Pattern Formation in a Reaction-Diffusion System with Space-Dependent Feed Rate

We develop novel mathematical techniques to study spot patterns in reaction-diffusion systems with space-dependent feed rate. The techniques are illustrated on the Schnakenberg model, which is a prototypical model of spot formation. Previous works have mostly addressed spot formation assuming the feed rate A is a constant, in which case the spots are uniformly distributed throughout the domain. By contrast, the spatially dependent feed rate $A(x)$ occurs naturally in applications, and very few studies exist in this case. In this work we determine analytically how this inhomogeneity affects spot creation, spot density, and spot death. We characterize the asymptotic density of spots and their heights for any spatially dependent feed rate $A(x)$. We also demonstrate a novel phenomenon which only happens when the feed rate is sufficiently inhomogeneous in space. Namely, new spots are continuously created in regions of high feed rate, travel toward regions of lower feed rate, and are destroyed there. This “creation-destruction loop” is only possible in the presence of heterogeneity. A key new technique is the use of the Euler--Maclaurin formula to estimate the effective spike density. For a sufficiently large feed rate, we find that the effective spot density scales like A2/3(x), whereas the spot mass scales like A1/3(x). We derive asymptotic bounds for the existence of N spots. As the feed rate is increased, new spots are created through self-replication, whereas the spots are destroyed as the feed rate is decreased. The thresholds for both spike creation and spike death are computed asymptotically.