A $p$-adic Reaction--Diffusion Model of Branching Coral Growth and Calcification Dynamics

TL;DR

The paper introduces a novel p-adic reaction–diffusion framework on $\mathbb{Z}_p$ to model internal coral calcification and branching, exploiting the ultrametric, hierarchical geometry of p-adic spaces. It combines simplified calcium carbonate chemistry with nonlocal diffusion via the restricted Vladimirov operator $\overline{D}^{\alpha}$, yielding a high-dimensional system on discretized p-adic balls that represents coral branches. After nondimensionalization, the authors analyze local stability of the reaction subsystem and implement an event-driven, level-by-level simulation to generate coral-like ramified morphologies, with branching triggered by CaCO3 accumulation and halting governed by saturation thresholds. The framework demonstrates how hierarchical diffusion and nonlinear kinetics collaborate to shape branching architectures and highlights parameter sensitivities to CO2-related inputs, while outlining limitations and future extensions toward irregular ultrametric trees and data-driven calibration. This work offers a mathematically rigorous bridge between non-Archimedean analysis and developmental morphogenesis, with potential applicability to other hierarchical biological networks.

Abstract

Coral colonies exhibit complex, self-similar branching architectures shaped by biochemical interactions and environmental constraints. To model their growth and calcification dynamics, we propose a novel p-adic reaction-diffusion framework defined over p-adic ultrametric spaces. The model incorporates biologically grounded reactions involving calcium and bicarbonate ions, whose interplay drives the precipitation of calcium carbonate (CaCO3). Nonlocal diffusion is governed by the Vladimirov operator over the p-adic integers, naturally capturing the hierarchical geometry of branching coral structures. Discretization over p-adic balls yields a high-dimensional nonlinear ODE system, which we solve numerically to examine how environmental and kinetic parameters, particularly CO2 concentration, influence morphogenetic outcomes. The resulting simulations reproduce structurally diverse and biologically plausible branching patterns. This approach bridges non-Archimedean analysis with morphogenesis modeling and provides a mathematically rigorous framework for investigating hierarchical structure formation in developmental biology.

Figures (11)

• Figure 1: First five levels of the binary tree representing $\mathbb{Z}_2$ via the expansion \ref{['expan']}. Each node corresponds to a coefficient $a_i \in \{0,1\}$.
• Figure 2: Hierarchical trees corresponding to $\mathbb{Z}_3$ and $\mathbb{Z}_5$. Each level encodes a digit $a_i \in \{0,\dots,p-1\}$.
• Figure 3: Fractal-like nesting of $p$-adic balls: each ball contains $p$ disjoint sub-balls at the next level.
• Figure 4: Solution of the system \ref{['Sdiscret']} without diffusion for $\beta = -0.2$, $\sigma = 1$, and $\eta = 1$.
• Figure 5: The plots show the temporal evolution of the system for different values of $\sigma$, representing the initial CO2 concentration. A higher $\sigma$ increases the initial availability of CO2, enhancing early HCO3 production and thus accelerating the precipitation of CaCO3, while the total yield remains fixed by the initial calcium and carbonate stocks.

Theorems & Definitions (6)

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