First passage percolation in hostile environment is not monotone

We study a natural growth process with competition, modeled by two first passage percolation processes, FPP 1 and FPP lambda , spreading on a graph. FPP( 1 )starts at the origin and spreads at rate 1, whereas FPP lambda starts from a random set of inactive seeds distributed as Bernoulli percolation of parameter /..1, is an element of (0 , 1). A seed of FPP lambda gets activated when one of the two processes attempts to occupy its location, and from this moment onwards spreads at some fixed rate A > 0. In previous works [17, 3, 7] it has been shown that when both /..1, or A are small enough, then FPP 1 survives (i.e., it occupies an infinite set of vertices) with positive probability. It might seem intuitive that decreasing /..1, or A is beneficial to FPP 1 . However, we prove that, in general, this is indeed false by constructing a graph for which the probability that FPP 1 survives is not a monotone function of /..1, or A , implying the existence of multiple phase transitions. This behavior contrasts with other natural growth processes such as the 2-type Richardson model.