Pattern Formation of a Pathway-Based Diffusion Model: Linear Stability Analysis and an Asymptotic Preserving Method

We investigate the linear stability analysis of a pathway-based diffusion model (PBDM), which characterizes the dynamics of the engineered Escherichia coli populations [X. Xue, C. Xue, and M. Tang, PLoS Comput. Biol., 14 (2018), e1006178]. This stability analysis considers small perturbations of the density and chemical concentration around two nontrivial steady states, and the linearized equations are transformed into a generalized eigenvalue problem. By formal analysis, when the internal variable responds to the outside signal fast enough, the PBDM converges to an anisotropic diffusion model, for which the probability density distribution in the internal variable becomes a delta function. We introduce an asymptotic preserving (AP) scheme for the PBDM that converges to a stable limit scheme consistent with the anisotropic diffusion model. Further numerical simulations demonstrate the theoretical results of linear stability analysis, i.e., the pattern formation, and the convergence of the AP scheme.