The 3D version treats the light-organ crypt as a single well-mixed compartment: one population number N and one autoinducer number A, shared instantly by every bacterium. This 2D version instead solves both quantities as spatial fields on a grid shaped like the crypt's actual cross-section (a capsule: a cylindrical body with two rounded ends), so the signal has to physically spread:
∂N/∂t = r·N·(1 − N) (local logistic growth, no migration)
∂A/∂t = D·∇²A + k_a·N − δ·A (Fick diffusion + local production − decay)
P_on(A) = A⁴ / (A⁴ + K_eff⁴) (per-pixel Hill switch, K_eff has spatial noise)
Because autoinducer now has to diffuse (∇²A term) instead of mixing instantly, colonization that starts unevenly produces a visible diffusion wave: the region that fills in first crosses the quorum threshold first and the "lit" patch spreads outward as signal reaches neighboring cells — a genuinely 2D phenomenon with no counterpart in the mean-field 3D model.
- Growth rate r, decay rate δ, threshold K — same meaning and same defaults as the 3D model, so the two are directly comparable.
- Signal diffusion D — new to this view: how fast autoinducer spreads through the crypt fluid. Low D lets separate colonized patches light up independently before merging; high D synchronizes the whole field almost as if it were well-mixed.
- Dawn Venting — expels ~95% of the population and dilutes the field by 75% everywhere. Verified numerically: with the default rates the glow only dims (it does not fully go dark) before bacterial regrowth outruns autoinducer decay — the same tension that governs the real diel cycle, where venting must be large enough, or growth slow enough, to actually reset the switch before the population recovers.
Same founding quorum-sensing system as the 3D pair (Nealson & Hastings) — the LuxI/LuxR loop that first put "quorum sensing" on the map, here shown as the traveling-wave phenomenon that real diffusion-limited signaling produces.