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A time- and space-discrete model for the growth of a rapidly saturating localbiological population $N(x,t)$ is derived from a hierarchical random depositionprocess previously studied in statistical physics. Two biologically relevantparameters, the probabilities of birth, $B$, and of death, $D$, determine thecarrying capacity $K$. Due to the randomness the population depends strongly onposition, $x$, 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 afunction of $B$ and $D$. The effective growth rate decreases with time, roughlyas in a Verhulst process. The model is possibly applicable, for example, tobacteria forming a "towering pillar" biofilm. The bacteria divide on randomlydistributed nutrient-rich regions and are exposed to random local bactericidalagent (antibiotic spray). A gradual overall temperature change away fromoptimal growth conditions, for instance, reduces bacterial reproduction, whilebiofilm development degrades antimicrobial susceptibility, causing stagnationinto a stationary state.Comment: 25 pages, 11 (9+2) figure

Hierarchical population model with a carrying capacity distribution for bacterial biofilms