Continuum and lattice models analysis of the aggregation diffusion cell movement

Abstract

The process by which one may take lattice models of a biophysical process and construct continuous models based on it is of mathematical interest as well as being of practical use. Such a process makes it possible to investigate how processes at the single-cell level affect collective dynamics. In this paper, we first study the singular limit of a class of reinforced random walks on a lattice which simulates cell migration on discrete points and a complete analysis of the existence and stability of solutions is possible. Combined with “the diffusion approximation” assumption, a continuous aggregation diffusion or backward forward parabolic equation is derived. Then the regularity estimate, the nonexistence of the solution with small initial data, and asymptotic behavior of the solution with large initial data are investigated. In contrast to the continuous model, the boundedness, asymptotic behaviors of the lattice model solution in the diffusion region with monotone initial date, and the interface behaviors of the aggregation, diffusion regions are explored. The numerical simulations agree well with theoretical results and biological observations.