This is the same Levin–Stewart–Rice mass-action model as the 3D liquid-culture version, but reframed as a 2D reaction-diffusion field — the way conjugation actually spreads across a bacterial lawn on an agar plate rather than through a stirred broth. Instead of tracking individual drifting cells, the plate is divided into a lattice of patches, each holding a local density of donors D, recipients R and transconjugants T (as a fraction of local carrying capacity):
∂D/∂t = ψ_D·D·(1−N/K) − d·D + D_c·∇²D
∂R/∂t = ψ_R·R·(1−N/K) − γ·R·(D+T) − d·R + D_c·∇²R
∂T/∂t = ψ_T·T·(1−N/K) + γ·R·(D+T) − τ·T − d·T + D_c·∇²T
ψ_X = ψ_max for R (no plasmid)
ψ_X = ψ_max·(1 − s) for D, T (plasmid carriage cost)
N = D + R + T, ∇² = discrete 2D Laplacian (patch diffusion)
Every patch integrates the exact same local reaction terms as the well-mixed ODE — you can verify this by setting the diffusion coefficient's effect to zero (a single isolated patch reproduces the liquid-culture trajectory exactly). What's genuinely 2D is the spatial term: donor and transconjugant density physically diffuses outward from the seeded source patch into the surrounding recipient-only lawn, so instead of a single well-mixed prevalence number you watch a traveling invasion wavefront — a phenomenon with no counterpart in the stirred 3D model.
- Conjugation rate γ — local mass-action transfer weight; higher γ steepens and speeds the wavefront.
- Fitness cost s — growth penalty for plasmid carriers; above a threshold set by γ and τ the front stalls and recedes.
- Segregational loss τ — rate a transconjugant patch reverts toward recipient, eroding the trailing edge of the wave.
- Source-patch donor fraction — donor share seeded at the lawn's center on Reset; the rest of the lawn starts as pure recipients near carrying capacity.
Real-world relevance: surface/lawn conjugation assays (donor spot on a recipient lawn) are a standard experimental counterpart to liquid mass-action assays, and produce exactly this kind of radial invasion front.