The influence of non-standard boundary conditions on the generation of spatial patterns
The influence of certain unilateral boundary or interior conditions to spatial Turing's patterns described by reaction-diffusion systems will be discussed. The conditions considered can model sources reflecting concentration in their neighbourhood in the following way. If the concentration exceeds a given threshold then the source is inactive, when the concentration is about to decrease below the threshold then the source either prevents it or at least decelerates the decrease by producing a morphogen (or ligand) and supplementing it into an extracellular space. Some interesting consequences follow. For instance, spatial patterns can arise in general for an arbitrary ratio of diffusion speeds, e.g. for fast diffusion of activator and slow diffusion of inhibitor (the opposite situation than in Turing's original idea), and can arise also for arbitrarily small domains. Simple numerical simulations using a model proposed for a description of pigmentation in animals (in particular felids) promise to describe patterns with spots and, moreover, with a darker strip along the spine, which are observed among some felids. The unilateral conditions can be described mathematically by variational inequalities or inclusions. A more detailed explanation of the theory should be a subject of the talk of Martin Vaeth.