Stochastic dynamics of chemotactic colonies with logistic growth

The interplay between cellular growth and cell-cell signaling is essential for the aggregation and proliferation of bacterial colonies, as well as for the self-organization of cell tissues. To investigate this interplay, we focus here on the collective properties of dividing chemotactic cell colonies by studying their long-time and large-scale dynamics through a renormalization group (RG) approach. The RG analysis reveals that a relevant but unconventional chemotactic interaction -- corresponding to a polarity-induced mechanism -- is generated by fluctuations at macroscopic scales, even when an underlying mechanism is absent at the microscopic level. This emerges from the interplay of the well-known Keller--Segel (KS) chemotactic nonlinearity and cell birth and death processes. At one-loop order, we find no stable fixed point of the RG flow equations. We discuss a connection between the dynamics investigated here and the celebrated Kardar--Parisi--Zhang (KPZ) equation with long-range correlated noise, which points at the existence of a strong-coupling, nonperturbative fixed point.

ON EXTREME VALUES OF BIRTH AND DEATH PROCESSES

We establish the convergence rate to exponential distribution in a limit theorem for extreme values of birth and death processes. Some applications of this result are given to processes specifying queue length.). We establish uniform estimates for the convergence rate in the exponential distribution in a limit theorem for extreme values of birth and death processes. This topic is closely related to the problem on the time of first intersection of some level u by a regenerating process. Of course, we assume that both time t and level u grow infinitely. The proof of our main result is based on an important estimate for general regenerating processes. Investigations of the kind are needed in different fields: mathematical theory of reliability, queueing theory, some statistical problems in physics. We also provide with examples of applications of our results to extremal queueing problems M/M/s. In particular case of queueing M/M/1, we show that the obtained estimates have the right order with respect to the probability q(u) of the exceeding of a level u at one regeneration cycle, that is, only improvement of the corresponding constants is possible.