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

Pattern formation of a pathway-based diffusion model: linear stability analysis and an asymptotic preserving method

“…Then, an asymptotic preserving (AP) scheme that connects the two models (PBDM) and (ADM) is designed. Due to the analytical difficulties, we only justify formally the limiting process (κ → ∞) from the model (PBDM) to the model (ADM) in [15]. But the numerical results on fine meshes in this work indicate the validity of this convergence (see Figure 2).…”

“…In [15], by taking the limit κ → ∞, i.e. the asymptotic behavior of the fast CheZ turnover rate, we f ormally get ρ(x, t, z) = (x, t)δ(L(h(x, t)) − z) and (x, t) satisfies the anisotropic diffusion model (ADM) with the following form:…”

“…Compared with (PBDM), the cell mobility of (ADM) is given by D(L(h(x, t))), depending directly on concentration of AHL. The key feature of ADM is the term ∆ x [D(L(h(x, t))) (x, t)] in the equation for , which is called anisotropic diffusion term in [15]. Since…”

“…There exist some analytical works for both PBDM and ADM in the literature. On the one hand the mathematical analysis of PBDM starts from our last paper [15]. This system is a highly nonlinear coupling of reaction-diffusion and kinetic-type equations.…”

“…The first step is to investigate the existence of smooth solutions near constant states, which are closely related to their stability. In [15], the global well-posedness around the trivial state (0, 0, 0) is proved. On the other hand, (ADM) is of the same type as the model introduced in [20], where the authors numerically reproduced some key features of experimental observations.…”