Instabilities in Random Media and Peaking Regimes

We consider a system of particles (bacteria) in a medium, in which the birth and death intensities are distributed in space at random. In this system, we study the increase in the average number of particles, which depends on the difference between the birth intensity and the death intensity and is referred to as the random potential. It is shown that if the potential decreases quite slowly at infinity, the explosive growth in the number of bacteria and their average population formally turns to infinity immediately after the beginning of system evolution. In addition, it is shown that the finiteness of the average numbers of bacteria for each specific realization of the medium does not guarantee the finiteness of the average numbers of bacteria in the averaging over all realizations of the medium. Finally, we describe the behavior of the average numbers of bacteria averaged over the medium for a wide class of potentials for long times.