An activator-inhibitor-substrate model of side-branching used in the context
of pulmonary vascular and lung development is considered on the supposition
that spatially localized concentrations of the activator trigger local
side-branching. The model consists of four coupled reaction-diffusion equations
and its steady localized solutions therefore obey an eight-dimensional spatial
dynamical system in one dimension (1D). Stationary localized structures within
the model are found to be associated with a subcritical Turing instability and
organized within a distinct type of foliated snaking bifurcation structure.
This behavior is in turn associated with the presence of an exchange point in
parameter space at which the complex leading spatial eigenvalues of the uniform
concentration state are overtaken by a pair of real eigenvalues; this point
plays the role of a Belyakov-Devaney point in this system. The primary foliated
snaking structure consists of periodic spike or peak trains with $N$ identical
equidistant peaks, $N=1,2,\dots \,$, together with cross-links consisting of
nonidentical, nonequidistant peaks. The structure is complicated by a multitude
of multipulse states, some of which are also computed, and spans the parameter...Continue Reading
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