This is the population-level (mean-field) limit of the same cyprid search-and-settle process, simulated as two coupled density fields on a grid instead of individual walking larvae: L(x,y,t), the density of still-searching cyprids, and S(x,y,t), the density already cemented down. Both evolve together:
∂L/∂t = ∇·( D(C) ∇L ) − R(C)·L
∂S/∂t = R(C)·L
Diffusion from klinokinesis. A larva doing a correlated random walk with speed v and turn rate ω has an effective spatial diffusion coefficient D ≈ v²/(2ω) (standard persistent-random-walk result). Plugging in the same cue-dependent kinematics as the walking model —
v(C) = v₀·vigor·(1 − 0.6·min(C,1))
ω(C) = ω₀·vigor·(1 + 1.5·min(C,1))
D(C) = v(C)² / (2·ω(C))
— gives a diffusivity that drops sharply in cue-rich patches, so the L field naturally piles up there: the same area-restricted-search effect the agent model produces, here as slowed spreading of a density cloud rather than a single larva's twistier path.
Dose-response settlement. Cue C combines a patchy biofilm field (static, reshaped by density/patch scale) with a gregarious field obtained by convolving the settled density S with the same Gaussian footprint every settled individual casts. The conversion rate follows the identical Hill function:
R(C) = k · Cⁿ / (Cⁿ + K²), k = 1.2 s⁻¹, K = 0.35, n = 2
Because ∂S/∂t exactly mirrors the sink term removed from L, total mass ∫(L+S) dA is conserved by construction (diffusion redistributes L but never creates or destroys it) — the same closed cohort as the agent model, just advected as a field.
- Biofilm cue density / Patch scale — reshape the static bacterial-film field under both layers.
- Gregariousness — strengthens the settled-density convolution feeding back into C.
- Exploration vigor — scales v₀ and ω₀ together, which scales D(C) linearly.