  In this paper, we study pattern formations in an aggregation and diffusion
cell migration model with Dirichlet boundary condition. The formal continuum
limit of the model is a nonlinear parabolic equation with a diffusivity which
can become negative if the cell density is small and spatial oscillations and
aggregation occur in the numerical simulations. In the classical diffusion
migration model with positive diffusivity and non-birth term, species will
vanish eventually with Dirichlet boundary. However, because of the aggregation
mechanism under small cell density, the total species density is conservative
in the discrete aggregation diffusion model. Also, the discrete system
converges to a unique positive steady-state with the initial density lying in
the diffusion domain. Furthermore, the aggregation mechanism in the model
induces rich asymptotic dynamical behaviors or patterns even with 5 discrete
space points which gives a theoretical explanation that the interaction between
aggregation and diffusion induces patterns in biology. In the corresponding
continuous backward forward parabolic equation, the existence of the solution,
maximum principle, the asymptotic behavior of the solution is also
investigated.
