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

TL;DR

This paper analyzes mathematical models describing stripe formation in engineered bacterial populations, proving well-posedness, positivity, and conservation laws for both kinetic and macroscopic models, and establishing their global existence near equilibrium.

Contribution

It provides the first rigorous analysis of well-posedness and global existence for both kinetic and derived macroscopic models of engineered bacterial stripe formation.

Findings

Proved local existence for the kinetic model with large initial data.

Established positivity and conservation laws for density and nutrient.

Derived a macroscopic anisotropic diffusion model and proved its global existence near equilibrium.

Abstract

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.