Mycelium Network (2D)

What is modelled

A fungal mycelium is a network of hyphae, thin tubes that grow only at their tips. This page grows one on a flat 40 x 40 cm dish of nutrient-poor soil with three buried wood patches. Each tip extends, sometimes branches, steers toward food (chemotropism) and away from crowded mycelium (self-avoidance), and may fuse with the existing network (anastomosis). Hyphae absorb nutrient from the soil into a shared reserve pool, and growth and upkeep are paid from that pool. It uses the same equations and constants as the 3D page, restricted to one soil layer.

Equations

• Tip extension (Monod kinetics): v = v0 [vmin + (1 − vmin) N/(K + N)] R/(R + KR), N the nutrient in the tip's soil cell, R the reserve pool, v_min = 0.6 the exploring speed in poor soil.
• Branching: probability per segment (ds = 0.35 cm) β (0.35 + 0.65 N/(K+N)) (1 − ρ/ρmax) R/(R+KR) ds. With free growth this is a Yule process: tips grow as eβ v t and the mean time to reach n tips is Hn−1/(β v).
• Uptake: dN/dt = − umax ℓ N/(K + N) in a cell holding hyphal length ℓ; nutrient diffuses between cells with D = 0.4 cm²/day (mass-conserving).
• Reserve: dR/dt = uptake − Y v · tips − m · L with growth cost Y and upkeep m per cm of hypha (illustrative values).
• Heading: Gaussian noise (persistence) plus a pull along the nutrient gradient χ |∇N|/(|∇N| + g0) and a push down the hyphal-density gradient.
• Network shape: the box-counting dimension D = − d ln N(s) / d ln s is measured live, with the hyphal growth unit L/tips (length per tip).

Checks run in the page

• A private copy of the model with every limit off grows from 1 to 100 tips 24 times; the mean time is compared with the Yule-process result H(99)/(β v).
• Nutrient bookkeeping of the live colony: reserve start + uptake − growth − upkeep must equal the pool, and soil nutrient + uptake must equal the initial nutrient.
• A single unbranched hypha in uniform nutrient N = K must advance at the Monod-predicted speed.

Honest simplifications

• All rate constants are typical or illustrative, not fitted to one species. Real mycelia move water and nutrient along hyphae in both directions, recycle old hyphae and respond to light, CO₂ and neighbours; none of that is modelled.
• Hyphae are drawn about 100 times thicker than real ones (5 to 10 micrometres). The model is limited to 16 000 segments and 2500 growing tips. The soil and wood pictures are Blender renders used as a backdrop only.

What to try

• Compare strong chemotropism with blind exploration: the first reaches the wood earlier and takes up more nutrient.
• Choose poor soil and watch the reserve pool collapse and growth stall.
• Raise branching: the hyphal growth unit falls and the network becomes denser and more fused.