Dynamics of self-replicating patterns in the one-dimensional Gray-Scott model

We study the self-replicating pattern (SRP) that is observed in the onedimensional Gray-Scott model from aglobal bifurcational view point. It is shown that the existence of the hierarchy structure of the limiting points of stationary Turing patterns causes SRP of static type as an aftereffect. The main difficulty lies in the fact that SRP is areal transient phenomenon and it can not be captured as an invariant set in afunction space. The aftereffect is the reflection of the fact that each element of hierarchy structure is connected by unstable manifolds.