First passage percolation in hostile environment is not monotone

We study a natural growth process with competition, modeled by two first passage percolation processes, FPP₁ and FPP_λ, spreading on a graph. FPP₁ starts at the origin and spreads at rate 1, whereas FPP_λ starts from a random set of inactive seeds distributed as Bernoulli percolation of parameter µ ∈ (0, 1). A seed of FPP_λ gets activated when one of the two processes attempts to occupy its location, and from this moment onwards spreads at some fixed rate λ > 0. In previous works [17, 3, 7] it has been shown that when both µ or λ are small enough, then FPP₁ survives (i.e., it occupies an infinite set of vertices) with positive probability. It might seem intuitive that decreasing µ or λ is beneficial to FPP₁. However, we prove that, in general, this is indeed false by constructing a graph for which the probability that FPP₁ survives is not a monotone function of µ or λ, implying the existence of multiple phase transitions. This behavior contrasts with other natural growth processes such as the 2-type Richardson model.