A discrete data assimilation algorithm for the reconstruction of Gray--Scott dynamics

The Gray--Scott model governs the interaction of two chemical species via a system of reaction-diffusion equations. Despite its simple form, it produces extremely rich patterns such as spots, stripes, waves, and labyrinths. That makes it ideal for studying emergent behavior, self-organization, and instability-driven pattern formation. It is also known for its sensitivity to poorly observed initial conditions. Using such initial conditions alone quickly leads simulations to deviate from the true dynamics. The present paper addresses this challenge with a nudging-based data assimilation algorithm: coarse, cell-averaged measurements are injected into the model through a feedback (nudging) term, implemented as a finite-volume interpolant. We prove two main results. (i) For the continuous problem, the nudged solution synchronizes with the true dynamics, and the $L^2$-error decays exponentially under conditions that tie observation resolution, nudging gains, and diffusion. (ii) For the fully discrete semi-implicit finite-volume scheme, the same synchronization holds, up to a mild time-step restriction. Numerical tests on labyrinthine patterns support the theory. They show recovery of fine structure from sparse data and clarify how the observation resolution, the nudging gain, and the frequency of updates affect the decay rate.