  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 and C. Xue and M. Tang, P LoS Computational Biology,
14 (2018), pp. e1006178]. This stability analysis considers small perturbations
of the density and chemical concentration around two non-trivial 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.
