Following up on our work on Turing pattern formation in a one-dimensional
reaction-transport model, we explore the emergence of patterns in a two dimensional model described by a system of PDEs for passively transported diffusing
particles (PT) and actively transported motor-driven particles (AT). We first
propose a model where the actively transported particles are taken to be functions of spatial locations x ∈ R and velocity v. We then consider two special
simplifying cases where particles are transported at (i) constant speeds v so that
the concentration of AT particles is taken to be a function of spatial location
and velocity angle θ ∈ S
1 and (ii) discrete velocities (still with constant speed)
in directions along a lattice tiling the plane. In the former case the system
is equivariant with respect to the so called shift-twist action of the Euclidean
group E(2) acting on functions on R
2 × S
1
, while in the latter case it is equivariant with respect to the group DN n R2 where DN (N = 4 for square and
N = 6 for hexagonal) is the holohedry group of the lattice. In both cases, we use
symmetric bifurcation theory to analyze the planforms emerging from a Turing
bifurcation, should it occur. In the discrete velocity square lattice case, we are
able to prove that a Turing bifurcation does indeed occur as the dimensionless
parameter γ = αD/v2
crosses some critical value. Here, D is the diffusion coefficient for the passively diffusing particles and α is the switching rate between
motor states.