Ice core measurements have revealed a highly asymmetric cycle in Antarctic temperature and atmospheric CO2 over the last 800 kyr. Both CO2 and temperature decrease over 100 kyr going into a glacial period and then rise steeply over less than 10 kyr at the end of a glacial period. There does not yet exist wide agreement about the causes of this cycle or about the origin of its shape. Here we explore the possibility that an ecologically driven oscillator plays a role in the dynamics. A conceptual model describing the interaction between calcifying plankton and ocean alkalinity shows interesting features: (i) It generates an oscillation in atmospheric CO2 with the characteristic asymmetric shape observed in the ice core record, (ii) the system can transform a sinusoidal Milankovitch forcing into a sawtooth-shaped output, and (iii) there are spikes of enhanced calcifier productivity at the glacial-interglacial transitions, consistent with several sedimentary records. This suggests that ecological processes might play an active role in the observed glacial-interglacial cycles.
 Anticipating the possible climatic and ecological impacts of the rapid increase in atmospheric CO2 now, and into the near future, demands a comprehensive understanding of the global carbon cycle. Our ability to interpret and model past CO2 changes constitutes a stringent test of that understanding. In particular, the striking glacial-interglacial variation in atmospheric CO2 provides a suitable but very challenging test.
 Ice-core records [Petit et al., 1999; Augustin et al., 2004; Jouzel et al., 2007; Lüthi et al., 2008] have revealed robust glacial-interglacial variations in temperature and CO2 over the last 800 kyr with a dominant periodicity of 100 kyr and minor contributions from periodicities of 41, 23, and 19 kyr. The latter can be interpreted as responses to orbital cycles [Milankovitch, 1941], but there is ongoing discussion about the origin of the 100 kyr component [Denton et al., 2010; Imbrie et al., 2011]. Furthermore, it is still not entirely clear what mechanisms underlie the almost 100 ppmv reduction in atmospheric CO2 during the glacial periods. The existence of a vast oceanic reservoir of carbon that equilibrates with the atmosphere on millennial timescales suggests that the ocean plays a key role [Broecker, 1982]. However, it appears necessary to model several combined mechanisms to obtain the full observed glacial-interglacial CO2 variation [Peacock et al., 2006; Brovkin et al., 2012]. The most conspicuous unresolved issue is the reverse sawtooth shape of the cycles. Both CO2 and temperature decrease slowly over almost 100 kyr going into a glacial period and then rise steeply over less than 10 kyr from a glacial to an interglacial period. Such nonharmonic oscillations are likely the result of nonlinear dynamics. Indeed, sawtooth cycles may be understood in general using dynamical systems theory [Crucifix, 2012].
 In Appendix A, we review nonlinear models for sawtooth oscillations from a wide variety of fields of study. Some models force an oscillation into a sawtooth shape by means of switches, whereas other models generate sawtooth cycles dynamically. In the latter class of models, at each of the steep drops in the sawtooth variable (or steep rises in the case of a reverse sawtooth), another variable exhibits a sharp spike. Generally, spikes arise due to a multiplicative term proportional to both the spiking and the sawtooth variables in the differential equation for the spiking variable. From a purely physical perspective, there is no general interpretation for a multiplicative term. In the context of chemistry, however, such a multiplicative term has a very specific interpretation as an expression of the phenomenon of autocatalysis: a reaction in which the product is at the same time a reactant involved in the generation of the product [Atkins and de Paula, 2009]. Recently, Fowler et al.  mentioned the relationship between autocatalysis and sawtooth dynamics, but they did not base their model for ice ages on an autocatalytic reaction system. Although not all autocatalytic systems have a biological origin (for example, certain inorganic chemical reaction systems can exhibit nonlinear oscillations due to autocatalysis [Belousov, 1959; Zhabotinsky, 1964]), the prime example of autocatalysis is self-assembly and reproduction by living organisms: A population of a certain organism is needed to produce more individuals of that organism. One geological record that shows maxima at the steep glacial-interglacial transitions is the calcite fraction of sediments from many different regions of the world ocean [Berger, 1977] (see section 3 for more details). One autocatalytic process that could generate spikes or maxima in the calcite fraction of marine sediments is the growth and reproduction of marine calcifying organisms.
 Here we present a dynamical model focused around autocatalysis by marine calcifiers that generates sawtooth cycles in ocean alkalinity (corresponding to reverse sawtooth cycles in atmospheric pCO2 because of the carbonate reaction equilibrium) and calcite spikes. In the next section, we first provide an overview of the general foundations of our calcifier-alkalinity model. Then, we provide the formulation and simulation results for a highly simplified and a somewhat more elaborate version of the model. We compare the simulation results with observations in section 3. However, we emphasize that this may not be the only possible autocatalytic system to generate cycles with the characteristic glacial-interglacial shape. In section 4, we focus on which general insights about sawtooth dynamics can be drawn from our study and how these insights could be used to develop alternative models and hypotheses for the glacial-interglacial cycles. To facilitate further exploration, we have made our model code available as online supporting information.
2 Model Formulation and Results
 Ocean carbonate chemistry is characterized through the dissolved inorganic carbon (DIC) concentration and the total titration alkalinity (ALK) that is a measure of the buffering capacity of the ocean for acids (such as atmospheric CO2) [Dickson, 1981]. and account for more than 95% of ALK and 99% of DIC [Williams and Follows, 2011] which means that to a reasonable approximation:
with A and C the ALK and DIC concentration, respectively. The carbonate reaction system equilibrates on much shorter timescales than considered here which implies that [Zeebe and Wolf-Gladrow, 2001]
with K a composite equilibrium constant for the carbonate system. Since variations in the bicarbonate ion concentration are relatively modest, pCO2 is approximately inversely proportional to the carbonate ion concentration [Broecker and Peng, 1982]. Proxy studies have indicated that the surface ocean pH and are elevated during glacial times [Sanyal et al., 1995; Hönisch and Hemming, 2005; Foster, 2008; Palmer et al., 2010], consistent with the observed lower atmospheric pCO2 [Lüthi et al., 2008]. This higher could correspond to either a higher ALK or a lower DIC at the ocean surface. However, because both the atmosphere and the terrestrial biosphere contain less carbon [Bird et al., 1994; Crowley, 1995], the ocean must contain more carbon. This means that glacial surface ocean DIC must be elevated, unless the organic carbon pump is primarily responsible for the glacial-interglacial pCO2 changes. As the observations do not suggest a strong glacial-interglacial change in the organic carbon pump [Kumar et al., 1995; Kohfeld et al., 2005, 2013], we focus on a scenario in which changes in surface ocean and atmospheric pCO2 are due to alkalinity variations.
2.1 Model Foundation: Conceptual View
 The central assumption is that the population growth rate of calcifying organisms increases with increasing surface ocean carbonate ion concentration. We are not aware of direct evidence for this assumption, but we do see two different lines of evidence that a higher carbonate ion concentration leads to enhanced calcification:
 Laboratory culture studies [Ohde and Hossain, 2004; Langdon and Atkinson, 2005; Schneider and Erez, 2006] as well as mesocosm experiments [Langdon et al., 2000; Leclercq et al., 2002] indicate an increase of the calcification rate of corals with increasing . However, such a relationship is not clear for all calcifying organisms [Fabry, 2008]. For coccolithophores, some laboratory studies indicate a positive correlation between and calcification [Riebesell et al., 2000; Zondervan et al., 2001; Bach et al., 2012], whereas others show a negative correlation [Langer et al., 2006; Iglesias-Rodriguez et al., 2008].
 The sizes of sedimented foraminiferal shells are large both in tropical oceans and during glacial periods [Barker and Elderfield, 2002]. Measured shell weights of planktonic foraminifera from the Atlantic increase by a factor of 2 going from 60°N to 30°N. These heavier shells are not likely due to the higher temperatures, since foraminiferal shell weights decrease by a factor of 2 going in time from a glacial to an interglacial period. Coccolithophores are also larger and more heavily calcified during glacial periods [Beaufort et al., 2011]. A general increase in calcification with increasing would provide an interpretation for these different trends, as the ocean surface increases from high to low latitudes [Key et al., 2004], while it decreases from a glacial to an interglacial period.
 An increase in calcification does not necessarily imply an increased population growth, but there likely exists some ecological benefit to calcification (although there is much discussion about what this benefit actually consists of). Thus, it appears reasonable that calcifiers grow more efficiently (or have reduced mortality) at higher , when they calcify more.
 The combination of calcifier growth, calcite sedimentation, and weathering input of ocean alkalinity constitutes the dynamics of the hypothesized calcifier-alkalinity cycles (schematic depiction in Figure 1a). As long as the number of calcifiers is small, calcite sedimentation is low and the alkalinity increases almost linearly due to the weathering input here assumed constant. The higher alkalinity leads to more calcifiers that can generate new calcifiers even faster, leading to a calcifier population spike and a rapid drop in alkalinity due to calcite sedimentation. This alkalinity drop then leads to a decline in the number of calcifiers, after which the cycle starts anew. Glacial-interglacial cycles have often been viewed as consisting of a glacial and an interglacial state between which the climate system switches [MacAyeal, 1979; Paillard, 1998; Rose et al., 2013]. In contrast, the two main states in our model correspond to glacial-interglacial transitions (when the sedimentation output of alkalinity is higher than the weathering input of alkalinity) and nontransition situations (when the sedimentation output is lower than the weathering input), as illustrated in Figures 1b and 1c.
 In principle, a sawtooth oscillation could also be generated through the interplay of a producer and a nutrient such as nitrogen or phosphorus with a constant input. However, if the producer is a noncalcifier, then we expect sawtooth cycles in atmospheric CO2 rather than the reverse sawtooth cycles that are observed: When the productivity spikes, nutrients, and carbon will be consumed and atmospheric CO2 will drop sharply, between these spikes, CO2 will increase slowly. Furthermore, its longer oceanic residence time [Broecker and Peng, 1982] makes alkalinity a better “fit” than the nutrients when considering the glacial-interglacial cycles.
 We now explore these interactions in two idealized model frameworks: first a highly abstracted view which illustrates the basic principles of a calcifier-based autocatalytic system. Then we expand the model, introducing some spatial and ecological resolution, showing that the mechanisms are robust.
2.2 Basic Model
 We consider a highly simplified, but transparent, zero-dimensional model for the interaction between alkalinity and calcifiers. We assume that CO2 simply responds to ocean alkalinity changes, and therefore, we do not explicitly model atmospheric CO2. Alkalinity A is added to the ocean through continental runoff at a fixed rate I=4×10−6 mol eq m−3 yr−1 which roughly corresponds to the estimated total river input of CaCO3 into the world ocean [Milliman et al., 1999]. The alkalinity is consumed by a population of calcifiers P with a per capita growth rate, kA, proportional to alkalinity. The rate constant k=0.05 (mol eq)−1 m3 yr−1 which implies that the effective population growth rate kA is about 0.1 yr−1; also, the calcifier sedimentation rate S is 0.1 yr−1. These rates are much lower than the actual per capita growth and death rates of calcifiers, but we are concerned with longer-term dynamics. Each year, most populations will go through a seasonal cycle of bloom and decline which is not represented by this highly idealized model; kA should be thought of as the rate at which the globally and annually averaged number of calcifiers can increase. Thus, the model consists of two simple differential equations
 The glacial-interglacial cycles may have emerged from an event, such as the closure of the Panama Seaway [Haug and Tiedemann, 1998; Haug et al., 1998; Lear et al., 2003], that has brought the Earth system away from equilibrium. While the system is returning back to equilibrium, it may be going through a series of transient nonlinear oscillations that are not externally forced. Alternatively, glacial cycles could reflect a nonlinear oscillation externally forced by orbital cycles [le Treut and Ghil, 1983; Huybers and Wunsch, 2005]. Evidence that variations in the relative abundance of calcifiers are, in fact, partly forced by orbital cycles is provided by the observation of a correlation between precession and coccolithophore productivity in the equatorial Indian Ocean over the past 900 kyr [Beaufort et al., 1997]. Furthermore, the sedimentary calcite content in cores from different locations exhibits various orbital periodicities throughout the past 100 Myr [Herbert, 1997]. Our calcifier-alkalinity system exhibits sawtooth cycles both with and without external forcing; we discuss each case in turn.
 Without external forcing, the alkalinity goes through sawtooth-shaped cycles with slow increases and steep declines, at which the calcifiers exhibit spikes (Figure 2). The amplitude of the illustrated alkalinity cycles is about 0.06 mol eq/m3, corresponding to variations in atmospheric pCO2 of about 75 ppmv [Omta et al., 2011]; the period of the cycles is about 20 kyr which is considerably shorter than the observed 100 kyr periodicity. The cycles slowly damp toward an equilibrium state with A=2.0 mol eq/m3 and P=4.0×10−5 mol eq/m3. The damping is intrinsic to the set of equations, as the equilibrium remains stable under variation of the parameter values. Approaching equilibrium, the cycles tend to smaller amplitudes, implying a shorter linear increase and thus a shorter period. Because the variations in P are weaker, the drops in the alkalinity are less dramatic which makes the alkalinity cycles less sawtooth-like close to equilibrium. Thus, the farther from equilibrium the initial conditions are chosen, the longer and more asymmetric the cycles are. Starting with an initial calcifier concentration P(0)=1.0×10−3 mol eq/m3 (farther away from equilibrium), with the standard parameter values, we obtain sawtooth alkalinity cycles with an amplitude of about 0.08 mol eq/m3 and a period of almost 30 kyr. The duration and amplitude of the cycles are also determined by the parameter values. For example, we can speed up the dynamics of the system by increasing the effective growth and mortality rates to 1 yr−1. Starting from the standard initial conditions, we then obtain alkalinity cycles with an amplitude of less than 0.03 mol eq/m3 and a period of less than 10 kyr. Transient oscillations occur over a very wide range of model parameter values: The system simply needs to start away from equilibrium. Nevertheless, the calcifier population growth parameter k does need to be relatively small to obtain oscillations with a long period without having implausibly large variations in the calcifier population.
 To investigate the effect of an interaction between a calcifier-alkalinity oscillation and an orbital cycle, we apply weak periodic variations to the calcifier growth parameter k. That is, , with period T=20 kyr and with β=2.5×10−3. In this forced setup, cycles can be sustained, as there emerges a stable attractor oscillation. This attractor oscillation has a sawtooth shape, even though the forcing is sinusoidal. As long as the initial conditions are within the basin of attraction, the final solution is independent of the initial state. Two simulations with different initial conditions give very different results during the first 100 kyr, but the solutions have become practically identical at 2.5 Myr (Figure 3).
2.3 Multibox Model
 The results from the highly idealized system are interesting and provocative, but does the behavior persist if we introduce more realism into the model? Previous studies have shown that resolving vertical and horizontal structure in the ocean can have a profound impact on carbon cycle dynamics [Volk and Hoffert, 1985; Sarmiento and Gruber, 2006]. Therefore, we have performed simulations with a multibox setup schematically depicted in Figure 4 and described in more detail in Appendix B. There is tracer transport between different oceanic boxes through a prescribed overturning circulation and through interbox mixing, but the setup is somewhat simpler than in, e.g., Peacock et al.  and Toggweiler  in that no distinction is made between North Atlantic Deep Water and Antarctic Bottom Water. All primary productivity takes place in the ocean surface boxes. We introduce explicit competition between calcifiers and noncalcifiers, which was previously implicit through the parameter k. The fraction of sedimenting calcite that is buried (Afrac) depends on the deep-ocean as proposed by Zeebe and Westbroek . If , then Afrac=0; if , then Afrac=1; and if is between 0.052 and 0.147 mM, then
Thus, a representation of the carbonate compensation feedback is included.
 Simulations are performed with initial conditions as given in Table 1 and parameter values as in Table 2. Without external forcing, the oscillation is very strongly damped: The system has reached equilibrium after ~30 kyr (Figure 5a). However, if the system is forced through a weak periodic variation in the maximum calcifier growth rate (and with the same initial conditions), sawtooth oscillations are sustained (Figure 5b). Atmospheric pCO2 exhibits reverse sawtooth oscillations with slow declines, followed by steep increases. Whenever the calcifiers spike, the noncalcifiers reach a minimum. Thus, the competition between the noncalcifiers and the calcifiers for phosphorus leads to variations in the abundance of calcifiers which, in turn, leads to variations in . In fact, the calcifier population grows when it has a competitive advantage over the noncalcifiers and declines when it is at a disadvantage.
Table 1. Description of Multibox Model Variables, With Their Respective Meanings, Units, and Initial Values
mol eq m−3
mol eq m−3
mol eq m−3
mol eq m−3
High-latitude calcifier soft-tissue detritus
Low-latitude calcifier soft-tissue detritus
Shelf calcifier soft-tissue detritus
Deep-ocean calcifier soft-tissue detritus
High-latitude noncalcifier detritus
Low-latitude noncalcifier detritus
Shelf noncalcifier detritus
Deep-ocean noncalcifier detritus
High-latitude calcite detritus
Low-latitude calcite detritus
Shelf calcite detritus
Deep-ocean calcite detritus
Table 2. Description of Multibox Model Parameters (Par), With Their Respective Units, Meanings, and Values
Maximum growth rate PC
Maximum growth rate PNC
P saturation constant PC
P saturation constant PNC
saturation constant PC
mol eq m−3/s
A:P ratio PC
C:P ratio soft tissue PC and PNC
Calcifier yield factor
Noncalcifier yield factor
Calcifier remineralization rate
Noncalcifier remineralization rate
Calcite redissolution rate
Calcifier soft-tissue detritus export
Noncalcifier detritus export
Calcite detritus export
Open-ocean burial fraction organic matter
Shelf burial fraction organic matter
Open-ocean burial fraction calcite
Shelf burial fraction calcite
Surface area high-latitude box
Surface area low-latitude box
Surface area shelf box
Surface area low-latitude box
Temperature high-latitude box
Temperature low-latitude box
Temperature shelf box
Temperature low-latitude box
Depth high-latitude box
Depth low-latitude box
Depth shelf box
Depth low-latitude box
Amount of gas in atmosphere
Water exchange rate between high-latitude and deep boxes
Water exchange rate between low-latitude and deep boxes
Water exchange rate between low- and high-latitude boxes
Water exchange rate between low-latitude and shelf boxes
Water exchange rate between deep and shelf boxes
Total meridional overturning transport
 Focusing on the autocatalytic process of calcifier growth and reproduction, our idealized model generates sawtooth cycles in alkalinity, consistent with the observed reverse sawtooth in atmospheric CO2, along with calcite accumulation spikes at the sharp transitions, broadly consistent with marine records. The asymmetric nature of the oscillations is the result of the vastly different timescales in the system. After a spike, the calcifier population declines to very low levels and the alkalinity starts to increase again. The duration of this phase of increasing alkalinity is much longer, because the alkalinity input is much slower than the dynamics of the calcifiers. It appears that sawtooth oscillations are a generic result of producer-resource dynamics which is why they were seen earlier in a different system with very different timescales [Monteiro and Follows, 2009]. There is a clear analogy to classical predator-prey cycles [Lotka, 1920; Volterra, 1926], with the producer playing the part of predator and the resource in the role of prey. However, in contrast to the living prey usually considered, the variations in the resource possess a sawtooth shape, because the resource does not reproduce itself.
 While some key observed features (reverse sawtooth in CO2, calcite spikes) are captured qualitatively, some quantitative features appear difficult to reconcile with the data. In each of the simulations, the amplitude of the CO2 cycles is similar to observations, but the period is significantly shorter than the observed 100 kyr. It is possible to decrease the calcifier population growth and sedimentation to obtain 100 kyr cycles, but then, the amplitude of the cycles becomes much larger than the observed ~90 ppmv variation in atmospheric CO2. To obtain 100 kyr cycles with a ~90 ppmv amplitude, the weathering input of alkalinity needs to be decreased to about 10−6 mol eq m−3 yr−1 which is below the estimated range of the actual river input of CaCO3 [Milliman et al., 1999]. The origin of this problem may be that in the simulations, there is almost no alkalinity output between the spikes; the rate at which atmospheric CO2 declines from an interglacial to a glacial period is completely set by the weathering input of alkalinity. In reality, there is likely some alkalinity output during the periods between the spikes which implies a slower decline in atmospheric CO2 during those periods. Moreover, our models produce variations in the number of calcifiers of many orders of magnitude which is clearly unrealistic. For different parameter values, it is possible to find solutions with significantly less variation in the calcifiers. However, sensitivity experiments (not shown for brevity) have suggested that the variations in the calcite accumulation need to be at least about a factor of 100 to obtain the sharp drops in alkalinity characteristic of the sawtooth cycles. Part of such a variation in accumulation may be accounted for by enhanced calcification of each individual calcifier, but it appears that the number of calcifiers still needs to change by at least 1 order of magnitude. North Pacific records indicate very large spikes in the calcite fraction of the sediment at the transitions, with almost no calcite during the time periods in between [Jaccard et al., 2005; Brunelle et al., 2010], which seems similar to our model results. At other locations, maxima in calcite accumulation are found around the transitions, but there is also a significant accumulation in between [Flores et al., 2003; Rickaby et al., 2010].
 Traditionally, the calcite spikes at the transitions have been ascribed to enhanced preservation due to a high deep-ocean carbonate ion concentration [Berger, 1977; Broecker et al., 1993; Marchitto et al., 2005]. For a compilation of 31 cores from the Pacific, Atlantic, and Indian Oceans, Mekik et al.  have attempted to tease apart the effects of accumulation and preservation at the last deglaciation, estimating CaCO3 accumulation rates with230Th normalization [François et al., 2004] and using the Globorotalia menardii fragmentation index [Ku and Oba, 1978; Mekik et al., 2002, 2010] as a proxy of preservation. Most of these cores exhibit a deglacial maximum in calcite accumulation, but not in preservation. Thus, at least for the majority of cores included in that compilation, it appears more likely that the accumulation spikes result from enhanced productivity than from enhanced preservation. Also, the calcite fraction of sediments that are accumulating in the current ocean could at least partly reflect productivity at the ocean surface. At low latitudes, core-top calcite sediments are found at locations where the local deep-ocean is 30 μM below saturation, whereas this is not the case at higher latitudes in either the Southern or the Northern Hemisphere [see Archer, 1996, Figure 9]. The simplest explanation for this feature appears to be the plankton community structure at the ocean surface: a relatively high abundance of calcifiers at low latitudes and a relatively high abundance of other plankton (e.g., diatoms) at high latitudes, as indicated by direct observations along a north-south transect throughout the Atlantic [Cermeno et al., 2008].
 If the dynamics are purely the result of competition for a limiting nutrient between calcifying and noncalcifying organisms, then at the transitions, one expects to observe a maximum in the calcite fraction of the sediments, but no maximum in the total marine productivity. Although the calcite fraction does generally peak around the transitions, the barium proxy [Kasten et al., 2001] indicates an overall maximum in marine productivity in the equatorial Atlantic at the last two transitions. This appears consistent with the observation of opal maxima in cores from various locations [Anderson et al., 2009; Hayes et al., 2011; Meckler et al., 2013]. However, the δ30Si proxy for diatom productivity indicates a lower diatom Si utilization at the Last Glacial Maximum increasing toward the modern day without a global maximum at the transition [de la Rocha et al., 1998; de la Rocha, 2006; Ellwood et al., 2010; Hendry et al., 2010]. An overall maximum in biological productivity at the transitions would indicate that maxima in the productivity of both the calcifiers and the noncalcifiers are triggered by a factor other than the carbonate ion concentration, such as nutrient supply or temperature. Such a different triggering factor would then need to have a stronger impact on the calcifiers than on the noncalcifiers to account for the maxima in the calcite fraction of the sediments. Whatever the specific trigger, our model results indicate that the spikes are able to transform a sinusoidal orbital forcing into a more asymmetric sawtooth output. A transformation of an orbital cycle into a calcifier-alkalinity cycle seems consistent with a generally inferred slight lag of CO2 with respect to temperature [Fischer et al., 1999; Siegenthaler et al., 2005], although this inferred lag has been challenged recently [Parrenin et al., 2013].
 In many models, the oceanic input and output of alkalinity are brought in balance on a timescale of 5–10 kyr [Sundquist, 1990; Zeebe and Westbroek, 2003; Goodwin and Ridgwell, 2010; Brovkin et al. 2012], because any increase in the carbonate ion concentration leads to a downward lysocline shift which then increases the alkalinity output. Although the Zeebe and Westbroek  representation of this carbonate compensation feedback is included in our multibox model, the alkalinity input and output are out of balance throughout most of each cycle (as in the model of Toggweiler ). Essentially, calcifier-alkalinity cycles can only exist if the feedback is not strong enough to bring the alkalinity input and output in balance after a calcite “dump.” Although it remains unclear whether this is the case on a global scale, we do want to bring the following into consideration:
 On a global scale, the impact of a change in the lysocline depth on the total CaCO3 sediment accumulation may actually be rather modest, since most sedimentation of CaCO3 takes place at depths (far) above the lysocline [Milliman, 1993].
 An enhanced calcite sediment preservation during glacial times, when the whole-ocean is (presumably) high, may be counteracted by other effects, such as less aragonite production by coral reefs [Berger, 1982] and more CaCO3 weathering from exposed continental shelves [Milliman, 1993].
 Various observations from the equatorial Pacific indicate currently ongoing chemical erosion of calcite sediments [Keir, 1984; Broecker et al., 1991; Anderson et al., 2008] which suggests that the calcite sediments have not yet equilibrated with the deep-ocean more than 10 kyr after the last glacial transition.
 Observations do not indicate large overall glacial-interglacial changes in the lysocline depth [Catubig et al. 1998; Sigman and Boyle, 2000] which has been interpreted as evidence against a significant change in the deep-ocean [Zeebe and Marchitto, 2010]. However, the validity of the lysocline depth as a proxy for the deep-ocean is uncertain. On a global scale, the correlation in space between the calcite content in core top sediments and the local deep-ocean is weak [see Archer, 1996, Figure 6], possibly due to variations in the calcite production at the ocean surface and the diluting effect of other sediment components. Yu and Elderfield  advocated the use of boron/calcium (B/Ca) ratios in benthic foraminifera to quantify changes. Estimates based on this proxy indicate a deep-ocean about 10–40 μM higher during glacial than during interglacial periods [Rickaby et al., 2010; Yu et al., 2010].
 We have explored how the interplay between calcifiers and alkalinity could give rise to reverse sawtooth oscillations in atmospheric CO2. This is one example of an autocatalytic mechanism to generate cycles with the characteristic glacial-interglacial shape, and other such mechanisms can probably be formulated. Regardless of any specific mechanisms, we believe that the following general insights are crucial:
 A system of autocatalytic reactions can generate sawtooth oscillations.
 Autocatalytic systems that produce sawtooth oscillations generally have at least two variables, one of which exhibits the sawtooth, whereas the other shows spiking behavior. Thus, clues for alternative mechanisms might possibly be found in the geologic record by looking for candidate spikes other than calcite accumulation.
 The most common form of autocatalysis is the growth and reproduction of living organisms.
 A spike in calcite sedimentation can lead to a sharp rise in atmospheric CO2 [see also Fowler et al., 2012]. Thus, marine calcifying organisms are a logical candidate for a key role, if the reverse sawtooth shape of the glacial-interglacial CO2 variations is due to biological autocatalysis.
 A model describing the interaction between calcifiers and alkalinity indeed produces oscillations in atmospheric CO2 with a reverse sawtooth shape.
Appendix A: Sawtooth Models
 Considering the large number of different processes involved, it seems very difficult to identify the essential mechanisms determining the dynamics of the glacial-interglacial cycles. However, the shape of the cycles does impose a rather severe constraint on the underlying dynamics: Not every dynamical system is able to generate the strongly characteristic sawtooth-shaped oscillations. Knowledge of the properties of dynamical systems that show such behavior could thus guide us in our search for the possible mechanisms behind glacial-interglacial cycles. Therefore, we take a look at different models that generate sawtooth oscillations, with a special emphasis on models for glacial-interglacial cycles. By comparing the structures of these different dynamical systems, we aim to obtain an understanding of the possible driving mechanisms behind sawtooth oscillations in general.
 Nonlinear oscillations characterized by short impulses have been termed “relaxation oscillations” [van der Pol, 1926]. They are thought to play a key role in neuronal dynamics [Hodgkin and Huxley, 1952; FitzHugh, 1961; Nagumo et al., 1962]. Of these oscillations, sawtooth cycles are a special case. They are known from observations and models in cell biology [Campás and Sens, 2006; Kirkman-Brown et al., 2004; Sherman and Rinzel, 1991] as well as the physics of plasmas [Bernabei et al., 2000; Bierwage et al., 2005; Thyagaraja et al., 1999] and the magnetosphere [Henderson et al., 2006; Kubyshkina et al., 2008; Pulkkinen et al., 2006]. Perhaps the simplest well-known set of equations that generates sawtooth oscillations is the Selkov-Higgins glycolysis model [Higgins, 1964; Selkov, 1968] well known in the field of systems biology. It describes the production of a doubly phosphorylated sugar called fructose-biphosphate (FP2) from a singly phosphorylated sugar named fructose-6-phosphate (F6P). The equations were given by Selkov  as
with v1, j, k2, and ks constants. Observe that the model very much resembles the basic predator-prey models by Lotka  and Volterra . The important features of this model are as follows:
 There are two variables (the minimum necessary to generate oscillatory behavior) of which one (F6P) exhibits sawtooth oscillations and the other (FP2) shows spiking behavior.
 In the differential equation for the dynamics of F6P, there is a constant input term that is responsible for the long linear ramp upward of the sawtooth.
 In the set of differential equations, there is a multiplicative (autocatalytic) term that is responsible for the sharp drop in the sawtooth variable F6P and a spike in the other variable FP2.
 Another very simple model generating asymmetric oscillations was formulated by Bora and Sarmah  to describe sawtooth disruptions observed in nuclear fusion reactor plasmas [von Goeler et al., 1974; Wroblewski and Snider, 1993]. Written in a generic form, the equations are
with a, α, and ρconstants. The variable x exhibits sawtooth oscillations, whereas the variable y shows spikes. The model has multiplicative terms, but it does not have a constant input term which is probably why the ramp upward in the sawtooth cycles is not entirely straight [see Bora and Sarmah, 2008, Figure 8].
 Models for the dynamics of glacial-interglacial cycles are generally more complicated than either the Selkov-Higgins or Bora and Sarmah models, but there exist some shared features. For example, the Oerlemans  model describing the interaction between an ice sheet with height H and the underlying bedrock with altitude h was formulated as follows:
Through an interplay of ice sheet instability and bedrock deformation, the model generates asymmetric 100-kyr oscillations, with a relatively slow ice buildup and a quick melting of the ice sheet. In the Oerlemans paper, the oscillations look rather irregular, probably because the model was forced by a sinusoidal function with a period of 20 kyr. Since the equilibrium line E is independent of time, equation (A3) can be rewritten into a form similar to the Selkov-Higgins model equation (A1) by making the substitution J=H−E
As the Selkov-Higgins model, this system has terms that are constant in time ( and ), as well as a multiplicative interaction term (2bJh).
 The model of Paillard and Parrenin  is composed of three variables
The system is linear, except for a discontinuity represented by a Heaviside function dependent on a parameter F, named the “salty bottom waters formation efficiency.” In effect, there is a switching mechanism driven by precessional forcing. The temporal dynamics of the ice volume and CO2 resemble a sawtooth, while F exhibits a spiking pattern. There is no multiplicative term, but instead, the oscillation is forced into a sawtooth shape by the switching parameter.
 The “sea ice switch” model of Gildor and Tziperman , with eight variables, is much more complex than any of the above mentioned models. We do not reproduce the model here. Instead, it suffices to mention that the model shares some characteristics with the other models: there are sawtooth-like variables (land ice and ocean temperature) as well as spiking variables (sea ice and land temperature), and the differential equation for land ice includes a multiplicative term.
Appendix B: Multibox Setup
 The model code is written in FORTRAN. On an 800 MHz AMD Athlon II processor, simulating 1 kyr takes approximately 35 min. The model resolves high-latitude, low-latitude, and shelf surface ocean boxes and a deep-ocean box, along with a well-mixed atmosphere. For the oceanic boxes, tracer concentrations are calculated through three modules: carbon exchange, oceanic transport, and ecology. For each surface ocean box, the carbon exchange module calculates the oceanic pCO2 using DIC, alkalinity, and the water temperature of the box using the Follows et al.  scheme. Then, the air-sea flux of carbon φ between the box and the atmosphere is calculated from the difference between pCO2,oce and pCO2,atm:
with Vp the piston velocity and ak0 an equilibrium constant. To have instantaneous equilibration of carbon between the sea surface and the atmosphere, Vp=0.5 m/s which is a few orders of magnitude higher than typical measured piston velocities [Wanninkhof, 1992].
 The oceanic transport module was adapted from Fennel et al. . It includes a unidirectional overturning q as well as bidirectional mixing between boxes i and j at rates mij. For each tracer X, it consists of the following equations (with the subscripts h, l, s, and d standing for the high-latitude, low-latitude, shelf, and deep boxes):
 We assume that dissolved inorganic phosphorus, P, is the limiting nutrient for which a calcifiers population, PC, and a noncalcifiers population, PNC, compete. However, the argument does not change essentially, if another nutrient, such as nitrogen or iron, is limiting primary production. The calcifiers die at a rate hC to form calcite detritus DA and organic detritus DC; organic detritus DNC is formed from dying noncalcifiers. Each form of detritus remineralizes at a rate remi and is exported into the deep ocean at a rate λi. We implicitly assume sufficiently large numbers of individuals for the populations not to go extinct, and no separation between internal reserves and structural biomass in the organisms. Alkalinity A and DIC C are added to the ocean at fixed rates IA and IC through river runoff in the form of dissolved calcium carbonate. Since the yield factors of the conversion of fixed nitrogen to calcifier and noncalcifier biomass yi are equal to 1.0 in our model, total phosphorus is conserved which implies that P+PC+PNC is constant. Furthermore, R (=100 mol eq A/mol P) is the alkalinity:phosphorus ratio of the calcifier uptake, RC (=106 mol C/mol P) is the carbon:phosporus ratio of organic tissue, and hC and hNC are the respective loss rates of calcifier and noncalcifier biomass (which are an aggregate of mortality and maintenance). For the ocean surface boxes, the dynamic equations are
The growth rate μC of the calcifiers depends on both P and (which is calculated in the carbon exchange module) according to a Synthesizing Unit (SU) formulation [Kooijman, 1998; Kuijper et al., 2003; Kooi et al., 2004] similar to the one used in the SU-based Internal Transformation Yield model [Omta et al., 2008, 2009], whereas the growth rate μNC of the noncalcifiers depends on P only according to Monod  kinetics:
Although we do not model the direct effect of temperature on metabolic rates, the calcifier growth rate is different in each surface box, because varies with temperature. Periodic forcing is applied through the maximum calcifier growth rate: , with period T and relative amplitude β.
 In the deep-ocean box, there is no growth of the organisms, and living organisms are only found there because of oceanic transport. Of the organic material exported from the low- and high-latitude boxes, a fraction pfrac is buried (the burial fraction from the shelf box is called pfrac,sh); t analogous definitions for the calcite burial fractions Afrac and Afrac,sh. The carbonate compensation feedback is modeled as in Zeebe and Westbroek ; that is, the burial fraction Afrac=0 if deep-ocean is below 0.052 mM, Afrac=1 if is above 0.147 mM, and Afrac increases linearly with increasing in between.
 We would like to thank Robbie Toggweiler, Wolfgang Koeve, and three anonymous reviewers for inspiring discussions and helpful comments. Anne Willem Omta is grateful for support from the Netherlands Organisation for Scientific Research (NWO) through a Rubicon fellowship and from the National Science Foundation (NSF). The work of George A. K. van Voorn was part of the strategic research program Knowledge Base IV (KBIV) “sustainable spatial development of ecosystems, landscapes, seas and regions” funded by the Netherlands Ministry of Economic Affairs. Rosalind E. M. Rickaby was supported through European Research Council (ERC) grant SP2-GA-2008-200915. Michael J. Follows was funded by the National Oceanic and Atmospheric Administration (NOAA), the National Aeronautics and Space Administration (NASA), and NSF.