  We study the formation of auxin peaks in a generic class of
concentration-based auxin transport models, posed on static plant tissues.
Using standard asymptotic analysis we prove that, on bounded domains, auxin
peaks are not formed via a Turing instability in the active transport
parameter, but via simple corrections to the homogeneous steady state. When the
active transport is small, the geometry of the tissue encodes the peaks'
amplitude and location: peaks arise where cells have fewer neighbours, that is,
at the boundary of the domain. We test our theory and perform numerical
bifurcation analysis on two models which are known to generate auxin patterns
for biologically plausible parameter values. In the same parameter regimes, we
find that realistic tissues are capable of generating a multitude of stationary
patterns, with a variable number of auxin peaks, that can be selected by
different initial conditions or by quasi-static changes in the active transport
parameter. The competition between active transport and production rate
determines whether peaks remain localised or cover the entire domain. We relate
the occurrence of localised patterns to a snaking bifurcation structure, which
is known to arise in a wide variety of nonlinear media but has not yet been
reported in plant models.
