WIAS Preprint No. 2642, (2019)

Phase transitions for chase-escape models on Gilbert graphs


  • Hinsen, Alexander
  • Jahnel, Benedikt
    ORCID: 0000-0002-4212-0065
  • Cali, Eli
  • Wary, Jean-Philippe

2010 Mathematics Subject Classification

  • 60J25 60K35 60K37


  • Interacting particle systems, random graphs, survival, extinction, percolation, Boolean model




We present results on phase transitions of local and global survival in a two-species model on Gilbert graphs. At initial time there is an infection at the origin that propagates on the Gilbert graph according to a continuous-time nearest-neighbor interacting particle system. The Gilbert 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.

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