  An understanding of the underlying mechanism of side--branching is paramount
in controlling and/or therapeutically treating mammalian organs, such as lungs,
kidneys, and glands. Motivated by an activator-inhibitor-substrate approach
that is conjectured to dominate the initiation of side--branching in pulmonary
vascular pattern, I demonstrate a distinct transverse front instability in
which new fingers grow out of an oscillatory breakup dynamics at the front
line, without any typical length scale. These two features are attributed to
unstable peak solutions in 1D that subcritically emanate from the Turing
bifurcation and that exhibit repulsive interactions. The results are based on a
bifurcation analysis and numerical simulations, and provide a potential
strategy toward developing a framework of side--branching also of other
biological systems, such as plant roots and cellular protrusions.
