Motivated by a very basic mechanism that leads to clustering of reproducing individuals (Young et al., Nature 412, 328 (2001)), we introduce a simple model of population dynamics which considers birth and death rates for individual bugs that depend on the number of other bugs in their neighbourhoods. The model leads to inhomogeneous quasistationary distributions with many different clusters of bugs coming from different lineages and arranged periodically. This phenomenon is investigated by deriving the equation for the macroscopic density of particles, and performing a linear stability analysis on it, which shows that there is a finite-wavelength instability leading to pattern formation. A weakly nonlinear analysis of the emerging pattern is also performed, and the influence of fluctuations discussed. When the population is inmersed in an aquatic medium, fluid stirring has important impact on the patterns and in the population dynamics.