Since early 90's, much attention has been paid to dynamic dissipative patte
rns in laboratories, especially, self-replicating pattern (SRP) is one of t
he most exotic phenomena. Employing model system such as the Gray-Scott mod
el, it is confirmed also by numerics that SRP can be obtained via destabili
zation of standing or traveling spots. SRP is a typical example of transien
t dynamics, and hence it is not a priori clear that what kind of mathematic
al framework is appropriate to describe the dynamics. A framework in this d
irection is proposed by Nishiurar-Ueyama [16], i.e., hierarchy structure of
saddle-node points, which gives a basis for rigorous analysis. One of the
interesting observation is that when there occurs self-replication, then on
ly spots (or pulses) located at the boundary (or edge) are able to split. I
nternal ones do not duplicate at all. For ID-case, this means that the numb
er of newly born pulses increases like 2k after k-th splitting, not 2(n)-sp
litting where all pulses split simultaneously. The main objective in this a
rticle is two-fold: One is to construct a local invariant manifold near the
onset of self-replication, and derive the nonlinear ODE on it. The other i
s to study the manner of splitting by analysing the resulting ODE, and answ
er the question "2(n)-splitting or edge-splitting?" starting from a single
pulse. It turns out that only the edge-splitting occurs, which seems a natu
ral consequence from a physical point of view, because the pulses at edge a
re easier to access fresh chemical resources than internal ones.