On kinetic and macroscopic models for the stripe formation in engineered bacterial populations

We study the well-posedness of the biological models with AHL-dependent cell mobility on engineered Escherichia coli populations. For the kinetic model proposed by Xue-Xue-Tang recently, the local existence for large initial data is proved first. Furthermore, the positivity and local conservation laws for density $\rho (t,x,z)$ and nutrient $n(t,x)$ with initial assumptions are justified. Based on these properties, it can be extended globally in time near the equilibrium $(0,0,0)$. Considering the asymptotic behaviors of faster response CheZ turnover rate (i.e.,$\varepsilon\rightarrow 0$), one formally derives an anisotropic diffusion engineered Escherichia coli populations model (in short, AD-EECP) for which we find a key extra a priori estimate to overcome the difficulties coming from the nonlinearity of the diffusion structure. The local well-posedness and the positivity and local conservation laws for density and nutrient of the AD-EECP are justified. Furthermore, the global existence around the steady state $(\varrho_a, h_a, 0)$ with $\varrho_a \in [0, \Lambda_b)$ is obtained.