Stability and dynamics of a chemotaxis system with deformed free-surface in a shallow chamber

Abstract

Complex bioconvection patterns have been studied analytically, experimentally, and numerically previously only for a flat free-surface of a suspension of chemotaxis bacteria in a shallow/deep chamber. In this paper, we have considered a two-dimensional chemotaxis-diffusion-convection system with a deformed free-surface. The influence of aggregation of chemotactic cells on the deformed free-surface of a shallow chamber is studied analytically. The aim of this paper is to investigate the nature of the instability in the chemotaxis-diffusion-convection system. We performed a detailed linear stability analysis of a steady-state cell and oxygen concentration distribution. The system becomes dominated by nonlinear convection terms beyond a critical Rayleigh number Raτ, which also depends on the critical wavenumber k as well as the other parameters. We have investigated that how the critical Rayleigh number in this system varies with three different sets of parameters. A weakly nonlinear analysis is carried out as well to determine the relative stability of the pattern formation at the onset of instability. A reactance between rolls, squares, hexagons, and mixed mode pattern is investigated in detail. Further research should link the weakly nonlinear analysis with the bifurcation analysis. Some important direct numerical simulation results have been presented in the support of linear stability analysis. Comparison of the analytical steady-state solution shows good agreement with the numerical result.