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The process by which one may take a discrete model of a biophysical processand construct a continuous model based on it is of mathematical interest aswell as being of practical use. In this paper, we first study the singularlimit of a class of reinforced random walks on a lattice for which a completeanalysis of the existence and stability of solutions are possible. In thecontinuous scenario, we obtain the regularity estimate of this aggregationdiffusion model. As a by-product, nonexistence of solution of the continuousmodel with pure aggregation initial data is proved. When the initial is purelyin diffusion region, asymptotic behavior of the solution is obtained. Incontrast to continuous model, bounded-ness of the lattice solution, asymptoticbehavior of solution in diffusion region with monotone initial date and theinterface behaviors of the aggregation, diffusion regions are obtained. Finallywe discuss the asymptotic behaviors of the solution under more general initialdata with non-flux when the lattice points $N\leq 4$.

Lattice and Continuum Models Analysis of the Aggregation Diffusion Cell Movement