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：我们研究了基于通路的扩散模型（PBDM）的线性稳定性分析，该模型描述了工程化大肠杆菌种群的动力学特征 [X. Xue 和 C. Xue 以及 M. Tang, PLoS Computational Biology, 14 (2018), pp. e1006178]。该稳定性分析考虑了密度和化学浓度围绕两个非平凡稳态的小扰动，并将线性化方程转化为广义特征值问题。通过形式分析，当内部变量对外部信号的响应足够快时，PBDM收敛于各向异性扩散模型，此时内部变量中的概率密度分布变为δ函数。我们为PBDM引入了一种渐近保持（AP）格式，该格式收敛于与各向异性扩散模型一致的稳定极限格式。进一步的数值模拟验证了线性稳定性分析的理论结果（即斑图形成）以及AP格式的收敛性。