  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.
