Phase differences in reaction-diffusion-advection systems and applications to morphogenesis

The authors study the effect of advection on reaction-diffusionpatterns. It is shown that the addition of advection to a two-variablereaction–diffusion system with periodic boundary conditionsresults in the appearance of a phase difference between thepatterns of the two variables which depends on the differencebetween the advection coefficients. The spatial patterns movelike a travelling wave with a fixed velocity which depends onthe sum of the advection coefficients. By a suitable choiceof advection coefficients, the solution can be made stationaryin time. In the presence of advection a continuous change inthe diffusion coefficients can result in two Turing-type bifurcationsas the diffusion ratio is varied, and such a bifurcation canoccur even when the inhibitor species does not diffuse. It isalso shown that the initial mode of bifurcation for a givendomain size depends on both the advection and diffusion coefficients.These phenomena are demonstrated in the numerical solution ofa particular reaction–diffusion system, and finally apossible application of the results to pattern formation inDrosophila larvae is discussed.