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 of the reactor as a bifurcation parameter. We define a parameter space Sigma of (, )for which the Turing instability may happen. Then, we show that it really occurs below the critical number0and obtain rigorous formula for the bifurcated stable patterns. When the critical eigen value is simple, the bifurcation leads to a continuous (resp. jump) transition for <(0)if(, )is negative (resp. positive).We prove that (, )>0when(, )lies near the Bogdanov-Takens point(1/16,1/16). When the critical eigenvalue is double, we have a supercritical bifurcation that produces an(1)-attractor Omega.We prove that Omega consists of four asymptotically stablestatic solutions, four saddle static solutions, and orbits connecting them. We also provide numerical results that illustrate the main theorems.