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We investigate the linear stability analysis of a pathway-based diffusionmodel (PBDM), which characterizes the dynamics of the engineered Escherichiacoli populations [X. Xue and C. Xue and M. Tang, P LoS Computational Biology,14 (2018), pp. e1006178]. This stability analysis considers small perturbationsof the density and chemical concentration around two non-trivial steady states,and the linearized equations are transformed into a generalized eigenvalueproblem. By formal analysis, when the internal variable responds to the outsidesignal fast enough, the PBDM converges to an anisotropic diffusion model, forwhich the probability density distribution in the internal variable becomes adelta function. We introduce an asymptotic preserving (AP) scheme for the PBDMthat converges to a stable limit scheme consistent with the anisotropicdiffusion model. Further numerical simulations demonstrate the theoreticalresults of linear stability analysis, i.e., the pattern formation, and theconvergence of the AP scheme.

Pattern formation of a pathway-based diffusion model: linear stability analysis and an asymptotic preserving method