Domain size driven instability: self-organisation in systems with advection

Models for self-organization have been used in complex systems across numerous disciplines, with a prime example given by the Turing instability. However, this instability is subject to several constraints and is restricted to relatively small regions of parameter space. This leads to parameter sensitivity and the Turing instability also exhibits sensitivity to initial conditions, domain geometries, the presence of immobile species, and, in biological contexts, receptor and gene expression dynamics. With many possible motivations, such as thermodynamic considerations that allow the coupling of transport with chemical and biochemical reactions, we include advection within the system description, which also highlights the need to consider many possible boundary conditions. Consequently, we use the Sturm--Liouville theory to analyze the conditions for pattern formation with the objective of assessing whether advection or different boundary conditions can induce self-organization, with the induction of patterning as the domain size exceeds a threshold but without the level of constraint of the Turing mechanism. Our results indicate that Dirichlet boundary conditions or advection with a variety of boundary conditions can lead to these patterning properties, which are characterized by the absence of the need for short-range activation and long-range inhibition. In the presence of advection, this instability mechanism also exhibits patterning that is distinct from the Turing instability, possessing a spatial modulation without additional model complexity.