Competitive exclusion by the slowest diffuser in a multispecies habitat
Let $\Omega \subset \mathbb{R}^n$ be a bounded connected domain with smooth boundary, and let $m \in C^\alpha(\overline\Omega)$, $0 < \alpha < 1$, be nonnegative and nonconstant. For $N \ge 3$ and $0 < d_1 < \cdots < d_N$ consider $N$ competing species, identical except for their diffusion rates, $$\partial_t u_i = d_i \Delta u_i + u_i \Bigl( m(x) - \sum_{j=1}^N u_j \Bigr) \ \text{ in } \Omega \times (0, \infty), \qquad \partial_\nu u_i = 0 \ \text{ on } \partial\Omega \times (0, \infty),$$ for $1 \le i \le N$, in an environment constant in time. For $d > 0$ let $\theta_d$ be the unique positive solution of $d \Delta \theta + \theta (m - \theta) = 0$ in $\Omega$ with $\partial_\nu \theta = 0$ on $\partial\Omega$. For $N = 2$, Dockery, Hutson, Mischaikow and Pernarowski proved that every solution with both initial densities nonnegative and not identically zero converges to $(\theta_{d_1}, 0)$ uniformly on $\overline\Omega$. Cantrell and Lam proved that the finite sets of diffusion rates for which the slowest diffuser wins, in the stronger form of a Morse decomposition, form an open set in the Hausdorff metric; in particular it wins whenever every $d_i$ is close enough to one of two fixed values. Lam and Lou proved that it wins whenever $d_1 \ge B_0$, for a constant $B_0$ depending on $m$ but not on $N$.
Problem. For every such $\Omega$ and $m$, every $N \ge 3$, all $0 < d_1 < \cdots < d_N$ and all continuous, nonnegative initial densities $u_i(0, \cdot)$, none identically zero, must $$u_1(t, \cdot) \to \theta_{d_1}, \qquad u_i(t, \cdot) \to 0 \quad (2 \le i \le N)$$ hold uniformly on $\overline\Omega$ as $t \to \infty$?
Source: Dockery, Hutson, Mischaikow and Pernarowski (1998); see Cantrell and Lam (2021), Conjecture 1, and Lam and Lou (2024), Conjecture 1. Stated for nonnegative m by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, competitive exclusion, evolution of dispersal, reaction-diffusion systems
References
@article{dockery1998,
author = {Dockery, Jack and Hutson, Vivian and Mischaikow, Konstantin and Pernarowski, Mark},
title = {The evolution of slow dispersal rates: a reaction diffusion model},
journal = {J. Math. Biol.},
volume = {37},
number = {1},
year = {1998},
pages = {61--83},
doi = {10.1007/s002850050120},
}
@article{cantrell2021,
author = {Cantrell, Robert Stephen and Lam, King-Yeung},
title = {On the evolution of slow dispersal in multispecies communities},
journal = {SIAM J. Math. Anal.},
volume = {53},
number = {4},
year = {2021},
pages = {4933--4964},
doi = {10.1137/20M1361419},
eprint = {2008.08498},
archivePrefix = {arXiv},
}
@article{lam2024,
author = {Lam, King-Yeung and Lou, Yuan},
title = {The principal {F}loquet bundle and the dynamics of fast diffusing communities},
journal = {Trans. Amer. Math. Soc.},
volume = {377},
number = {1},
year = {2024},
doi = {10.1090/tran/8975},
}
@misc{aim-dockery-slowest-diffuser,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Competitive exclusion by the slowest diffuser in a multispecies habitat}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/352-dockery-slowest-diffuser.md},
note = {Entry 352 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00586, "Competitive exclusion by the slowest diffuser in a multispecies habitat", clopen.solutions.