Top, typical density profiles in the wave. Bottom, BARE wave model defined by two coupled partial differential equations for n (red, growing-tip density) and (blue, hyphal-filament density), describing the dynamics of branching at rate b (orange), anastomoses at rate a (green) and tip movement by spatial flux J (pink), and yields wave speed c . Middle, wave propagation driven by fast tip growth of puller hyphae at the front (red dots, cyan tails) and densification through slower growth of tips behind the front (red dotsand pink tails). Tips duplicate by branching (orange dots) close to growing tips and annihilate by anastomosis (green) with existing hyphae.
