  In this paper, we analyse a model for the growth of three-dimensional walled
cells. In this model the biomechanical expansion of the cell is coupled with
the geometry of its wall. We consider that the density of building material
depends on the curvature of the cell wall, thus yield-ing possible anisotropic
growth. The dynamics of the axisymmetric cell wall is described by a system of
nonlinear PDE including a nonlin-ear convection-diffusion equation coupled with
a Poisson equation. We develop the linear stability analysis of the spherical
symmetric config-uration in expansion. We identify three critical parameters
that play a role in the possible instability of the radially symmetric shape,
namely the degree of nonlinearity of the coupling, the effective diffusion of
the building material, and the Poisson's ratio of the cell wall. We also
investigate numerically pattern selection in the nonlinear regime. All the
results are also obtained for a simpler, but similar, two-dimensional model.
