The Great Oxidation Event (GOE), approximately 2.4 billion years ago, marks a pivotal moment in Earth's history, signifying the transition from an anoxic to an oxygenated atmosphere and ocean. Geological and geochemical evidence strongly supports this transition, including the decline in banded iron formations, the emergence of red beds and sulfate evaporites, and changes in the isotopic signatures of sulfur. However, the precise mechanisms driving this dramatic shift remain a subject of ongoing debate. Several hypotheses attempt to explain the GOE. Some focus on physical drivers, such as changes in volcanic gas composition, alterations in hydrothermal iron fluxes, or hydrogen escape to space. Others emphasize biological factors, particularly the evolution and proliferation of oxygenic photosynthesis by cyanobacteria. However, the presence of anoxygenic photoautotrophs that thrived in anoxic, light-rich environments raises questions about the inherent competitive advantage of cyanobacteria before the GOE. Existing models often assume cyanobacterial abundance prior to the GOE, attributing the transition primarily to physical triggers. Some studies suggest that methane oxidation, limited by factors like nickel availability, played a significant role in controlling atmospheric oxygen levels prior to the GOE. Another line of inquiry focuses on the changes in the carbon isotope record, suggesting that biological innovations triggered the GOE. However, the persistence and success of anoxygenic photosynthetic bacteria in various environments, even today, challenge the assumption of an inherent cyanobacterial advantage. This success likely relates to their lower metabolic cost of electron acquisition compared to water-splitting oxygenic photosynthesis, or perhaps to the direct inhibition of oxygenic photosynthesis by substances like sulfide or ferrous iron. Geochemical data indicates the presence of oxygen “oases” in the Archean biosphere, suggesting oxygenic photosynthesis may have existed for a substantial period before the widespread oxygenation of the planet. Therefore, a more nuanced understanding of the interplay between biological and geological factors is necessary to fully explain the GOE. Previous research by Knoll and Nowak utilized a simple mathematical model to describe the competition between anoxygenic and oxygenic photoautotrophs on a changing Earth. This early model demonstrated that a decrease in the planetary removal rate of oxygen could trigger the GOE. Ozaki et al. provided further support for the idea that oxygenic photoautotrophs existed long before the GOE, arguing that geochemical conditions favored anoxygenic photosynthesis until a shift in environmental conditions allowed for cyanobacterial dominance. This study aims to build upon this foundation to explore the interaction between ecological and geological dynamics in detail.

Existing literature presents diverse perspectives on the GOE's causes. Some studies emphasize geophysical factors like changes in volcanic gas composition (Gaillard et al., 2011; Holland, 2002), hydrothermal iron fluxes (Kump & Seyfried Jr., 2005), or hydrogen escape (Catling et al., 2001). Others point to biological drivers, particularly the evolution of oxygenic photosynthesis (Ward et al., 2016; Knoll & Nowak, 2017). However, these models often struggle to reconcile the observed dominance of anoxygenic phototrophs in suitable environments with the assumed competitive advantage of cyanobacteria before the GOE. The role of methane oxidation in regulating pre-GOE oxygen levels has been highlighted (Goldblatt et al., 2006; Konhauser et al., 2009). Models suggest a low-level oxygen steady state maintained by methane oxidation, with a transition to a high-level steady state (with an ozone layer limiting methane oxidation) representing the GOE. Investigations into the carbon isotope record (Bjerrum & Canfield, 2004; Krissansen-Totton et al., 2015) suggest biological innovations as triggers for oxygenation. However, this view is challenged by the observed prevalence of anoxygenic photoautotrophs in various environments. The inhibitory effects of sulfide (de Beer et al., 2017) and ferrous iron (Swanner et al., 2015) on oxygenic photosynthesis have also been explored, suggesting that the presence of these reductants limited cyanobacterial proliferation. Studies on the limitations of phosphorus availability also play an important role (Jones et al., 2015; Bjerrum & Canfield, 2002). Geochemical findings point to the existence of spatially and temporally limited oxygen oases in the Archean (Anbar et al., 2007; Satkoski et al., 2015; Albut et al., 2019).

This research develops a mathematical model building upon the work of Knoll and Nowak (2017), incorporating the abundance of phosphate. This expansion accounts for the influence of the ratio of alternative electron donors (like Fe2+, H2S, or H2) to phosphorus on the competition between anoxygenic photosynthetic bacteria (APB) and cyanobacteria. The model tracks the abundances of APB (x1), cyanobacteria (x2), iron(II) (y1), phosphate (y2), and dioxygen (z). The model assumes a competitive advantage for APB in early environments with electron donors other than water. Cyanobacteria are assumed to exist concurrently in environments where they can thrive, potentially serving as a seedbed for migration. The model uses five differential equations to represent the dynamics: APB: ẋ₁ = x₁y₁y₂ − x₁ + μ₁ Cyanobacteria: ẋ₂ = cx₂ − x₂ + μ₂ Fe²⁺: ẏ₁ = f₁ − x₁y₁y₂ − by₁z PO₄³⁻: ẏ₂ = f₂ − ax₂y₂ − by₂z − x₂y₂ O₂: ż = ax₂y₂ − by₁z − by₂z Here, parameters *c* represents cyanobacteria reproduction rate; *f₁* and *f₂* are influxes of iron(II) and phosphate; *a* is biogenic oxygen production; *b* is geochemical oxygen consumption; and *μ₁* and *μ₂* are small migration rates for APB and cyanobacteria. The model implicitly satisfies redox balance. The study analyzes the stability of three equilibria: E₁ (APB dominance), E₂ (cyanobacteria dominance), and Ê (coexistence). Conditions for the stability of each equilibrium are derived. The transition between E₁ and E₂ is investigated, considering changes in iron(II) and phosphate influxes, as well as changes in cyanobacterial reproduction rates, oxygen production, and oxygen consumption rates. The effects of migration rates on the transition are also assessed. Numerical integration using the fourth-order Runge-Kutta method is employed to support analytical solutions. The paper investigates the impact of limited bacterial growth rates and organic carbon presence (in supplementary notes).

The model reveals that a shift in ecological dominance from APB to cyanobacteria can trigger the GOE. The key parameters determining the timing of the GOE are: the competitive advantage of cyanobacteria (*c*), the influx of suitable reductants (*f₁*), and the influx of phosphate (*f₂*). The study finds that a decrease in *f₁* (reductant influx) and/or an increase in *f₂* (phosphate influx) are robust mechanisms for initiating the GOE. The timing of the GOE is determined by the difference *f₁ − f₂*, not by the individual values. This functional dependence remains consistent even when considering limited bacterial growth rates. The transition between APB and cyanobacteria dominance can be either gradual or sudden, depending on the relative rates of oxygen production and consumption. A gradual transition occurs when the geochemical consumption of oxygen (*b*) is high relative to cyanobacterial reproduction and oxygen production. In this case, the system passes through a stable coexistence equilibrium (Ê) before reaching cyanobacterial dominance. This gradual transition is reversible; an increase in *f₁* can lead to APB regaining dominance. In contrast, a sudden transition happens when *b* is low. Here, cyanobacterial dominance (E₂) becomes stable before APB dominance (E₁) loses stability, resulting in bistability. Once the world is oxygenated, moderate fluctuations in *f₁* will not revert the system to the anoxic state. Similarly, increasing *c* (cyanobacterial reproductive rate) or *a* (oxygen production rate), or decreasing *b* (oxygen consumption rate) can also trigger the GOE. However, for the latter two scenarios, the effects are significantly dependent on the magnitude of cyanobacteria migration rates. Migration rates (*u₁* and *u₂*) primarily affect the magnitude of the transition, influencing the levels of APB and cyanobacteria during the shift. They have negligible effects on equilibrium abundances of APB in E₁ and of cyanobacteria and oxygen in E₂. However, they do influence the transition magnitude during the GOE. The analysis also explored the effect of varying other key parameters. Supplementary analysis extends the core model to consider various complexities, including bounded bacterial growth rates and explicit modelling of organic carbon, without changing the core conclusion that the GOE arises from a balance between the influxes of phosphate and reductants.

This research highlights the importance of ecological dynamics in understanding the GOE. The model demonstrates that a switch in ecological dominance from APB to cyanobacteria is sufficient to trigger the GOE. While previous studies focused heavily on sources and sinks of oxygen, this research shows that their influence is contingent on the competition between APB and cyanobacteria. If cyanobacteria were initially scarce and ecologically subordinate, changes in oxygen sources and sinks would have had minimal impact. The model's findings emphasize that the GOE wasn't solely caused by geophysical processes or biological innovations, but rather by their intricate interaction. The parameters *c*, *f₁*, and *f₂* are crucial in determining the timing of the GOE. Extant cyanobacteria's limited fitness in sunlit anoxic environments suggests limitations on potential mechanisms for increasing *c* around the GOE. In contrast, decreasing *f₁* and/or increasing *f₂* offer robust mechanisms for initiating oxygenation, consistent with predictions from secular mantle cooling, continental emergence, and increasing oxidant supply. The analysis suggests that the difference (*f₁ − f₂*) in influxes, not the individual fluxes, governs when the GOE started. This study provides a more complete and nuanced explanation of the GOE, emphasizing the complex interplay between biological and geophysical processes.

This study presents a mathematical model that successfully links ecological dynamics to the GOE. The model demonstrates that a shift in dominance from anoxygenic to oxygenic photoautotrophs, influenced by changes in reductant and phosphate influxes, could have triggered the GOE. The transition could be gradual or sudden depending on the balance between oxygen production and consumption. This integrated approach emphasizes the interconnectedness of biological and geological factors in shaping Earth's history. Future research could focus on refining the model to incorporate additional environmental factors, such as the impact of trace elements or more detailed geochemical processes. Further investigations into the precise mechanisms driving changes in reductant and phosphate influxes during the Archean would also enhance our understanding of the GOE.

The model makes simplifying assumptions, such as the assumption of steady state in the bacterial and chemical abundances during the transition. While this is a reasonable approximation given the slow rates of planetary change compared to bacterial generation times, it might not perfectly capture the dynamic interplay during the actual GOE. The model also simplifies the complex geochemical cycling of iron and other elements. A more detailed representation of these processes could provide additional insights. Finally, uncertainties remain about the precise values of the model parameters, particularly the migration rates of APB and cyanobacteria in the Archean. Further research to constrain these parameters would improve the model's predictive power. Despite these limitations, the model provides a valuable framework for understanding the interplay between ecological and geological factors driving the GOE.