Ocean Presentation

Reaction-Diffusion Models for Coral Reef Growth

Kenneth Lopez

Introduction

Reaction-diffusion equations describe the spread of chemicals through a combination of diffusion and reaction between the different particles involved to describe the spread of the two. Various forms of reaction-diffusion models exist in ways that describe population dynamics (predator-prey), chemical reactions, and biological pattern flow.

01

Building a Reaction-Diffusion Equation

Adding more particles to start at the origin and working their way out gives diffusionTo avoid having no particles at the origin in the first step, diffusion assumes only a percentage of the particles get diffused in the stept

The Random Walker

• We will have a particle move 1 unit step for every time period
• Average distance from origin of any particle proportional to the square root of the number of steps it took

Now we introduce 2 Substances, A and B

• Both diffuse through space
• A diffuses twice as fast as B
• A gets replenished (feed) into the system at a constant feed rate, f
• B gets removed from the system at a kill rate, k
• A and B react in the following way:
• A + 2B -> 3B

The Model

Grey-Scott Reaction Diffusion and Coral

• some factors that impact the rate of this growth are light and flow
• light impacts photosynthesis
• flow impacts nutrient transfer
• autocatalytic growth of solid matrix as more interactions occur with the larger skeleton

Reaction-Diffusion Models for Coral Growth

• Coral create skeletal structures through a process called calcification

A = dissolved nutrientsB = dissolved calcium carbonate (solid material) u and v are respective biomasses

then the time rate of change of the Nutrient concentration is:(time rate of change of the Nutrient concentration) = (anomalous diffusion of u) + (constant supply rate of u) - (wasting rate of u) - (reactive loss of u) and the time rate of change of dissolved Calcium Carbonate Ions is: (time rate of change of dissolved Calcium Carbonate Ions) = (anomalous diffusion of v) - (loss of v due to deposition) + (reactive production of v)

This gives us the following system of equations:

Mathematical Results

• There are three space homogeneous steady-states S1, S2, and S3 where Turing instability only occurs for S2.
• Coral growth patterns can be expected when sufficient perturbations from S2 occur to make the system unstable
• dispersion relation and the growth rate depend on the fractional power of our fractional reaction-diffusion model
• the range of unstable wave modes corresponding to the maximum growth rate increases as d decreases from 𝑑𝑐.

Mathematical Results contd.

• At a fixed d, as sqrt(b1us2/b) increases from its minimum value to a maximum value in the Turing space, the growth rate decreases from its maximum value to 0.
• As d decreases, the growth rate, increases when sqrt(b1us2/b) and b2/b1 are fixed.
• Adjusting the nutrient supply rate can cause an adjustment in parameters so they are in the Turing instability region

Qualitative Conclusions

Coral growth can be guided by a few factors, driven by flow and nutrient availability

• coral grows perpendicular to the direction of nutrient flow (current)
• increased coral density increases porosity
• increased porosity increases the heterogeneity of coral patterns
• as the turbulence of the media increases, the heterogeneity of coral patterns decreases

References

Compeau, P. (2022). Biological Modeling: A Short Tour. Philomath Press .Mistr, S., & Bercovici, D. (2003). A Theoretical Model of Pattern Formation in Coral Reefs. Ecosystems, 6(1), 0061–0074. https://doi.org/10.1007/s10021-002-0199-0 Olovsson, N. (2023). Reaction-diffusion models and Turing patterns. Nils-Olovsson.se. https://nils-olovsson.se/articles/reaction_diffusion_models_and_turing_patterns/#figure_header Somathilake, L. W., Burrage, K., & Rogovchenko, Y. (2018). A space-fractional-reaction-diffusion model for pattern formation in coral reefs. Cogent Mathematics & Statistics, 5(1), 1426524. https://doi.org/10.1080/23311835.2018.1426524 A special thanks to Dr. Anthony Bellantuono for his helpful conversations on coral functions and his ideas on Turing patterns

When looking at biological patterns, the results of reaction-diffusion equations have been theorized to describe animal patterning (zebrafish-Turing), leaf structure, and animal morphology

General Form

flow through porus medium when gamma less than 2 and diffusion process is slower

fractional diffusion process