Competitive displacement in the grass-bushes-trees process on a scale-free network
Abstract
The grass-bushes-trees process is a two-type contact process in which the types face asymmetric birth mechanisms. Vertices of a graph can either be occupied by a tree, occupied by a bush, or empty (containing grass). The trees die with rate $1$ and give birth with rate λ𝒯 on any neighbouring vertex (possibly taking the place of a bush). Bushes die with rate $1$ and give birth with rate λℬ on any empty neighbouring vertex; in particular, they are forbidden from taking the place of trees. We study this process on a power law random graph, namely the erased configuration model. We study which graph parameters lead to the possibility of coexistence versus the necessity of competitive displacement, i.e. joint metastability or fast extinction of the bushes.
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BibTeX
@misc{fernley2026competitive,
title = {{Competitive displacement in the grass-bushes-trees process on a scale-free network}},
author = {John Fernley and Daniel Valesin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20984},
eprint = {2610.20984},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20984},
}
RIS
TY - UNPB
TI - Competitive displacement in the grass-bushes-trees process on a scale-free network
AU - John Fernley
AU - Daniel Valesin
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20984
UR - https://arxiv.org/abs/2610.20984
AB - The grass-bushes-trees process is a two-type contact process in which the types face asymmetric birth mechanisms. Vertices of a graph can either be occupied by a tree, occupied by a bush, or empty (containing grass). The trees die with rate $1$ and give birth with rate λ_𝒯 on any neighbouring vertex (possibly taking the place of a bush). Bushes die with rate $1$ and give birth with rate λ_ℬ on any empty neighbouring vertex; in particular, they are forbidden from taking the place of trees. We study this process on a power law random graph, namely the erased configuration model. We study which graph parameters lead to the possibility of coexistence versus the necessity of competitive displacement, i.e. joint metastability or fast extinction of the bushes.
ER -