dlacorals 2024.0.0

DLA model for simulating coral growth

Catching the Drift:
Modelling Coral Growth with Diffusion Limited Aggregation (DLA)

Course Project | Complex System Simulation | MSc Computational Science (UvA/VU)

This repository contains the Python framework for modelling coral growth using Diffusion Limited Aggregation (DLA)¹¹. It was constructed during a two-week university project for the course Complex System Simulation in January 2024.

Introduction

Corals are colonial marine organisms of great environmental importance. Understanding their behaviour, morphogenesis and lifecycle are essential for their preservation. It has been recognised² that their growth can be modelled using DLA - an algorithm which generates branching clusters that can closely mimic various natural phenomena, from urban structures¹⁰ to fluid interface instabilities⁷.

The building blocks of corals are polyps - small tentacled invertebrates that live in a symbiotic relationship with photosynthesising microalgae called zooxanthellae. The algae hosted in their tissues absorb sunlight to produce most of the energy required for the coral to survive. The rest is acquired by the polyps by capturing and digesting nutrients carried by the underwater currents, which can range from zooplankton to small fish. Throughout their lifecycle, corals excrete calcium carbonate, which becomes the main component in coral reefs. This branching skeleton remains a long-lasting trace of the organisms that once inhabited it, allowing scientists to identify species by the morphological properties of the calcified structures³.

As any living system, corals have a complex lifecycle, with a lot of environmental and biological factors determining their growth and proliferation. Nonetheless, in a highly abstracted depiction of the conditions under which coral skeletons form, it is easy to spot some parallels with the elementary principles of the DLA algorithm. A diffuse transport of elements required for sustenance (analogical to random walks of particles in a discretised space) ultimately results in the permanent deposition of solid matter (assigning a fixed value to regions of this discrete space) and the formation of dendritic aggregations (the DLA clusters).

Existing research has gone into great depth in incorporating more realistic representations into coral growth models, including the modelling of fluid dynamics⁵ or alternative, more sophisticated growth principles⁶. Additionally, various measures have been highlighted for quantifying the emergent morphological properties of actual corals, including fractal dimension⁴, compactness³ and curvature⁹.

On the other hand, quantitative measures of similar nature have been applied to the phenomenon of DLA as a complex system, analysing its fractal dimension¹ and the distribution of branch lengths⁸. The latter is evidently governed by a power-law, characteristic for many emergent phenomena found in natural systems and in computational models.

The Model

Given the short time frame of the course, we aimed to produce a toy model which builds up on the elementary implementation of DLA and encompasses many of these aspects in a relatively simple form. The goal was to have a versatile simulation framework which enables a range of inputs - either spatial / topological parameters of the simulation space or special variables which can be seen as analogies to environmental influences. As an output, the framework is meant to provide, on one hand, qualitative visuals of the emerging DLA-made "corals", and on the other, a comparative analysis of the variation in selected morphological measures. This framework was used to address the main research questions posed in this project:

• Are there consistent measures to evaluate the emerging morphological complexity of DLA-generated “coral” growth?
• How do drift, growth bias (preferential growth toward sunlight) and surrounding “nutrition” density influence the outcomes of the growth model?
• How can inhibitory influences (e.g. macroalgae blocking the sunlight, competing species, local spatial constraints) affect the outcomes of the growth model - as variants in the environmental parameterisations?

The code is structured in four .py modules. The DLA model containing all functions defining the movement, aggregation, initialisation and regeneration of particles can be found in dla_model.py. The analysis of the fractal dimension and the branch distribution of DLA-generated "corals" are designated functions in cs_measures.py. The experimental setups for executing the simulations and analysing the results are defined as functions in dla_simulation.py. Matplotlib functions tend to create a bit of a mess, which is why most of the plotting procedures are contained in a separate module, vis_tools.py.

A more detailed explanation of the code can be found, together with a documentation of the experimental process, in the Project Notebook.

Prerequisites

The code makes use of the following packages:

• numpy (https://numpy.org/);
• scipy (https://scipy.org/);
• pandas (https://pandas.pydata.org/);
• matplotlib (https://matplotlib.org/);
• seaborn (https://seaborn.pydata.org/);
• networkx (https://networkx.org/);
• powerlaw (https://pypi.org/project/powerlaw/).

The relevant versions are indicated in requirements.txt.

Additionally, to render the animations it may be necessary to install ffmpeg (https://ffmpeg.org/download.html).

References

[1] Jiang, Minhui, and Zhouqi Zhong. ‘Research on Fractal Dimension and Growth Control of Diffusion Limited Aggregation’. International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021), vol. 12163, SPIE, 2022, pp. 778–86.
[2] Kaandorp, Jaap A., Christopher P. Lowe, et al. ‘Effect of Nutrient Diffusion and Flow on Coral Morphology’. Physical Review Letters, vol. 77, no. 11, 1996, p. 2328.
[3] Kaandorp, Jaap A., Peter MA Sloot, et al. ‘Morphogenesis of the Branching Reef Coral Madracis Mirabilis’. Proceedings of the Royal Society B: Biological Sciences, vol. 272, no. 1559, 2005, pp. 127–33.
[4] Martin-Garin, Bertrand, et al. ‘Use of Fractal Dimensions to Quantify Coral Shape’. Coral Reefs, vol. 26, no. 3, 2007, pp. 541–50.
[5] Merks, RMH, et al. ‘Diffusion-Limited Aggregation in Laminar Flows’. International Journal of Modern Physics C, vol. 14, no. 09, 2003, pp. 1171–82.
[6] Merks, Roeland, et al. ‘Models of Coral Growth: Spontaneous Branching, Compactification and the Laplacian Growth Assumption’. Journal of Theoretical Biology, vol. 224, no. 2, 2003, pp. 153–66.
[7] Paterson, Lincoln. ‘Diffusion-Limited Aggregation and Two-Fluid Displacements in Porous Media’. Physical Review Letters, vol. 52, no. 18, 1984, p. 1621.
[8] Pastor-Satorras, Romualdo, and Jorge Wagensberg. ‘Branch Distribution in Diffusion-Limited Aggregation: A Maximum Entropy Approach’. Physica A: Statistical Mechanics and Its Applications, vol. 224, no. 3–4, 1996, pp. 463–79.
[9] Ramírez-Portilla, Catalina, et al. ‘Quantitative Three-Dimensional Morphological Analysis Supports Species Discrimination in Complex-Shaped and Taxonomically Challenging Corals’. Frontiers in Marine Science, 2022.
[10] Rybski, Diego, et al. ‘Modeling Urban Morphology by Unifying Diffusion-Limited Aggregation and Stochastic Gravitation’. Findings, 2021.
[11] Witten Jr, Thomas A., and Leonard M. Sander. ‘Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon’. Physical Review Letters, vol. 47, no. 19, 1981, p. 1400.