  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.
