On biological pattern formation by contact inhibition

Many patterning events in multi-cellular organisms rely on cell-to-cell contact signaling. We study a model which employs a graph to describe which cells are in contact, and examine its spatio-temporal dynamics. We first give an instability condition for the homogeneous steady-state. We then show that, for bipartite graphs, this instability condition also guarantees the existence and asymptotic stability of steady-states that exhibit a pattern of alternating high and low values in adjacent cells. Finally, we establish a strong monotonicity property of this model for bipartite graphs, which implies that almost every bounded solution converges to a steady-state.