Coexistence for every positive cyclic invasion rate
Fix $\beta_1, \beta_2, \beta_3 > 0$ and consider the continuous-time Markov process $(\eta_t)$ on $\{1, 2, 3\}^{\mathbb{Z}^2}$ in which, writing $f_i(x, \eta)$ for the fraction of the four nearest neighbours of $x$ in state $i$, a site in state $3$ changes to $1$ at rate $\beta_1 f_1$, a site in state $1$ changes to $2$ at rate $\beta_2 f_2$, and a site in state $2$ changes to $3$ at rate $\beta_3 f_3$, with no other transitions. This is the cyclic case of the multitype biased voter model studied by Durrett and Levin. Its mean-field ODE has an interior equilibrium, with densities proportional to $(\beta_3, \beta_1, \beta_2)$, surrounded by concentric periodic orbits. Simulations of the spatial model, by Durrett and Levin and by Durrett, show the three types persisting together in a spatially structured state.
Problem. For every $\beta_1, \beta_2, \beta_3 > 0$, does this process have 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 types must be present in $\nu$-almost every configuration; a mixture of the three constant configurations does not qualify.
Source: Durrett (2009), Problem 7, after the simulations of Durrett and Levin (1998); repeated as Open Problem 4.5.2 in Durrett's book manuscript Interacting Particle Systems (2026). Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, coexistence, cyclic competition, 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 4.5.1},
}
@article{durrett1998,
author = {Durrett, Rick and Levin, Simon},
title = {Spatial aspects of interspecific competition},
journal = {Theor. Popul. Biol.},
volume = {53},
number = {1},
year = {1998},
pages = {30--43},
doi = {10.1006/tpbi.1997.1338},
}
@misc{aim-cyclic-spatial-coexistence,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Coexistence for every positive cyclic invasion rate}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/181-cyclic-spatial-coexistence.md},
note = {Entry 181 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-00422, "Coexistence for every positive cyclic invasion rate", clopen.solutions.