A generalized light-driven model of community transitions along coral reef depth gradients

Aim: Coral reefs shift between distinct communities with depth throughout the world. Yet, despite over half a century of research on coral reef depth gradients, researchers have not addressed the driving force of these patterns. We present a theoretical, process-based model of

Mesophotic reefs could be considered marginal habitat, because of their low light levels. However, these reefs thrive world-wide (Baker et al., 2016), and contain their own diversity (Muir, Wallace, Pichon, & Bongaerts, 2018). Though we must be careful when generalizing, a number of ecological patterns have been observed down coral reef depth gradients, from the shallows to MCEs. There is now broad agreement that the mesophotic zone is subdivided into the 'upper mesophotic' and 'lower mesophotic' , with a transition at approximately 60 m, depending on water clarity and depth of the thermocline (Lesser, Slattery, & Mobley, 2018;Slattery, Lesser, Brazeau, Stokes, & Leichter, 2011). This '60 m rule' has been linked to reductions in shallow species richness with increasing depth, and holds across ocean basins (Fricke & Meischner, 1985;Kahng et al., 2010;Lesser, Slattery, Laverick, Macartney, & Bridge, 2019;Slattery & Lesser, 2012). The upper mesophotic zone is an area of overlap between shallow and mesophotic taxa, while the lower mesophotic is characterized by taxa adapted to low-light environments (Kahng et al., 2019). These changes are in parallel to zonation in Symbiodinium phylotypes in some places (Lesser et al., 2010).
The lower depth limit of the mesophotic zone, meanwhile, is linked to the deepest occurrence of zooxanthellate corals (Kahng et al., 2010).
Yet, significant gaps remain in our knowledge. A lack of information on the functioning of these ecosystems has caused researchers to rely on a fixed depth limit of 30 m as a boundary between shallow and mesophotic reefs (Laverick et al., 2018). This 30-m limit is based on SCUBA regulations and difficulties of access (Pyle, 2019), and fails to explicitly accommodate environmental variation. Recently the boundary was updated to 30-40 m, and to be theoretically rooted in ecology (Baker et al., 2016), but still remains a largely untested assumption with little empirical support. In the Gulf of Eilat/Aqaba in the Red Sea, shifts in community structure were detected at 40 and 70 m (Tamir, Eyal, Kramer, Laverick, & Loya, 2019). However, the same approach when applied to the reefs off Utila in the Caribbean returned boundaries at approximately 15 and 40 m (Laverick, Andradi-Brown, & Rogers, 2017). For more examples of varying boundary depths, see Pyle and Copus (2019). The underlying causes, and variability, in these faunal breaks requires more research.
The principal candidate driver of change along coral reef depth gradients is light. In their review,  recognized that optical properties of the water column are likely to have the strongest correlation with MCE community patterns. Muir et al. (2018) reported mesophotic species are more likely to be found in shaded microhabitats in the brighter portions of their depth range, with the same being true for juvenile corals . This can blur trends in physiology with depth for mesophotic corals (Laverick, Green, Burdett, Newton, & Rogers, 2019). Kahng et al. (2010) found notable correlations between light attenuation with depth (K dPAR ) and regional maximum depth records for zooxanthellate corals world-wide. This in turn could be linked to the small-scale optics of corals at the cellular level. Highly organized skeletal geometries have been observed in some mesophotic species, consisting of ridges and furrows, in contrast to typically shallow species (Kahng et al., 2012). The skeletal scattering of photons can increase path lengths and the potential for photons being absorbed (Wangpraseurt et al., 2014). This highlights how species physiology may need to be linked to community models through an understanding of the light field. Linking mesophotic taxa to abiotic conditions is, therefore, a research priority for the field (Costa et al., 2015;Turner et al., 2019).
When doing this, it is important to remember that depth range alone is a poor descriptor of a species' niche. Individuals at maximum and minimum observed depths may exist at the limits of their physiology, inflating our sense of where species are likely to occur (Roberts, Bridge, Caley, Madin, & Baird, 2019). When assessing these physiological envelopes, we must look at the whole distribution of abundance with depth.
The number of studies that include community characterization through the mesophotic zone, combined with the optics of the water column, have increased . While MCEs share species with shallow water reefs (Laverick et al., 2018;Muir et al., 2018), the dominant habitat forming species largely differ (Kahng et al., 2010). This means the relationships between different taxa and light across depth gradients need to be quantified (Edmunds, Tsounis, Boulon, & Bramanti, 2018;Kleypas, Mcmanus, & Meñez, 1999;Muir et al., 2018). Recent studies in the Red Sea have shown this can be achieved at a community level by linking light data  to cluster analyses of co-occurring coral taxa (Laverick et al., 2017;Lin & Denis, 2019). Similarly, in upper mesophotic sites throughout the central Indo-Pacific, light controls community structure in tandem with hydrodynamic effects (Turak & DeVantier, 2019). Substrate type, rugosity, and slope angle all alter the light environment and were found to lead to zoned communities when interacting with the habitat preferences of hard corals.

K E Y W O R D S
depth gradients, light, marine, mesophotic coral ecosystems, reef zonation, tropical reefs Modelling exercises are particularly powerful when they direct our current knowledge towards answering an overarching question. The last 10 years has seen the mesophotic literature increase threefold (Laverick et al., 2018;Pyle & Copus, 2019;Turner et al., 2017), allowing us to present a framework for thinking of mesophotic reefs as ecological entities (Baker et al., 2016). However, the empirical work in mesophotic ecology has rarely been supported by theoretical studies (see Roberts, Keith, et al., 2019 for an exception). With this in mind,  used an existing model (Mobley, Zhang, & Voss, 2003) to combine a simple 3D reef geometry with features that affect the underwater light field. In doing so, they created a quantitative representation of the current candidate structuring force across coral reef depth gradients.  suggested that by uniting a community approach to defining MCEs (Laverick et al., 2017) with optical data describing the underwater light field, it may be possible to provide a reef-to-reef definition of where community boundaries occur.
Though they are right to call for more studies through the entire depth range of photosynthetic reefs linked to optical data, a working quantitative theory is valuable to compare against data as they become available. We believe it is already possible to theoretically link these approaches at a community level.
Here, we present a generalized, mechanistic, light-driven model of the shallow to mesophotic reef transition. In doing so we unify key ecological patterns from coral reef depth gradients into a single quantitative framework. Our model assumes community-level light relationships, and reproduces plausible patterns. We can express current ecological definitions as mathematical conditions, and can now predict how the boundaries between reef zones will shift under different light regimes.

| MATERIAL S AND ME THODS
We take a community level approach to create a theoretical, mechanistic, model of the shallow to mesophotic reef transition. Machinelearning algorithms have already been applied in the Caribbean (Laverick et al., 2017) and the Red Sea  to show mesophotic and shallow reefs can be grouped as two assemblages of Scleractinia (Hartigan & Wong, 1979). By assessing the similarity of field observations to these two communities across light gradients, it is apparent that shallow taxa may be light-limited with depth, while mesophotic taxa occur in an optimal light envelope .
We now construct a light-driven model to predict community values (D) for reef patches down a generalized depth gradient. We refer to community values as D because the two studies previously mentioned co-opted the equations of Defrene and Legendre traditionally used for indicator species identification (Dufrene & Legendre, 1997). We highlight that the Caribbean and Red Sea studies only inspire this model, and that the machine-learning and calculation of D for indicator values are not required for the construction and use of this model. To build our model, we begin by assuming shallow communities (D Shallow ) are light-limited beneath a threshold, and that community values (how similar a reef patch is to the typical shallow community) increase asymptotically with increasing light (Figure 1 middle). Values will plateau as habitat heterogeneity and the size of reef patches considered will prevent all shallow taxa occurring in the same place, even under optimal light levels. We assume mesophotic communities (D Mesophotic ) centre on a preferred light value, between light limitation and light-induced stress (Figure 1 bottom). Finally we assume a tradeoff, where an observed patch of reef can approach a mesophotic or shallow community. A patch of reef cannot simultaneously look like the exemplar mesophotic and shallow reef patch (D Reef ). The functions to represent these assumptions, and create the curves in Figure 1, are reported below (Equations 1-3). A detailed description of model parameters is available in Table 1. In brief, V max is the ceiling of the relationship between shallow communities and light, %PAR surface is the percentage of photosynthetically active radiation from the surface, K is the point of light limitation of shallow taxa, a is a scalar term and b is the shape term in the light relationship for the mesophotic community.
The D Shallow and D Mesophotic functions were recently validated on mesophotic reefs in the Red Sea .
To transform the community-light relationships into depth ranges we use the Lamberte-Beer law (Kirk, 2011) We find the upper and lower depths of the upper mesophotic zone by finding the root of D Reef . We do this numerically using the {rootSolve} package (Soetaert, 2016) in R (R core team, 2020). This light value is then passed to the Depth function under maximum and minimum shading.
To test the performance of our model, we predicted where community boundaries occur for 11 different reefs and compared these to the depth limits currently used as proxies in the field. Kahng et al. (2010) collated light attenuation coefficients (K dPAR ) and photosynthetic coral depth records from around the world. We used these K dPAR values to parameterize the underwater light field functions in our model. For this analysis the community relationships were parameterized to Tamir et al. (2019).
We explore the sensitivity of the model, to varying parameter values, in terms of the change in the predicted depths of the upper Depth PAR surface Photosynthetically active radiation (µmol/m 2 /s) just below the air-water interface. Also found in D Shallow and D Mesophotic .
K dPAR A light attenuation coefficient, indicating water clarity. Bounded between 0 and 1. Lower numbers indicate clearer water, which allows the passage of more light.

Shade
A parameter to penalize the light available because of slope angle. Vertical walls are shaded in comparison to horizontal planes. Bounded between 0 and 1, Shade can be interpreted as the proportion of light available. A value of 1 means no shading, a value of 0 means no light reaches the shaded area.
The maximum similarity of a reef patch to the idealized shallow reef assemblage. Bounded by 0 and 1. Factors such as the size of the reef patch, and spatial heterogeneity of taxa will affect the maximum similarity a reef patch can achieve, that is, a small 1-m 2 quadrat is unlikely to contain all shallow coral species.

K
The light value where D Shallow = V max /2. This can be interpreted as the light level at which shallow reef communities become light limited, on a percentage scale.
The scale parameter of a Weibull distribution. a describes the spread of the data, or 'the peakyness'. Larger numbers create a flatter, wider, curve. Varying a will simultaneously change the width of the preferred light environment of mesophotic taxa, and the maximum similarity a reef patch is expected to achieve in comparison to the idealized mesophotic assemblage. b The shape parameter of a Weibull distribution. A value of 3 approximates a normal distribution. Values larger than 3 introduce a left-skew to the curve. This means mesophotic taxa can be light-limited, while keeping mesophotic communities in low-light environments. Varying b will therefore shift the preferred light environment of mesophotic communities.  . The sensitivity of the model to these values is shown in Figure 4. The results of these sensitivity analyses are not site specific.

| RE SULTS
Our model succeeds in returning key ecological features, such as a shallow to upper mesophotic and upper to lower mesophotic boundary ( Figure 2). We can now predict the depths of these features at different sites (Figure 3), and find support for the heuristics used in mesophotic ecology. We can define these ecological features in a single unified, quantitative, framework. To achieve this, we characterized coral photo-physiology and the underwater light field with a series of equations. Combining these equations gives us a model of community transitions down a general coral reef gradient. Our model is comprised of two curves that represent coral communities at the two extremes of light variability for a given depth (Figure 2).
We introduced this variability through bathymetry, varying slope angle from horizontal to vertical. A full explanation of the model parameters, and the values used, can be found in the methods and in Table 1. shallow reef patches than mesophotic patches, in keeping with existing ecological data (Laverick et al., 2017;Tamir et al., 2019).
In addition to the fixed representation of our model (Figure 2), an interactive web app can be accessed at https://laver ick.shiny apps.io/a_light -driven_mesop hotic_model /. We provide the R script (version: 1.2.1335) in Supporting Information Code S1 to safeguard against stability issues. By opening the app, the reader is able to vary parameter values by moving sliders, and see how this changes community transitions with depth. Figure 1  Our sensitivity analysis revealed that changes in the underwater light field may have a stronger influence on community structure than community level physiology. Changing the parameters controlling the preference of shallow and mesophotic taxa for different light levels can be thought of as changing their physiology (Table 1).
Varying these parameters introduced only minor shifts in reef boundaries, and no change in the depth range of the upper mesophotic zone ( Figure 4). Meanwhile, light parameters (i.e. light attenuation as K dPAR and shading) can be changed to predict community transitions at sites with differing underwater light fields. These parameters produced the largest changes in community boundaries in our model.

| D ISCUSS I ON
Our model unifies the broad community patterns from coral reef depth gradients into a mathematical framework. By specifying community-level light relationships, and combining these with an aries (Baldwin, Tornabene, & Robertson, 2018;Fricke & Knauer, 1986;Kahng et al., 2019;Kinzie, 1973;Laverick et al., 2017;Lesser et al., 2019;Rocha et al., 2018;Semmler, Hoot, & Reaka, 2016). When the term mesophotic was adopted at a 2008 workshop (Puglise et al., 2009) -although the word had been published earlier (Ginsburg, 2008 Our model shows we can expect to find mesophotic communities in relatively shallow waters when light attenuation is high. We should, therefore, embrace shallow-water turbid reefs as mesophotic coral ecosystems. This is not to say there will not be deviations be- When these studies are conducted, it is important to use the distribution of abundances instead of depth ranges to avoid extending the physiological niche of communities with depth . We stress that this does not impact our model's abil-  (Appeldoorn et al., 2015;Sherman et al., 2016) are not yet in the model framework, and could reduce the depths reefs extend to. A missing factor important for Bermuda is the absence of seasonal changes in irradiance, which are greater at higher latitudes.
We have identified two major processes missing from our theoretical framework. As well as environmental seasonality, heterotrophy (Houlbrèque & Ferrier-Pagès, 2009;Lesser et al., 2010 could introduce error into model predictions. Some corals have been seen to increase their reliance on heterotrophy under low irradiance at depth (Lesser et al., 2010;Muscatine, Falkowski, Dubinsky, Cook, & McCloskey, 1989). Similarly, light levels are increasingly seasonal at higher latitudes. Corals may also vary their reliance on heterotrophy in response to this variation in irradiance throughout the year (Nir, Gruber, Shemesh, Glasser, & Tchernov, 2014).
Heterotrophic subsidy may permit corals to exist on reef patches (Anthony & Fabricius, 2000), or to survive disturbance events (Grottoli et al., 2014), which a model based on light alone may fail to predict. Seasonal changes in irradiance could lead to shifts in the depths at which transitions between autotrophy and heterotrophy might occur (Brandtneris, Brandt, Glynn, Gyory, & Smith, 2016 (Brandtneris et al., 2016).
Energy budgets must, therefore, balance across the course of a year, and could explain observations of seasonal coral bleaching on MCEs (Nir et al., 2014). Phenological investigations of MCEs are, therefore, necessary to identify which periods of the year for different sites are key to the persistence of MCEs. This will allow us to target research effort to key points in the year that lock in long-term trends, and will allow us to design effective management strategies.
A third missing factor worthy of mention, but less connected to the discussion of light so far, is hydrodynamics. Violent waters may keep some low-light specialist taxa at mesophotic depths (e.g. Acropora pichoni and Acropora tenella). These species may only be found in shallow low-light environments when sheltered, such as lagoons in Kimbe Bay (Micronesia; Rowley et al., 2019), and the Great Barrier Reef (Bridge et al., 2012). Similarly cyclones may impact on MCEs to varying degrees. Mesophotic coral communities in areas exposed to tropical storms such as the Great Barrier Reef (Bongaerts, Muir, Englebert, Bridge, & Hoegh-Guldberg, 2013) and Okinawa (White et al., 2013) differ from MCEs where storms are less frequent and weaker [e.g.
Papua New Guinea: Smith, Holstein, & Ennis, 2019] despite minimal differences in water quality and, therefore, light quality.
Theoretical work has been largely neglected in mesophotic ecology. We have made a contribution to rectifying this, as theoretical work is needed to complement and direct empirical studies. Models such as ours are a valuable tool for generating questions and formulating hypotheses. Though a number of processes are not included in our framework, this model represents a much needed successor to the strawman argument of the 30-m depth boundary for mesophotic reefs. We have shown that we can get remarkably far in explaining ecological pattern using light as a single abiotic factor, but we require more empirical studies before we can include additional processes.
Creating this model was a crucial step in the iterative process that allows us to better understand the structure of coral communities with depth.

ACK N OWLED G M ENTS
We would like to thank the Interuniversity Institute for Marine

CO N FLI C T O F I NTE R E S T
The authors declare that they have no conflicts of interest.

DATA AVA I L A B I L I T Y S TAT E M E N T
The authors declare there are no data to archive. However, the R code behind this model is provided in the Supporting Information, and an interactive version of the model framework can be found at https://laver ick.shiny apps.io/a_light -driven_mesop hotic_model /.