Spatial Predator-Prey — Turing Instability
Adding spatial diffusion to Lotka-Volterra dynamics can destabilize uniform equilibria, producing spatial patterns. Slow predator diffusion relative to prey creates patches — diffusion-driven (Turing) instability.
Predator density (low→high)
Spatial Lotka-Volterra: ∂u/∂t = ru(1−u) − auv + D_u∇²u, ∂v/∂t = eauv − mv + D_v∇²v. Turing condition: the homogeneous steady state is stable without diffusion, but when D_v ≪ D_u, diffusion destabilizes it — fast prey "run away" from growing predator patches, amplifying spatial heterogeneity.