Studies on the Existence of Unstable Oscillatory Patterns Bifurcating from Hopf Bifurcations in a Turing Model

We revisit a homogeneous reaction-diffusion Turing model subject to the Neumann boundary conditions in the one-dimensional spatial domain. With the help of the Hopf bifurcation theory applicable to the reaction-diffusion equations, we are capable of proving the existence of Hopf bifurcations, which suggests the existence of spatially homogeneous and nonhomogeneous periodic solutions of this particular system. In particular, we also prove that the spatial homogeneous periodic solutions bifurcating from the smallest Hopf bifurcation point of the system are always unstable. This together with the instability results of the spatially nonhomogeneous periodic solutions by Yi et al., 2009, indicates that, in this model, all the oscillatory patterns from Hopf bifurcations are unstable.