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.

翻译：我们研究了一种路径基础扩散模型（PBDM）的线性稳定性分析，该模型描述了工程大肠杆菌群体的动力学[X. Xue and C. Xue and M. Tang, P LoS Computational Biology, 14（2018）, pp. e1006178].此稳定性分析考虑密度和化学浓度围绕两个非平凡稳态的微小扰动，线性化方程被转化为广义特征值问题。通过形式分析，当内部变量的响应足够快时，PBDM收敛于一个各向异性扩散模型，其中内部变量的概率密度分布成为一个delta函数。我们为PBDM引入了一种渐近保留（AP）方案，该方案收敛于一个与各向异性扩散模型一致的稳定极限方案。进一步的数值模拟证明了线性稳定性分析的理论结果，即模式形成和AP方案的收敛。