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

Abstract

The authors study the effect of advection on reaction-diffusion patterns. It is shown that the addition of advection to a two-variable reaction-diffusion system with periodic boundary conditions results in the appearance of a phase difference between the patterns of the two variables which depends on the difference between the advection coefficients. The spatial patterns move like a travelling wave with a fixed velocity which depends on the sum of the advection coefficients. By a suitable choice of advection coefficients, the solution can be made stationary in time. In the presence of advection a continuous change in the diffusion coefficients can result in two Turing-type bifurcations as the diffusion ratio is varied, and such a bifurcation can occur even when the inhibitor species does not diffuse. It is also shown that the initial mode of bifurcation for a given domain size depends on both the advection and diffusion coefficients. These phenomena are demonstrated in the numerical solution of a particular reaction-diffusion system, and finally a possible application of the results to pattern formation in Drosophila larvae is discussed.

Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments II: patterns with the largest possible propagation speeds

“…For a number of specific chemical systems, conditions for patterning have been derived, and in some cases tested against experiment (Merkin et al 2000;Nekhamkina & Sheintuch 2003;Taylor 2003). There has also been some work on pattern formation in advection-reactiondiffusion equations applied to developmental biology (Perumpanani et al 1995;Bernasconi & Boissonade 1997;Satnoianu & Menzinger 2002); in that context, a key issue is the phase difference between two interacting morphogens, which depends on the advection coefficients. In oscillatory reaction-diffusion equations, periodic travelling waves are important both in their own right, and because they play a key role in the transition to spatio-temporal chaos (Petrovskii & Malchow 2001;Sherratt et al 2009).…”

Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes

“…Hence, there exists an upper bound on the wavenumber of destabilizing perturbations, independent of c, 42 determined by the activator only. For future reference, we note that as a consequence of (15), it holds that…”

“…The existence of 1D patterns is equivalent to the existence of striped patterns in 2D. Below, g is a scaled version of the second spatial dimension y the same way as n relates to x, see (42).…”

“…The initial steps are as in Ref. 42 2, a listing of ecohydrological models with and without differential flow is given. The general results can be applied to various other disciplines, in particular, to differential flow models where the reactants have different advection coefficients.…”

A Mathematical Biologist’s Guide to Absolute and Convective Instability

“…In contrast to the Turing mechanism, they do not require activator-inhibitor type of interactions, but they critically depend on other conditions, e.g. a significant difference in the flow experienced by species (Rovinsky and Menzinger 1992;Perumpanani et al 1995;Malchow 2000) or resource-dependent dispersal (Anderson et al 2012). However, they all have in common that the steady states in the kinetic ODEs are locally stable.…”