Spatial coexistence despite extinction in the mean-field game
Each site $x \in \mathbb{Z}^2$ holds $H_x \in \mathbb{N}_0$ hawks and $D_x \in \mathbb{N}_0$ doves. Each individual jumps at rate $\mu$ to a uniformly chosen nearest neighbour of its site, and each individual at $x$ dies at rate $\kappa(H_x + D_x)$. For $L \ge 1$ let $N_L = \{z \in \mathbb{Z}^2 : \|z\|_\infty \le L\}$ and $$p_x = \frac{\sum_{z \in N_L} H_{x+z}}{\sum_{z \in N_L} (H_{x+z} + D_{x+z})},$$ with $p_x = 0$ when the denominator vanishes. In addition, each hawk at $x$ has game rate $r_H = -0.6\, p_x + 0.9\, (1 - p_x)$ and each dove at $x$ has game rate $r_D = -0.9\, p_x + 0.37\, (1 - p_x)$: a positive rate is the rate at which the individual gives birth to an offspring at $x$, and a negative rate is an additional death rate equal to its absolute value.
This is the spatial hawk–dove game of Durrett and Levin, with payoffs $a = -0.6$, $b = 0.9$, $c = -0.9$, $d = 0.37$. Durrett notes that for these payoffs the solution of the mean-field equations $$\frac{du}{dt} = u\Bigl(\frac{a u + b v}{u + v} - \kappa(u + v)\Bigr), \qquad \frac{dv}{dt} = v\Bigl(\frac{c u + d v}{u + v} - \kappa(u + v)\Bigr)$$ for the mean numbers $u$ of hawks and $v$ of doves per site converges to $(0, 0)$ whenever $u(0) > 0$, while in the spatial model a dove with no hawks in its neighbourhood has positive game rate $r_D = 0.37$.
Problem. Are there a finite $L \ge 1$ and rates $\mu, \kappa > 0$ for which the process has a translation-invariant stationary distribution with finite mean number of individuals per site, under which almost every configuration contains infinitely many hawks and infinitely many doves?
Source: Durrett, Interacting Particle Systems: Ideas, Techniques, Applications (book manuscript, version of 2 September 2026), Open Problem 4.2.3 (Open Problem 3.2.3 in the version of 19 November 2025), for an example in the spatial game model of Durrett and Levin (1994). Stated with an explicit neighbourhood and notion of coexistence by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, coexistence, evolutionary games, interacting particle systems
References
@unpublished{durrett2026,
author = {Durrett, Rick},
title = {Interacting Particle Systems: Ideas, Techniques, Applications},
note = {Book manuscript, version of 2 September 2026, Section 4.2},
year = {2026},
url = {https://sites.math.duke.edu/~rtd/PASTA/PASTA0902.pdf},
}
@article{durrett1994,
author = {Durrett, Richard and Levin, Simon},
title = {The importance of being discrete (and spatial)},
journal = {Theor. Popul. Biol.},
volume = {46},
number = {3},
year = {1994},
pages = {363--394},
doi = {10.1006/tpbi.1994.1032},
}
@misc{aim-spatial-hawk-dove-coexistence,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Spatial coexistence despite extinction in the mean-field game}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/282-spatial-hawk-dove-coexistence.md},
note = {Entry 282 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-00474, "Spatial coexistence despite extinction in the mean-field game", clopen.solutions.