Modelling formation of stationary periodic patterns in growing population of motile bacteria

Biological pattern formation is one of the most intriguing phenomena in nature. Simplest examples of such patterns are represented by travelling waves and stationary periodic patterns which occur during various biological processes including morphogenesis and population dynamics. Formation of these patterns in populations of motile microorganisms such as Dictyostelium discoideum and E. coli have been shown in a number of experimental studies. Conditions for formation of various types of patterns are commonly addressed in mathematical studies of dynamical systems containing diffusive and advection terms. In this work, we do mathematical study of spatio-temporal patterns forming in growing population of chemotactically active bacteria. In particular, we perform linear analysis to find conditions for formation of stationary periodic patterns, and nonlinear (Fourier) analysis to find characteristics, such as amplitude and wavelength, of these patterns. We verify our analytical results by means of numerical simulations.