Lattice and Continuum Models Analysis of the Aggregation Diffusion Cell Movement

TL;DR

This paper analyzes the relationship between lattice-based and continuum aggregation-diffusion models, providing existence, stability, regularity, and asymptotic behavior results for various initial conditions and model types.

Contribution

It offers a comprehensive analysis of the singular limit from reinforced random walks to continuous models, including stability, regularity, and asymptotic behavior insights.

Findings

Complete analysis of existence and stability for lattice models.

Regularity estimates and nonexistence results for continuous models.

Asymptotic behavior characterized for different initial conditions.

Abstract

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 $N\leq 4$.