  Highly concentrated patterns have been observed in a spatially heterogeneous,
nonlocal, model of BGK type implementing a velocity-jump process.
  We study both a linear and a nonlinear case and describe the concentration
profile. In particular, we analyse a hyperbolic (or high frequency) regime that
can be interpreted both as a local (microscopic) or as a nonlocal (macroscopic)
rescaling. We consider a Hopf-Cole transform and derive a Hamilton-Jacobi
equation. The concentrations are then explained as a consequence of the
stationary points of the Hamiltonian that is spatially heterogeneous like the
velocity-jump process. After revising the classical hydrodynamic limits for the
aggregate quantities and the eikonal equation that can be derived from those
with a Hopf-Cole transform, we find that the Hamilton-Jacobi equation is a
second order approximation of the eikonal equation in the limit of small
diffusivity. For nonlinear turning kernels, the Hopf-Cole transform allows to
study the stability of the possible homogeneous configurations and of patterns
and the results of a linear stability analysis previously obtained are found
and extended to a nonlinear regime. In particular, it is shown that instability
(pattern formation) occurs when the Hamiltonian is convex-concave.
