A simple two-component reaction-diffusion system showing rich dynamic behavior: spatially homogeneous aspects and selected bifurcation scenarios

We present a simple reaction-diffusion model for two variables. The model was originally designed to have a stable localized solution for a range of parameters as a consequence of the coexistence of a stable limit cycle and a stable fixed point. We classify the spatially homogeneous solutions of the model. In addition we describe several bifurcation scenarios for particle-like solutions as a function of two of the parameters.