Abstract
We present results on phase transitions of local and global survival in a two-species model on Poisson–Gilbert graphs. Initially, there is an infection at the origin that propagates on the graph according to a continuous-time nearest-neighbor interacting particle system. The graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finite-degree approximations of the underlying random graphs.
Highlights
We present results on phase transitions of local and global survival in a two-species model on Poisson–Gilbert graphs
There is an infection at the origin that propagates on the graph according to a continuous-time nearest-neighbor interacting particle system
We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival
Summary
We pick up a line of research that very recently has attracted some attention, about the survival of some species when chased by another species, see [DJT18] and references therein. Theorem 1.2 says that if the graph of susceptible nodes is insufficiently connected, i.e., μS < μcr, global survival is impossible for any infection rate and even without any white knights in the system. If the infection is too weak with respect to the intensity of susceptible nodes, i.e., λI ≤ ρ(μSκr), global survival is impossible for any positive intensity of white knights. Theorem 1.3 states that there is a positive chance for global survival if the underlying graph of susceptible nodes is sufficiently connected and the infection is strong enough to overcome the chasing white knights. If the process of susceptible nodes is supercritical but the other parameters guarantee global extinction as described in Theorem 1.2, for the survival probability exp(−μκr) ≤ P(Ec) ≤ (1 − θ(μS)) exp(−μWκr).
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