On the natural speed of pattern propagation in reaction-diffusion systems #

Vaclav Klika, Eamonn A Gaffney, Philip K Maini

13:50 Tuesday in 4Q04.

Abstract #

Motivated by a recent article [Liu et al. “Control of diffusion-driven pattern formation behind a wave of competency.” Physica D: Nonlinear Phenomena 438 (2022): 133297], we explore the idea of the existence of a natural speed of pattern propagation in reaction-diffusion systems. We do so by analysing a situation when a stationary in time pattern develops from a localised perturbation behind a travelling wave front in a system lying in a parameter regime of Turing instability. Instead of invoking the marginal instability analysis approach, as done in similar problems previously, we resort to weakly nonlocal analysis and identify the envelope equation of the pattern near a bifurcation point. This real Ginsburg-Landau equation is then used to estimate the amplitude and speed of the travelling wave analytically. Finally, we compare the numerical solutions to our analytical results and also to those generated by the marginal instability approach. As a result, we hereby lay the foundation for the concept of the natural speed of pattern formation in reaction-diffusion systems.