Relevance of coral geometry in the outcomes of the coral-algal benthic war

Corals have built reefs on the benthos for millennia, becoming an essential element in marine ecosystems. Climate change and human impact, however, are favoring the invasion of non-calcifying benthic algae and reducing coral coverage. Corals rely on energy derived from photosynthesis and heterotrophic feeding, which depends on their surface area, to defend their outer perimeter. But the relation between geometric properties of corals and the outcome of competitive coral-algal interactions is not well known. To address this, 50 coral colonies interacting with algae were sampled in the Caribbean island of Curaçao. 3D and 2D digital models of corals were reconstructed to measure their surface area, perimeter, and polyp sizes. A box counting algorithm was applied to calculate their fractal dimension. The perimeter and surface dimensions were statistically non-fractal, but differences in the mean surface fractal dimension captured relevant features in the structure of corals. The mean fractal dimension and surface area were negatively correlated with the percentage of losing perimeter and positively correlated with the percentage of winning perimeter. The combination of coral perimeter, mean surface fractal dimension, and coral species explained 19% of the variability of losing regions, while the surface area, perimeter, and perimeter-to-surface area ratio explained 27% of the variability of winning regions. Corals with surface fractal dimensions smaller than two and small perimeters displayed the highest percentage of losing perimeter, while corals with large surface areas and low perimeter-to-surface ratios displayed the largest percentage of winning perimeter. This study confirms the importance of fractal surface dimension, surface area, and perimeter of corals in coral-algal interactions. In combination with non-geometrical measurements such as microbial composition, this approach could facilitate environmental conservation and restoration efforts on coral reefs.


INTRODUCTION
Corals use energy derived from photosynthesis and heterotrophic feeding to build reefs. This has 65 enabled corals to dominate the battle for light and space on the reef benthos for millennia 66 (Kaandorp & Kubler, 2001). However, the combination of overharvesting of herbivorous fish, 67 increased nutrient runoff from land (eutrophication), and ocean warming is stimulating the 68 growth of non-calcifying algae at the expense of corals world-wide (Alevizon & Porter, 2015). 69 The increase in algal coverage is re-routing the energy to alternative trophic pathways that are is crucial to understand the key factors that determine the outcomes of these interactions. 76 While there has been significant study into the effects of nitrification and changes in 77 herbivore biomass on coral-algal interactions, results have been somewhat equivocal (Smith et  increases so does its potential for nutrient acquisition and distribution (Oren et al., 2001). Thus, 100 coral surface area should be another key factor in determining the coral-algal interaction 101 outcome.

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The resource availability hypothesis (RAH) (Endara & Coley, 2011) predicts that fast 103 growing corals will rely on clonal growth strategies to indirectly outcompete the invading algae.

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This explains the resilience observed among branching corals, which invest the resources 105 5 acquired from their large surface areas to grow new polyps rather than to protect their small 106 perimeters (Swierts & Vermeij, 2016). In contrast, RAH predicts that slow growing species tend 107 to face more encounters with competitors and will invest more resources in protecting their 108 perimeters. This has been confirmed for slow growing corals like encrusting and massive corals 109 (Swierts & Vermeij, 2016). 110 The morphology and size of these slow growing corals have been linked to corals' natural 111 competitive edge against most algal groups (Porter, 1976;Tanner, 1995 won more often than small and large corals (Swierts & Vermeij, 2016). Thus, the influence of 118 the geometrical properties in the outcome of the coral-algal interaction remains unclear.

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The accurate measurement of the perimeter and surface area in natural objects, however, 120 is usually challenged by the presence of fractality (Mandelbrot, 1967(Mandelbrot, , 1977(Mandelbrot, , 1983 Here we hypothesize that larger fractal dimensions and smaller perimeter-to-surface area      (Table S1). The upper value of this range was 171 used as the theoretical error for the fractal dimension. The perimeter fractal dimension (DP) was 172 calculated from the 2D high-resolution models, which allowed a minimum of ten bisections in 173 the algorithm. The surface fractal dimension (DS) was calculated from the 3D high-resolution 174 models, which allowed a minimum of five bisections in the algorithm. The null hypotheses DP ≠   (Table S2). This discrepancy was reasonable taking into the account the projection 185 on the model and the measurement field error. The 2D perimeter used in the analysis was 186 obtained using a 1 mm ruler in the Richardson algorithm. Polyp diameters were also measured 187 from the 2D models using ImageJ 1.47v and averaging 10 polyp diameters per colony. See      The surface areas and surface fractal dimensions were measured for 50 corals within the 251 1 mm to 1 m range using the 3D coral models (Figure 3a). The 5% to 95% confidence intervals

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This is due to %W being negatively correlated with %L (slope = -0.9 ± 0.1, R 2 = 0.8, p-value = 277 2.2x10 -16 ***) ( Figure S4). The percentage of neutral perimeter (%N) was discarded due to its   Figure S6a). The variance explained using all variables was 4.3 ± 0.6 % (SE). The surface 288 fractal dimension was the most important predictor, and the only one selected statistically against 289 the null hypothesis by rfPermute (p-value < 0.05). The following variables-listed with 290 decreasing importance-were the 3D perimeter, surface area, and perimeter-to-surface ratio. The 291 lowest ranked predictor was the mean perimeter fractal dimension.

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The surface area had a similar value but the p-value was slightly larger (p-value = 0.059 ± 315 15 0.010). The geometrical properties of corals explained ~25% of the variability of %Winning 316 outcomes, and the perimeter to surface area ratio was the strongest predictor.   (Figures S10 and 4b). Thus, the outcome averages for