Interior smoothness of carrying simplices in competitive population maps
For an integer $d \ge 3$ let $C = [0,\infty)^d$ and consider the population map $F_i(x) = x_i f_i(x)$, where $f$ is $C^1$ on an open neighbourhood of $C$ and $f_i(x) > 0$ and $\partial f_i/\partial x_j(x) < 0$ on $C$ for all $i, j$. Suppose each positive coordinate axis has a fixed point $q_i e_i$ with $q_i > 0$, put $q = (q_1, \dots, q_d)$ and $[0,q] = \prod_i [0,q_i]$, and assume $$\rho(M(x)) < 1 \quad (x \in [0,q]), \qquad M_{ij}(x) = -\frac{x_i}{f_i(x)} \frac{\partial f_i}{\partial x_j}(x),$$ where $\rho$ is the spectral radius. Baigent proved that $F$ then has a carrying simplex $\Sigma \subset C \setminus \{0\}$: a compact invariant hypersurface, unordered (no two distinct points are comparable coordinatewise), to whose trajectories every nonzero trajectory is asymptotic. It is the graph of a Lipschitz function, and $\Sigma = \{R(u)u : u \in \Delta\}$ for a continuous $R\colon \Delta \to (0,\infty)$, where $\Delta = \{u \in C : \sum_i u_i = 1\}$. For a class of competitive maps, Mierczyński proved that a convex carrying simplex is a $C^1$ submanifold-with-corners, neatly embedded in $C$.
Problem. Is $\Sigma \cap (0,\infty)^d$ a $C^1$ embedded $(d-1)$-dimensional submanifold for every such map? Equivalently, must $R$ be $C^1$ on the relative interior of $\Delta$? No smoothness or transverse embedding at the extinction faces is assumed or required.
Source: Baigent (2019), Section 6, who asks for conditions for C^r smoothness of the interior of the carrying simplex; stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, carrying simplex, competitive maps, invariant manifolds
References
@incollection{baigent2019,
author = {Baigent, Stephen},
title = {Carrying simplices for competitive maps},
booktitle = {Difference Equations, Discrete Dynamical Systems and Applications},
editor = {Elaydi, Saber and P{\"o}tzsche, Christian and Sasu, Adina Lumini{\c{t}}a},
series = {Springer Proc. Math. Stat.},
volume = {287},
publisher = {Springer},
address = {Cham},
year = {2019},
pages = {3--29},
doi = {10.1007/978-3-030-20016-9_1},
}
@article{mierczynski2020,
author = {Mierczy{\'n}ski, Janusz},
title = {The {$C^1$} property of convex carrying simplices for competitive maps},
journal = {Ergodic Theory Dynam. Systems},
volume = {40},
number = {5},
year = {2020},
pages = {1335--1350},
doi = {10.1017/etds.2018.85},
eprint = {1801.01032},
archivePrefix = {arXiv},
}
@misc{aim-carrying-simplex-interior,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Interior smoothness of carrying simplices in competitive population maps}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/300-carrying-simplex-interior.md},
note = {Entry 300 at commit a153f46, 2026-10-13},
}
Resolution claims
A carrying simplex reduces the nonzero long-term dynamics of a competitive population map to an invariant hypersurface. We prove that its positive interior is a $C^1$ embedded manifold under the standard $C^1$ per-capita competition and spectral-radius assumptions. Equivalently, its radial function is $C^1$ in the interior of the probability simplex. The proof uses logarithmic coordinates. Although inverse derivatives lose strict …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/interior_smoothness.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00743, "Interior smoothness of carrying simplices in competitive population maps", clopen.solutions.