of organic carbon (10, 11). Such disruptions have also been interpreted as geochemical responses to other sources of environmental change, including variations in Earth’s orbital parameters (12); dissociation of methane hydrate (13); bolide impacts (14); biogeochemical innovations (15, 16); and changes in chemical weathering (17), organic carbon burial (18), and volcanic emissions (19). These interpretations typically treat the marine carbon cycle as a passive recorder of an externally imposed stress. The response imprinted in the geochemical record is then implicitly assumed to be proportional to the magnitude of the forcing (8, 16, 20). Fig. 1. Fluctuations of the isotopic composition of carbonate carbon ( δ 13 C ) during the Eocene period, about 54 Ma (26). Time advances to the right and is given with respect to the minimum of the first abrupt downswing, an event known as Eocene Thermal Maximum 2 (ETM2) or H1. The second event, about 100 ky later, is called H2. The timescale is derived from astrochronology (26). Given the variety of possible forcing mechanisms, it comes as no surprise that the size and timescale of disruptions vary immensely, by 2 orders of magnitude during the Phanerozoic (0 to 542 Ma) (7). Yet the same data show that major fluctuations of the carbon cycle exhibit a characteristic rate of change. Events associated with mass extinction tend to exceed the characteristic rate, whereas others appear bounded by it (7). It seems unlikely that a rich diversity of stressors, expressed as proportionate responses in the geochemical record, would exhibit such uniformity. Here I formulate and analyze an elementary mathematical model that shows how the characteristic rate can instead emerge within the carbon cycle. The model portrays the upper ocean as a chemical reactor open to an incoming flux of dissolved calcium carbonate from rivers and an outgoing flux representing carbonate burial in sediments. Within the reactor, the concentrations of carbonate species respond not only to imbalances in inputs and outputs, but also to imbalances in the biological consumption and production of CO 2 . This framework reveals a mechanism for autocatalytic amplification of a small but finite perturbation of a stable steady state followed by relaxation back to the same steady state. The process is analogous to the excitation of an action potential (nerve impulse) in a neuron (21). Here, excitations manifest themselves as transient ocean acidification; i.e., a temporary increase in the concentration of carbon dioxide in the upper ocean. This change is distinguished by a characteristic rate. These results suggest that the magnitude of a carbon cycle’s disruption is determined not by the strength of the cycle’s perturbation but rather by the intrinsic dynamics of the system itself. Once the addition of CO 2 to the oceans passes a threshold, the rate of amplification and the eventual severity of the resulting environmental change should be independent of the detailed history of the perturbation. Moreover, the model indicates that the consequences of fast forcing at human timescales may be similar to the outcome of slow forcing at geologic timescales (7). Within this framework, the greater rates of change associated with mass extinction events are straightforwardly interpreted as the consequence of forcing sustained beyond the threshold, due, for example, to massive volcanism (22⇓⇓–25). This paper is organized as follows. The first section presents a 2-component dynamical system that represents important features of the marine carbonate system. Next, this paper analyzes the stability of the model’s steady state. Three dynamical regimes receive focus: The limit cycle that results when the steady state is unstable, the excitations that result when the steady state is stable, and the model’s behavior when both the steady state and the limit cycle are stable. Particular attention is given to the case of excitations and their size and timescale. This paper then discusses the implications of these findings for interpreting the geochemical record of past disruptions and predicting the carbon cycle’s response to modern anthropogenic perturbations. The conclusion highlights strengths and weaknesses of this paper’s principal contributions and points the way toward future progress.

Conclusion Major disruptions of Earth’s carbon cycle are typically interpreted as a proportionate response to an environmental perturbation (8, 16, 20). This paper suggests instead that many of these events represent the nonlinear amplification of processes that operate within the carbon cycle. To illustrate how this could work, this paper constructs and studies a simple dynamical-system model of the marine carbon cycle. The results suggest that disruptions can follow when a stable steady state is perturbed beyond a threshold. A likely source of perturbations is an increased flux of CO 2 into the oceans. The autocatalytic amplification of model disruptions exhibits a characteristic increase and rate of growth of the ocean’s store of carbon before returning to the original steady state. Such characteristic sizes and rates, independent of the history of external perturbations, are classical features of nonlinear systems (29). The historical record of the real carbon cycle also exhibits similar characteristics (7). These shared properties suggest that the model displays important features of the real carbon cycle. These findings suggest that a characteristic excitation of upper-ocean CO 2 levels follows the injection of CO 2 into the oceans at an above-threshold rate. If injection continues over a time greater than the timescale τ w over which the oceans homeostatically adjust to changes in pH, the threshold’s upper bound is roughly equivalent to the higher estimates of the rate at which CO 2 degasses from major flood-basalt eruptions associated with mass extinction (24, 76). If injection is instead limited to a shorter duration t i , the threshold increases by a factor of τ w / t i . Thus, the relatively slow rate of CO 2 injection commensurate with massive volcanism at geologic timescales turns out to be roughly equivalent, in terms of its potential to reach the threshold, to the much stronger but briefer perturbation of the modern carbon cycle. Mass extinction events appear to be associated with excitations well above threshold. These conclusions rely in part on the assumption that the dynamics of a 2D dynamical system represent a subset of the complex behavior exhibited by the real carbon cycle. This assumption does not require that the mechanisms encoded by the terms within the dynamical system be strictly correct. However, it does demand that the model’s qualitative dynamical properties, such as the possible existence of a limit cycle, be realistic. This may not be true. One could argue, for example, that phenomena at submillenial timescales, such as ecological reorganization, will act as strong negative feedbacks in local environments, impeding global autocatalytic self-organization. The present state of the geochemical record makes such notions difficult to rule out observationally. On the other hand, numerical simulation of more detailed carbon-cycle models (44) should allow one to test whether excitations are expressed in such frameworks. Likewise, careful analysis of individual disruptions or specific periods in the geochemical record should allow one to test whether quantitative signatures of excitable systems exist. The possibility of resonance with the fluctuations of external perturbations (21), such as those induced by orbital variations (12), offers one such route. The work reported here identifies why the carbon cycle may be excitable, how excitability may have been expressed in the past, and why an excitation may occur in the future. In a curious twist of geological and biological evolution, dynamical mechanisms underlying environmental upheaval and mass extinction may be similar to those that make neurons spike.

Acknowledgments I thank C. Arnscheidt, O. Devauchelle, R. Ferrari, and J. Weitz for helpful discussions and B. Schoene and B. Keller for providing data. This work was supported by NASA Astrobiology Grant NNA13AA90A and NSF Grant EAR-1338810.

Footnotes Author contributions: D.H.R. designed research, performed research, analyzed data, and wrote the paper.

The author declares no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1905164116/-/DCSupplemental.