  A time- and space-discrete model for the growth of a rapidly saturating local
biological population $N(x,t)$ is derived from a hierarchical random deposition
process previously studied in statistical physics. Two biologically relevant
parameters, the probabilities of birth, $B$, and of death, $D$, determine the
carrying capacity $K$. Due to the randomness the population depends strongly on
position, $x$, and there is a distribution of carrying capacities, $\Pi (K)$.
This distribution has self-similar character owing to the imposed hierarchy.
The most probable carrying capacity and its probability are studied as a
function of $B$ and $D$. The effective growth rate decreases with time, roughly
as in a Verhulst process. The model is possibly applicable, for example, to
bacteria forming a "towering pillar" biofilm. The bacteria divide on randomly
distributed nutrient-rich regions and are exposed to random local bactericidal
agent (antibiotic spray). A gradual overall temperature change away from
optimal growth conditions, for instance, reduces bacterial reproduction, while
biofilm development degrades antimicrobial susceptibility, causing stagnation
into a stationary state.
