  The process by which one may take a discrete model of a biophysical process
and construct a continuous model based on it is of mathematical interest as
well as being of practical use. In this paper, we first study the singular
limit of a class of reinforced random walks on a lattice for which a complete
analysis of the existence and stability of solutions are possible. In the
continuous scenario, we obtain the regularity estimate of this aggregation
diffusion model. As a by-product, nonexistence of solution of the continuous
model with pure aggregation initial data is proved. When the initial is purely
in diffusion region, asymptotic behavior of the solution is obtained. In
contrast to continuous model, bounded-ness of the lattice solution, asymptotic
behavior of solution in diffusion region with monotone initial date and the
interface behaviors of the aggregation, diffusion regions are obtained. Finally
we discuss the asymptotic behaviors of the solution under more general initial
data with non-flux when the lattice points $N\leq 4$.
