Mathematical model for collective migration on a viscoelastic collagen network

Abstract

In this paper, we study a model of self-generated directional cell migration on viscoelastic substrates in the absence of apparent intrinsic polarity. This model, first proposed in \cite{Clark}, was observed numerically to manifest traveling pulse solutions for sufficiently large collagen stiffness, leading to a persistent collective migration. Here we provide a rigorous mathematical framework for the model, finding the exact stationary states and conditional traveling pulse. We also prove global well-posed in $W^{k,\infty }$ spaces, local stability of the traveling pulse for high stiffness, and exponential convergence to the stationary state for low stiffness.