Maximum vertical accretion rates in the simulation are user-defined. For shallow assemblages on exposed margins, maximum vertical accretion rates (11 m/kyr) are chosen to reflect known average rates for robust branching coral facies in high-energy environments established for the Indo-Pacific . Moderate–deep assemblages represent slightly higher maximum accretion rates (15 m/kyr) with the lowest accretion rates (9 m/kyr) for deep assemblages. These were chosen to reflect the average accretion rates for Indo-Pacific tabular-branching and massive coral facies found in high-energy conditions .

pyReef-Core requires knowledge of the intrinsic rate of assemblage population growth/decline (ϵi) and the matrix coefficients (αij) of interactions between distinct assemblages. However, inferring ecological dynamics from ecological studies is challenging. Empirical studies of coral competition and growth are often focused at the species rather than assemblage level and explain competitive relationships qualitatively rather than quantitatively e.g.. Moreover, GLVEs have not been used to model coral population dynamics at the temporal resolution (centennial to millennial) we are interested in. Based on an initial sensitivity analysis, we define a set of values for the Malthusian parameter (ϵi) and interaction coefficients among assemblages (αij) which are summarised in Table . Chosen coefficients define small competitive interactions between assemblages.

The coral assemblages defined in this study largely do not share the same environmental setting and optimal growth conditions. Therefore, competitive interactions are restricted to only those assemblages that may reasonably co-exist due to overlapping depth, sediment flux or flow velocity thresholds. This translates to an interaction matrix with values only along the main diagonal and the super- and sub-diagonals. Everywhere else, interactions are set to 0. Associated with these interactions, we define a series of critical threshold response functions for each assemblages (Fig. ).

Based on a statistical analysis of the depth and environmental distribution of modern coral communities at One Tree Reef (GBR), calibrated the palaeo-water depositional environments of six fossil coral assemblages (three in protected and three in exposed environments). This calibration was also based on quantitative measurements of crustose coralline algae thickness and vermetid gastropod abundance which are reliable palaeo-depth indicators, allowing for the depth intervals to be more accurately constrained. These assemblages are broadly consistent with shallow- and deep-water coral facies of the Indo-Pacific .

Here, three assemblages typically occurring on exposed slopes are modelled according to the estimated depth intervals defined by (Table ) and represent shallow-water (< 6 m), moderate-to-deep-water (6–20 m) and deep-water (20–30 m) assemblages.

The water flow function is constructed according to the theoretical relationship defined by whereby wave stress decreases exponentially with depth (Fig. ). Here, we rely on the velocity–depth relationships on wave-exposed reef slopes from the field study by . A maximum velocity of 25 cm s-1 in region ≤ 1 m and an exponential decrease up to 25 m below which flow velocity is set to 0. This is consistent with direct observations from exposed algal flat and maximum velocities (> 50 cm s-1) beyond which branching corals are susceptible to breakage .

With specific data on the optimal flow environment for specific corals lacking, assumptions about thresholds for distinct coral assemblages are inferred from boundary conditions. That is, the water flow exposure threshold range for each assemblage reflects the attenuation of water flow with depth (Fig. ).

pyReef-Core can model the vertical sedimentation rate (m day-1) as a function of either time or depth. When sediment flux is dependent on depth, it implies that sediments are autochthonous (loose carbonate materials), in contrast to terrigenous sediments transported from outside the reef system (siliclastic materials), which may be represented by sediment flux varying with time. In our case studies, we use a depth-dependent sedimentation rate input curve to approximate the temporal variations in sediment accumulation along the core (Fig. ).

Sediment tolerance thresholds for each coral assemblage (Fig. ) are informed by before receiving maximum and minimum sedimentation rates corresponding to the sediment input boundary condition (Fig. ). The boundary condition provides a broad indicator of the sediment load expected at certain depths and thus what would be tolerated for each depth-specific assemblage. With alternate sediment input boundary conditions, the upper and lower tolerance thresholds can be adjusted to represent how coral communities respond differently to site-specific suspended sediment levels.

Considering the simulated temporal scale, neither subsidence nor uplift are considered to be important in this experiment. Instead, accommodation is simulated as a function of Holocene sea-level changes and vertical coral reef growth only. The Holocene relative sea-level (RSL) curve from is used to represent sea-level change (Fig. ). The data suggest a RSL history that is characterised by a mid-Holocene highstand of 1.8 m at ∼ 4 ka before returning slowly to present sea level, matching other estimates of RSL .

Simulation begins at 8.5 ka, which is within the take-off envelope for Holocene growth of outer-platform GBR reefs . At 8.5 ka, RSL is 15 m below sea level and substrate is at 20 m depth in order to simulate a catch-up growth strategy from a deep substrate. We compute the GLVEs at time intervals of 2.5 years and combine each accumulated assemblage as a stratigraphic unit within the core for every 50 years.

Figure  presents the GBR-representative assemblages summarised by as well as the simulated core by pyReef-Core. The modelled core is 35 m long and is composed of three assemblages characteristic of an exposed margin and carbonate sediments. The simulation portrays two distinct assemblage transitions from massive assemblages representing deep (20–30 m), low-flow conditions to a faster-growing, tabular-and-branching assemblage characteristic of the 6–20 m depth interval, which is succeeded in shallow water (< 6 m) by a robust-branching assemblage representing higher-energy conditions (Figs. , ).

As sea level rises from 8.5 to 6.5 ka, the deeper assemblages have sufficient accommodation space (> 20 m) and low-flow to thrive. However, greater sediment input at depth is inhibitive in the early part of the simulation at the base of the core (32–35 m) (Fig. ). As sea level begins to stabilise (Fig. b), accommodation space decreases and moderate–deep assemblages start to dominate the sequence up to 4.7 ka (Fig. c). Following stabilisation from 4.7 to 3.2 ka, shallow assemblages develop as a result of the decreased accommodation space (∼ 6 m at 4.7 ka), high-velocity hydrodynamic conditions and reduced sediment input. Assemblage growth rates (Fig. d) show a pattern similar to the population number curves with values lower than assemblage maximum production rates (Table ) indicative of the effects of environmental factors (sediment input and flow velocity) on the growth of each assemblage. The deeper assemblage is 15 m thick and is composed of 30–60 % loose sediment and is succeeded by ∼ 12 m of moderate–deep assemblages with a lesser proportion of sediment (Fig. ). The last 6–7 m of core are predominantly formed by shallow assemblages with on average less than 20 % of carbonate sediments (Fig. ). The simulated shallowing-up sequence accurately reflects expected shift from deep to moderately deep assemblages at ∼ 15–20 m depth and from moderately deep to shallow assemblages at ∼ 6 m depth proposed by and . The simulated sequence relates well to the description proposed by and reproduces the distinct assemblages defined in the idealised reef sequences found on exposed margin along the GBR (Fig. ).

The modelled core reaches sea level at around 2.5 ka (Fig.  ) which also correlates well with values reported for several reefs in the GBR . Average vertical accretion rate implied by the model is around 4.1 m kyr-1 (Fig. ), again in the range of actual drill cores average rates, which varies by around 3 to 5 m kyr-1 on exposed reef margins . It is also worth noting that coral growth becomes predominant within the sequence at ∼ 7.8 ka in the modelled core, which coheres with the observed delay in reef initiation of approximately 1 kyr after initial flooding of the substrate during the Holocene transgression. We also notice that the transitions between assemblages also correspond to periods where the proportion of carbonate sediment deposited increases (Fig. ). It mimics a lag between optimal conditions from one assemblage to the other and relates to the choice of environmental threshold functions that were imposed in our simulation (Fig. ). Overall, the model reproduces the details of the formation of shallowing-upward sequences both in terms of assemblages succession, accretion rates, deposited thicknesses and timing of initiation. It can be applied to estimate the impact of changing environmental conditions on growth rates and patterns under many different settings and initial conditions.

We reconstruct using pyReef-Core the evolution of an ideal coral reef sequence since the last interglacial (LIG). LIG is represented by marine isotope stage (MIS) 5e, which is a proxy record of low global ice volume and high sea level . It is arbitrarily set to begin at approximately 130 ka before present and our simulation runs over 140 kyr. The GLVEs which control the coral productions dynamic are updated every 25 years and stratigraphic layers are recorded at time interval of 100 years.

Here we use the sea-level curve proposed by , who estimate sea-level records based on the timing of past ice-volume changes, relative to polar climate change. The relative sea-level change over the simulated period has rates of rise reaching 12 cm yr-1 during all major phases of ice-volume reduction, with values below 7 mm yr-1 when sea level exceeded present mean sea level . The applied sea-level curve is shown in Fig. b.

The karstification of Pleistocene reef limestone has been identified as a controlling factor on variations in antecedent topography, which in turn is thought to influence the morphology of modern reefs . Rates of karstification are a function of exposure time, rainfall, porosity and original topography of exposed carbonate reefs. Summary of karstification rates from both the Indo-Pacific and Caribbean shows values ranging from 0.01 m kyr-1 (Barbados, ) to 0.14 m kyr-1 (mid–outer platform reefs, southern GBR; ). Here we impose a karstification rate of 0.07 m kyr-1 consistent with estimates from Ribbon Reef 5 and outer central GBR shelf .

Over such a period of time, sea-level fluctuations are not the only factor controlling the accommodation change and uplift–subsidence evolution has to be considered . Based on a comprehensive study of GBR reefs, Dechnik (personal communication, 2017) estimates that a subsidence rate of ∼ 0.083 to 0.13 m kyr-1 is required to explain the observed elevation of the upper surface of the LIG reef that provides the antecedent topography of the modern mid–outer platform reefs in the GBR. The proposed range is consistent with values found for other reefs along the GBR . In our model, we use a constant rate of subsidence set to 0.1 m kyr-1 that corresponds to 14 m of subsidence over the duration of the simulation. In addition, the initial elevation is set 20 m above sea-level position at the start of the simulation (140 ka), corresponding to a depth of ∼ 70 m below the current sea-level position.

Prior to 135 ka, the model shows a first stage of reef growth characterised by shallow-water, high-energy coral community colonisation (Fig. a,c and Fig. ), following the flooding of the antecedent platform. The cumulative thickness for this phase is < 10 m (Fig. d) and is compatible with values estimated for Ribbon Reef 5 and Heron Island .

Following this initial phase, a deepening-upward sequence occurs up to 132 ka (Fig. ). Again, this sequence has also been identified in a similar time interval at One Tree Reef (southern GBR) and Stanley Reef (central GBR) . A lack of significant reef framework (< 30 %) characterises the stratigraphic sequence during this interval.

The rapid sea-level rise during the end of the penultimate deglaciation explains the drowning event observed in the core from 128 to 118 ka (Fig. c). During this period, the accommodation increase is mainly driven by sea-level fluctuations and to a small extent (∼ 1 m) by the imposed subsidence rate.

From 118 to 107 ka, during the first stage of the regression phase, a shallowing-upward sequence (∼ 30 m thick) is identified with three distinct community populations modelled over time (Fig. c). During this time interval, the maximum population number for the moderate–deep assemblages is relative lower (< 3) than for the two other assemblages (> 7). Consequently, the percentage of accumulated thickness for this assemblage is below 7 %. These assemblage transitions are primarily controlled by high-frequency sea-level variations observed in the curve (Fig. b). Minor events of karstification (< 2 cm of erosion) are triggered by short episodes of subaerial exposure around 110 ka. From 107.5 to 104 ka, high-energy coral communities (shallow assemblages) dominate the sequence with a maximum growth rate above 8 mm yr-1 (Fig. d).

The following stage from 107 to 12 ka is characterised by a period of subaerial exposure due to sea-level fall (Fig. d). Both subsidence and karstification occur and account for nearly 11 m of elevation offset with about 1 m attributed to karstification processes (Fig. ). Applied to a real case, pyReef-Core can be used to test several scenarios with different rates of subsidence and karstification in order to explain for example the discrepancy in age–elevation data of LIG deposits observed in the GBR . It can also be used to estimate the contribution of karst dissolution and subsidence with a more quantitive approach.

By 13 ka, sea level re-floods the LIG reef, and Holocene reef growth initiates ∼ 10.5 ka in the experiment (Fig. ). The lag (2.5 kyr) between flooding and reef growth initiation matches well with observations for the GBR . However, the timing of the initial flooding occurs 3 kyr earlier than what is expected for the GBR. This temporal difference is related to both sea-level variations and chosen initial starting elevation of the model. The Holocene reef sequence is around 46 m thick (Fig. ), which is above most of the GBR reef maximum vertical accretion thicknesses (usually < 30 m) but correlates with thicknesses found in reefs from Tahiti and Huon Peninsula . This Holocene sequence is first composed of more than 30 m of moderate–deep assemblage, which corresponds to the catch-up phase discussed in the first study case and is associated with the rapid sea-level rise. The reef accretion rate during this time interval is maximal and reaches values above 8.2 mm y-1 (Fig. d). The remaining ∼ 15 m of the uppermost sequence is built of shallow assemblages that become predominant after 6 ka when sea-level rise decreases. It is also worth noting the presence of short periods of subaerial exposure which coincide with two small karstification events (karst dissolution < 1 cm; Fig. ).

The total simulated core has an overall thickness > 86 m. A complete sequence such as the one modelled here is unlikely to be found in a natural reef complex mainly due to the 3-D nature of such a system . Nevertheless the predicted sequence represents in 1-D the idealised succession of coral assemblages produced for a given set of initial and forcing conditions. Therefore, it can be compared to series of drill cores at different positions along a given region and used as a quantitative approach to analyse stratigraphic responses of coral reefs to a combination of physical, biological and sedimentological processes.