Growth of a population of bacteria in a dynamical hostile environment

We study the growth of a population of bacteria in a dynamical hostile environment corresponding to the immune system of the colonised organism. The immune cells evolve as subcritical open clusters of oriented percolation and are perpetually reinforced by an immigration process, while the bacteria try to grow as a supercritical oriented percolation in the remaining empty space. We prove that the population of bacteria grows linearly as soon as it survives. In this perspective, we build general tools to study dependent percolation models issued from renormalization processes. 1. A growth model in dynamical hostile environment We consider the following discrete time interacting particle system: at time n = 0, a particularly fertile bacterium (represented here by a type 1 particle) is submerged in a population of immune cells (type 2 particles) that are going to impede its development. The immune cells are not very fertile but benefit from a constant immigration process. Our aim is to find conditions that ensure, when the bacteria survive, that their growth is linear. Our system is described by a discrete time Markov chain taking its values in {0, 1, 2}Zd, depending on 3 parameters p, q, α ∈ (0, 1). The time is indexed by N = {0, 1, 2, . . .} and we also note N∗ = {1, 2, 3, . . .}. The transition between two states is in two steps. First, between time n and time n+1/2, each particle tries to colonize its neighbor sites: it succeeds with probability p if it is a type 1 particle, and with probability q if it is a type 2 particle. All events are independent, and in case of conflict, the type 2 particle wins. Next, between time n + 1/2 and time n + 1, the immigration of type 2 particles occurs: on each site, a type 2 particle appears with probability α > 0, possibly taking the place of the particle previously occupying the site. Once again, all events are independent. In the degenerate case where q = 0 and α = 0, we recover independent oriented percolation with parameter p, which provides a simple model for the spread of an infection. By classical arguments, there exists a critical probability −→pcalt(d+ 1) for the possibility for independent oriented percolation on Z × N to grow infinitely. Of course, we choose p > −→pcalt(d + 1) to avoid the almost sure extinction of the bacteria in the absence of immune cells. Hence, if q = 0 and α = 0, we know that the bacteria survive with positive probability, and when they survive, their growth is linear. These results have been proved for the supercritical contact process by 2000 Mathematics Subject Classification. 60K35, 82B43.