Turing Instability and Dynamic Bifurcation for the One-Dimensional Gray–Scott Model

We study the dynamic bifurcation of the one-dimensional Gray–Scott model by taking the diffusion coefficient (Formula presented.) of the reactor as a bifurcation parameter. We define a parameter space (Formula presented.) of (Formula presented.) for which the Turing instability may happen. Then, we show that it really occurs below the critical number (Formula presented.) and obtain rigorous formula for the bifurcated stable patterns. When the critical eigenvalue is simple, the bifurcation leads to a continuous (resp. jump) transition for (Formula presented.) if (Formula presented.) is negative (resp. positive). We prove that (Formula presented.) when (Formula presented.) lies near the Bogdanov–Takens point (Formula presented.). When the critical eigenvalue is double, we have a supercritical bifurcation that produces an (Formula presented.) -attractor (Formula presented.). We prove that (Formula presented.) consists of four asymptotically stable static solutions, four saddle static solutions, and orbits connecting them. We also provide numerical results that illustrate the main theorems.