Existence and Asymptotic Properties of Solutions of a Nonlocal Evolution Equation Modeling Cell-Cell Adhesion

Abstract

In this paper we consider some fundamental properties of a new type of nonlocal reaction-diffusion equation originally proposed a few years ago in [N. J. Armstrong, K. J. Painter, and J. A. Sherratt, J. Theoret. Biol., 243 (2006), pp. 98–113] as a possible continuum mathematical model for cell-cell adhesion. The basic model is on an infinite domain and contains a nonlocal flux term which models the component of cell motion attributable to the cell having formed bonds with nearby cells within its sensing radius, and the nonlocal term is both nonlinear and involves spatial derivatives, making the analysis challenging. We establish the local existence of a classical solution working in spaces of uniformly continuous functions. We then establish that the model has a positivity preserving property and we find bounds on the solution, and we then establish the existence of a unique global solution in each of the biologically realistic cases when the cell density $n(x,t)$ satisfies $n(x,0)\rightarrow0$ and $n(x,0)\rightarrow N_1$ as $|x|\rightarrow\infty$, where $N_1$ is the spatially uniform steady state. Finally, we establish that under certain conditions solutions approach the spatially uniform state $N_1$. This nonlinear result complements the findings of other investigators who showed that the simple scalar model can develop spatial patterns in other parameter regimes. The model is illustrated by simulations that can be applied to in vitro wound closure experiments.