A generalized numerical model for clonal growth in scleractinian coral colonies

Abstract

Coral reefs, vital ecosystems supporting diverse marine life, are primarily shaped by the clonal expansion of coral colonies. Although the principles of coral clonal growth, involving polyp division for spatial extension, are well-understood, numerical modelling efforts are notably scarce in the literature. In this article, we present a parsimonious numerical model based on the cloning of polyps, using five key parameters to simulate a range of coral shapes. The model is agent-based, where each polyp represents an individual. The colony's surface expansion is dictated by the growth mode parameter (s), guiding the preferred growth direction. Varying s facilitates the emulation of diverse coral shapes, including massive, branching, cauliflower, columnar and tabular colonies. Additionally, we introduce a novel approach for self-regulatory branching, inspired by the intricate mesh-like canal system and internode regularity observed in Acropora species. Through a comprehensive sensitivity analysis, we demonstrate the robustness of our model, paving the way for future applications that incorporate environmental factors, such as light and water flow. Coral colonies are known for their high plasticity, and understanding how individual polyps interact with each other and their surroundings to create the reef structure has been a longstanding question in the field. This model offers a powerful framework for studying these interactions, enabling a future implementation of environmental factors and the possibility of identifying the key mechanisms influencing coral colonies' morphogenesis.

Keywords: agent-based modelling; clonal growth; coral growth dynamics; generalized minimal model; scleractinian coral colonies.

Figure 1.

The pictures represent five of the most common and representative shapes of coral colonies found in reefs worldwide [1,31]. Credits for the images to Hugo Mann, johnandersonphoto and Rostislavv.

Figure 2.

Table summarizing the relevant parameters of the coral colony model (top) and a graphic representation of its growth steps (bottom), where the vertices on the triangles represent the individual polyps. At each iteration, the following steps are repeated: (a) Surface growth: selected vertices will expand along their normal vector ${\displaystyle \hat{n}}$ , a distance ${\displaystyle \alpha =\nu \mathrm{\Delta }t}$ each time step ${\displaystyle \mathrm{\Delta }t}$ . The growth mode ${\displaystyle s}$ determines which vertices are more likely to expand. (b) Polyp cloning: new clones are generated if the distances between vertices, ${\displaystyle d}$ in the figure, are higher than ${\displaystyle {\delta }_{\mathrm{s}\mathrm{u}\mathrm{b}}}$ . The positions of the new vertices are located at the center of Bezier interpolation curves that unite the original vertices (represented by the black curved line), created using the Casteljau algorithm [32]. (c) Colony branching: polyps are selected and displaced a distance ${\displaystyle 4\alpha }$ over the vector ${\displaystyle {\hat{n}}_{\mathrm{b}\mathrm{r}}}$ , which forms an angle ${\displaystyle \theta }$ with the primary branch. This perturbation generates new branches in the colony in the following iterations.

Figure 3.

Time evolution of the structures generated with our model for different choices of the growth mode ( ${\displaystyle s_{\mathrm{m}\mathrm{i}\mathrm{n}}}$ , ${\displaystyle s_{\mathrm{m}\mathrm{a}\mathrm{x}}}$ ), the inter-branching length ( ${\displaystyle l_{\mathrm{b}\mathrm{r}}}$ ) and the branching angle ( ${\displaystyle \theta }$ ). The drawings are organized into columns (a)-(e), each representing a specific combination of the parameters as detailed in the table at the bottom of the figure. The rows correspond to snapshots taken at three different times: ${\displaystyle t=12,22,32\mathrm{y}\mathrm{r}}$ , for the branching and ${\displaystyle t=2,4,6\mathrm{y}\mathrm{r}}$ for the rest of the shapes. The other parameters of the model have been fixed to ${\displaystyle \nu =10\mathrm{m}\mathrm{m}\mathrm{y}{\mathrm{r}}^{-1}}$ and ${\displaystyle {\delta }_{sub}=10\mathrm{m}\mathrm{m}}$ in all cases. Exceptionally, the right-most structure is produced considering two growth regimes: an initial growth at ${\displaystyle t_1=0\mathrm{y}\mathrm{r}}$ guided by ${\displaystyle s=\left[0.30,1\right]}$ , which at time ${\displaystyle t_2=3\mathrm{y}\mathrm{r}}$ is modified to ${\displaystyle s=\left[0,0.24\right]}$ . This shift reproduces the characteristic transition from vertical to horizontal expansion that is observed in table corals. For a dynamic visualization of the evolution of the coral structures collected in this figure, see the electronic supplementary materials, videos S1–S5.

Figure 4.

Sensitivity of the model to changes in the subdivision distance, ${\displaystyle {\delta }_{\mathrm{s}\mathrm{u}\mathrm{b}}}$ . (a) Linear relationship between the average inter-polyp distance and ${\displaystyle {\delta }_{\mathrm{s}\mathrm{u}\mathrm{b}}}$ for all colony shapes. (b) Branch diameters hold a linear relation with ${\displaystyle {\delta }_{\mathrm{s}\mathrm{u}\mathrm{b}}}$ . The diameters have been measured by averaging them at different positions and for different branches of branching colonies. (c) The average number of neighbours is close to six for all colony shapes, independent of the choice of ${\displaystyle {\delta }_{\mathrm{s}\mathrm{u}\mathrm{b}}}$ . (d) Snapshots exemplify different surface-filling distributions of polyps depending on ${\displaystyle {\delta }_{\mathrm{s}\mathrm{u}\mathrm{b}}}$ . All four depicted colonies in (d) have grown for a time of 10 yr. On the contrary, in graphics (a)–(c), the colonies have evolved until the measured value stabilizes. The elongation rate ${\displaystyle \nu }$ has been set to ${\displaystyle 10\mathrm{m}\mathrm{m}\mathrm{y}{\mathrm{r}}^{-1}}$ , and the growth modes ${\displaystyle s_{\mathrm{m}\mathrm{i}\mathrm{n}}}$ and ${\displaystyle s_{\mathrm{m}\mathrm{a}\mathrm{x}}}$ are chosen following figure 3 to reproduce all colony shapes.

Figure 5.

Branching coral structures reproduced with different choices of the model parameters. Columns correspond to a change in inter-branching distance ${\displaystyle l_{\mathrm{b}\mathrm{r}}}$ and the column varies the branching angle $\theta$ . The rest of the parameters have been set to: ${\displaystyle s=\left[0,0.375\right],\nu =10\mathrm{m}\mathrm{m}\mathrm{y}{\mathrm{r}}^{-1},{\delta }_{sub}=10\mathrm{m}\mathrm{m}}$ and run for ${\displaystyle t=32\mathrm{y}\mathrm{r}}$ . The three-dimensional objects representing these colonies are displayed in the electronic supplementary material, named as Object ${\displaystyle Sl}$ _ $\theta$ , where ${\displaystyle l}$ labels the inter-branching distance and ${\displaystyle \theta }$ the branching angles fixed for the specific shape.