Effect of time delay on pattern formation: Competition between homogenisation and patterning

We study a mathematical model for juxtacrine signalling in a discrete lattice of cells. As the signalling is assumed to be under transcriptional control, and transcription is a time-consuming process, we incorporate time delays in the equations and study the effect of this on the pattern forming potential of the model. Previous models without time delays have shown that the mechanism is able to generate spatial patterns. The analysis of the delay-model reveals a transient competition between patterning and homogeneous oscillations. A fine-grained pattern eventually appears over the whole lattice, but the duration of the oscillatory behaviour increases as the time delay increases. The results illustrate the importance of including the known delays in a model and of studying transients, as these may not be favourable to the system. In addition, the results may suggest that there are other mechanisms regulating the signalling than transcription, for example protein-protein interactions, which would render the patterning process much faster.