Delayed Pattern Formation in Two-Dimensional Domains

This study investigates how the interaction between gene expression time delay and domain size governs spatio-temporal pattern formation in a reaction-diffusion system. To investigate these phenomena, we utilize a modified version of the Schnakenberg model called the LI (ligand internalisation) model. In a one-dimensional domain, a linear relationship has been observed between the gene expression time delay and the time it takes for patterns to form. We extend the model to the two-dimensional domain and confirm that a similar relationship holds there as well. However, our exploration reveals a non-monotonic correlation between domain size and the time required for pattern emergence. In our attempt to unravel these complex dynamics, we expand our analysis by considering a diverse range of initial conditions, including random perturbations of the spatially homogenous steady state as well as initial functions from its unstable manifold, to observe how pattern formation unfolds. To understand the role of these key parameters, we compute a two-parameter chart of patterns, with respect to time delay and domain size.