Competitive exclusion with unequal death rates
Each site of $\mathbb{Z}^2$ is vacant or occupied by one individual of type $1$ or $2$. A vacant site becomes occupied by type $i$ at rate $\beta_i f_i$, where $f_i$ is the fraction of its four nearest neighbours of type $i$, and an individual of type $i$ dies at rate $\delta_i > 0$. Let $\lambda_c$ be the critical value of the one-type contact process on $\mathbb{Z}^2$ in which occupied sites die at rate one and a vacant site becomes occupied at rate $\lambda f$, with $f$ the fraction of occupied neighbours. The upper stationary distribution of the type-$2$ contact process is the limit of the process started with every site of type $2$; it charges type $2$ when $\beta_2/\delta_2 > \lambda_c$. The mean-field ODE predicts that the type with the larger ratio $\beta_i/\delta_i$ takes over. Neuhauser proved this for $\delta_1 = \delta_2$: if $\beta_2 > \beta_1$ and type $2$ is supercritical, then from every translation-invariant initial law with a positive density of type $2$ the process converges to that upper stationary distribution. Stover noted that, by monotonicity in the parameters, the same holds when $\beta_1 < \beta_2$ and $\delta_1 > \delta_2$.
Problem. Assume $\beta_2/\delta_2 > \max\{\beta_1/\delta_1, \lambda_c\}$. Starting from any i.i.d. initial configuration that gives positive probability to each type, must the law of the process converge on finite sets to the upper stationary distribution of the type-$2$ contact process?
Source: Durrett (2009), Problem 1; repeated as Open Problem 4.1.3 in Durrett's book manuscript Interacting Particle Systems (2026). Lanchier (2010) and Stover (2020) describe it as a conjecture of Neuhauser (1992). 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, competitive exclusion, multitype contact process
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},
}
@article{neuhauser1992,
author = {Neuhauser, Claudia},
title = {Ergodic theorems for the multitype contact process},
journal = {Probab. Theory Related Fields},
volume = {91},
number = {3-4},
year = {1992},
pages = {467--506},
doi = {10.1007/BF01192067},
}
@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 4.1},
}
@article{stover2020,
author = {Stover, Joseph P.},
title = {A stochastic comparison result for the multitype contact process with unequal death rates},
journal = {Statist. Probab. Lett.},
volume = {162},
year = {2020},
pages = {108763},
doi = {10.1016/j.spl.2020.108763},
eprint = {1908.06628},
archivePrefix = {arXiv},
}
@misc{aim-competing-contact-unequal-deaths,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Competitive exclusion with unequal death rates}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/183-competing-contact-unequal-deaths.md},
note = {Entry 183 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-00416, "Competitive exclusion with unequal death rates", clopen.solutions.