Stationary periodic patterns in a model of growing population of motile bacteria

Biological pattern formation is one of the most intriguing phenomena in nature. Simple examples include travelling waves and stationary periodic patterns, which arise in various biological processes, including morphogenesis and population dynamics. The emergence of such patterns in populations of motile microorganisms, such as Dictyostelium discoideum and E. coli, has been demonstrated in numerous experimental studies. The conditions required for different types of pattern formation are commonly explored in mathematical studies of dynamical systems incorporating diffusion and advection terms. In this work, we introduce a prototype model for a growing population of motile bacteria that move in response to a chemical signal. We conduct a linear analysis of this model to determine the conditions for the formation of stationary periodic patterns, followed by a nonlinear (Fourier) analysis to characterise their properties, such as amplitude and wavelength. Our analytical findings are further validated through numerical simulations.