  To explore the coupling between a growing population of microorganisms such
as E. coli and a nonuniform nutrient distribution, we formulate a minimalistic
model. It consists of active Brownian particles that divide and grow at a
nutrient-dependent rate following the Monod equation. The nutrient
concentration obeys a diffusion equation with a consumption term and a point
source. In this setting the heterogeneity in the nutrient distribution can be
tuned by the diffusion coefficient.
  In particle-based simulations, we demonstrate that passive and weakly active
particles form proliferation-induced clusters when the nutrient is localized,
without relying on further mechanisms such as chemotaxis or adhesion. In
contrast, strongly active particles disperse in the whole system during their
lifetime and no clustering is present. The steady population is unaffected by
activity or nonuniform nutrient distribution and only determined by the ratio
of nutrient influx and bacterial death. However, the transient dynamics
strongly depends on the nutrient distribution and activity. Passive particles
in almost uniform nutrient profiles display a strong population overshoot, with
clusters forming all over the system. In contrast, when slowly diffusing
nutrients remain centred around the source, the bacterial population quickly
approaches the steady state due to its strong coupling to the nutrient.
Conversely, the population overshoot of highly active particles becomes
stronger when the nutrient localisation increases. We successfully map the
transient population dynamics onto a uniform model where the effect of the
nonuniform nutrient and bacterial distributions are rationalized by two
effective areas.
