What is chemotactic collapse in this model?

When χ is large enough relative to Dₙ and total mass, the continuum Keller–Segel equations can blow up in finite time: most cells pile into an ever-sharper peak. On a fixed grid you see max(n) surge and a bright hotspot—an discrete analogue of that singularity formation.

Why periodic boundaries?

They simplify the Laplacian and chemotactic divergence at edges so the demo focuses on bulk aggregation without boundary layers. Real petri dishes use no-flux walls; patterns are qualitatively similar for teaching purposes.

What does increasing α or decreasing β do?

Both raise steady attractant levels for a given n, steepening ∇c and strengthening effective chemotactic drift. Very strong secretion with slow decay can accelerate clumping; high β dissipates signal faster and spreads guidance more evenly.

Is the simulation stable for all slider values?

Explicit Euler can destabilize if Δt or χ is too large; defaults and presets are chosen to show collapse without immediate numerical blow-up from the scheme itself. If the field becomes noisy, lower χ or Δt and reseed.

Keller–Segel Chemotaxis

The Keller–Segel system is the standard minimal model of chemotaxis: cells with density n move up gradients of a self-secreted attractant c. In this simulator, ∂ₜn = Dₙ∇²n − χ∇·(n∇c) combines random spreading with directed drift along ∇c, expanded explicitly as −χ(∇n·∇c + n∇²c) on a periodic 96×96 grid. The attractant obeys ∂ₜc = D_c∇²c + αn − βc: diffusing, produced proportionally to local cell density, and linearly degraded. Bacteria seeded in a central bump secrete c, climb its gradient, and aggregate. When the chemotactic sensitivity χ exceeds a mass-dependent critical level, the continuum model predicts finite-time blow-up of density—chemotactic collapse—visible here as a sharpening peak in n (log-scaled colors). Lower χ yields milder clumps without a sharp spike. The scheme is explicit Euler with periodic boundaries; it is pedagogical, not calibrated to laboratory strains. Presets include high-χ collapse, moderate aggregation, and mild chemotaxis. Optional blue tint overlays attractant c.

Who it's for: Students of mathematical biology, PDEs, or pattern formation exploring how directed motion plus diffusion produces aggregation and collapse.

Key terms

Keller–Segel model
Chemotaxis
Attractant gradient
Chemotactic collapse
Reaction–diffusion
Cell density
Critical chemotactic parameter
Finite-difference grid

How it works

Minimal Keller–Segel chemotaxis on a 2D grid: cell density n follows attractant c with diffusion and sensitivity χ. Secretion αn and decay βc close the loop. High χ drives aggregation and finite-time collapse (blow-up in the continuum limit); lower χ yields milder clumping.

Key equations

∂ₜn = Dₙ∇²n − χ∇·(n∇c) = Dₙ∇²n − χ(∇n·∇c + n∇²c) · ∂ₜc = D_c∇²c + αn − βc

Frequently asked questions

What is chemotactic collapse in this model?: When χ is large enough relative to Dₙ and total mass, the continuum Keller–Segel equations can blow up in finite time: most cells pile into an ever-sharper peak. On a fixed grid you see max(n) surge and a bright hotspot—an discrete analogue of that singularity formation.
Why periodic boundaries?: They simplify the Laplacian and chemotactic divergence at edges so the demo focuses on bulk aggregation without boundary layers. Real petri dishes use no-flux walls; patterns are qualitatively similar for teaching purposes.
What does increasing α or decreasing β do?: Both raise steady attractant levels for a given n, steepening ∇c and strengthening effective chemotactic drift. Very strong secretion with slow decay can accelerate clumping; high β dissipates signal faster and spreads guidance more evenly.
Is the simulation stable for all slider values?: Explicit Euler can destabilize if Δt or χ is too large; defaults and presets are chosen to show collapse without immediate numerical blow-up from the scheme itself. If the field becomes noisy, lower χ or Δt and reseed.

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