Coexistence in a three-strain colicin model
Each site of $\mathbb{Z}^2$ is vacant (state $0$) or occupied by an individual of strain $1$, $2$ or $3$, and $f_i(x, \eta)$ denotes the fraction of the four nearest neighbours of $x$ in state $i$. A vacant site becomes occupied by strain $i$ at rate $\beta_i f_i$. Individuals of strains $1$ and $2$ die at rate one, and individuals of strain $3$ at rate $1 + \gamma_1 f_1 + \gamma_2 f_2$: strains $1$ and $2$ produce colicin, which kills strain $3$. This is the three-strain model of Durrett and Levin, whose simulations show a colicin-sensitive strain, a strong producer and a resistant weaker producer sustained together in a spatial population. For $\beta = (3, 3.2, 4)$ and $\gamma = (3, 0.5)$, Durrett observes that in pairwise competition strain $2$ beats strain $1$, strain $3$ beats strain $2$ and strain $1$ beats strain $3$, as in rock–paper–scissors, and his simulations show coexistence. Kirkup and Riley observed such a cycle among sensitive, colicin-producing and resistant strains of E. coli in mice.
Problem. Do there exist parameters $$0 < \beta_1 < \beta_2 < \beta_3, \qquad 0 < \gamma_2 < \gamma_1$$ for which this process has a stationary distribution $\nu$, invariant under translations of $\mathbb{Z}^2$, such that $$\nu\bigl(\#\{x \in \mathbb{Z}^2 : \eta(x) = i\} = \infty \text{ for every } i = 1, 2, 3\bigr) = 1?$$ All three strains must be present in $\nu$-almost every configuration; mixtures of laws in which one or more strains are absent do not qualify.
Source: Durrett (2009), Problem 8, for the three-strain model of Durrett and Levin (1997); repeated as Open Problem 4.5.3 in Durrett's book manuscript Interacting Particle Systems (2026). Stated by AIM — Open Applied Problems with unit natural death rates and ordered parameters.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, allelopathy, coexistence, interacting particle systems
References
@article{durrett2009,
author = {Durrett, Rick},
title = {Coexistence in stochastic spatial models},
journal = {Ann. Appl. Probab.},
volume = {19},
number = {2},
year = {2009},
pages = {477--496},
doi = {10.1214/08-AAP590},
eprint = {0906.2293},
archivePrefix = {arXiv},
}
@misc{durrett2026ips,
author = {Durrett, Rick},
title = {Interacting Particle Systems: Ideas, Techniques, Applications},
howpublished = {Book manuscript},
year = {2026},
url = {https://sites.math.duke.edu/~rtd/PASTA/PASTA0902.pdf},
note = {Version of 2 September 2026, Section 4.5.2},
}
@article{durrett1997,
author = {Durrett, Rick and Levin, Simon},
title = {Allelopathy in spatially distributed populations},
journal = {J. Theoret. Biol.},
volume = {185},
number = {2},
year = {1997},
pages = {165--171},
doi = {10.1006/jtbi.1996.0292},
}
@article{kirkup2004,
author = {Kirkup, Benjamin C. and Riley, Margaret A.},
title = {Antibiotic-mediated antagonism leads to a bacterial game of rock--paper--scissors in vivo},
journal = {Nature},
volume = {428},
number = {6981},
year = {2004},
pages = {412--414},
doi = {10.1038/nature02429},
}
@misc{aim-three-strain-colicin-coexistence,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Coexistence in a three-strain colicin model}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/182-three-strain-colicin-coexistence.md},
note = {Entry 182 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00483, "Coexistence in a three-strain colicin model", clopen.solutions.