A p-adic reaction-diffusion model of branching coral growth and calcification dynamics

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 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.