Pattern formation in bacterial colonies with density-dependent diffusion

Abstract

Recent experiments have shown that patterns can emerge in bacterial colonies genetically modified to have a drop in diffusion when population densities (detected via a quorum sensing mechanism) are sufficiently large. We examine one PDE model of this system, and construct its non-constant stationary solutions in the form of an interface for the bacterial density. We derive the equations for the interface motion and demonstrate analytically that such interface solution is stable when the diffusion rate of bacteria is large and the diffusion rate of signalling molecules, Dh, is small. We further demonstrate that increasing Dh induces a Hopf bifurcation, resulting in a loss of stability, which can lead to either complex spatio-temporal dynamics or extinction of pattern altogether. These results are confirmed by numerical simulations.

Boundedness, Stabilization, and Pattern Formation Driven by Density-Suppressed Motility

“…This is confirmed by our numerical simulation shown in Figure 1(a) where spatio-temporal aggregation patterns are observed but seem to be unstable. We remark that similar so-called chaotic patterns to Figure 1(a) have been numerically found in [24] for a different smoothedout step function of γ(v). Next we choose μ = 0.2 closer to the critical number 0.5, and check that n 1 = k1l π = 0.0000577 and n 2 = k2l π = 18.8439, which allow many integers between n 1 and n 2 and pattern formation is therefore expected.…”

“…For the system (1.1), not many rigorous mathematical results have been available either. When γ(v) is a piecewise decreasing function, formal analysis of (1.1) has been performed in [6] to explain the mechanism of stripe formation, and the dynamics of the interface where γ(v) jumps was recently studied in [24]. The purpose of this paper is to derive some qualitative behaviors for the system (1.1) with a smooth motility function γ(v) in a bounded domain with Neumann boundary conditions.…”

“…are two positive constants), Tao and Winkler [29] recently established the existence of global classical solutions in two dimensions and global weak solutions in three dimensions for (1.4) with μ = 0. To our knowledge, no much rigorous results for the system (1.4) with μ > 0 have been available to date except some formal analysis work in [6,24]. The goal of this paper is to establish the global existence and asymptotic behavior of solutions to (1.4) with μ > 0 which is the original model constructed in [6].…”

On the parabolic‐elliptic Keller–Segel system with signal‐dependent motilities: A paradigm for global boundedness and steady states

“…When γ ( v ) is a constant step‐wise function, the dynamics of discontinuity interface was studied in a previous work. 23 …”

Stochastic phenomena in pattern formation for distributed nonlinear systems

“…Self-organization of matter has attracted the attention of researchers from various areas for many years. Diverse phenomena related to self-organization are encountered in physics, chemistry, biology, ecology and other fields of science [1][2][3][4][5]. However, the recreation of self-organization in laboratory conditions for the experimental study of underlying mechanisms is very complicated and extremely expensive.…”