Abstract
This paper investigates a non-homogeneous two-dimensional model for reproducing chemotactic bacteria, immersed in a porous medium that experiences non-uniformly imposed flows. It is shown that independently of the form of the fluid velocity field, the compressible/incompressible nature of the fluid significantly shifts the Turing stability-instability transition line. In dry media, Gaussian perturbations travel faster than the hyperbolic secant ones, yet the latter exhibit better stability properties. The system becomes highly unstable under strong flows and high surface tension. Approximated solutions recovered by injecting Gaussian perturbations overgrow, in addition to triggering concentric breathing features that split the medium into high and low-density domains. Secant perturbations on the other hand scatter slowly and form patterns of non-uniformly distributed peaks for strong flows and high surface tension. These results emphasize that Gaussian perturbations strongly modulate the activity of bacteria, hence can be exploited to perform fast spreading in environments with changing properties. In this sense, Gaussian profiles are better candidates to explain quick bacterial responses to external factors. Secant-type approximated solutions slowly modulate the bacterial activity, hence are better alternatives to dive into weak bacterial progressions in heterogeneous media.