Global attraction in D-stable Lotka–Volterra systems
Let $n \ge 1$, $A = (a_{ij}) \in \mathbb{R}^{n \times n}$ and $x^* \in (0,\infty)^n$, and consider the Lotka–Volterra system $$\dot x_i = x_i \sum_{j=1}^n a_{ij}(x_j - x_j^*), \qquad i = 1, \dots, n,$$ which has the positive equilibrium $x^*$. Suppose that $A$ is D-stable: for every diagonal matrix $D$ with positive diagonal entries, every eigenvalue of $DA$ has negative real part. Then $x^*$ is locally asymptotically stable, its Jacobian being $\operatorname{diag}(x^*)A$. If $A$ is Volterra–Lyapunov stable, that is, $DA + A^{\mathsf T}D$ is negative definite for some positive diagonal $D$, then $x^*$ attracts every solution in $(0,\infty)^n$; Lu and Takeuchi proved global stability when $A$ is qualitatively stable.
Problem. For every $n \ge 1$, every D-stable $A$ and every $x^* \in (0,\infty)^n$, must every solution with $x(0) \in (0,\infty)^n$ exist for all $t \ge 0$ and satisfy $\lim_{t \to \infty} x(t) = x^*$?
Source: Hofbauer and Sigmund (1988; 1998, p. 200), as stated by Lu and Takeuchi (1993), §2 (the Hofbauer–Sigmund conjecture), and by Hong and Pego (2021), §2.4. Stated by AIM — Open Applied Problems with forward existence included.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, D-stability, global stability, Lotka-Volterra systems
References
@book{hofbauer1998,
author = {Hofbauer, Josef and Sigmund, Karl},
title = {Evolutionary Games and Population Dynamics},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {1998},
doi = {10.1017/CBO9781139173179},
}
@article{lu1993,
author = {Lu, Zhengyi and Takeuchi, Yasuhiro},
title = {Global dynamical behavior for {L}otka--{V}olterra systems},
journal = {S\=urikaisekikenky\=usho K\=oky\=uroku},
volume = {828},
year = {1993},
pages = {171--178},
url = {https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/0828-16.pdf},
}
@article{hong2021,
author = {Hong, Won Eui and Pego, Robert L.},
title = {Exclusion and multiplicity for stable communities in {L}otka--{V}olterra systems},
journal = {J. Math. Biol.},
volume = {83},
number = {2},
year = {2021},
pages = {16},
doi = {10.1007/s00285-021-01638-7},
eprint = {2102.00611},
archivePrefix = {arXiv},
}
@misc{aim-lotka-volterra-d-stable-global-attraction,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global attraction in D-stable Lotka–Volterra systems}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/489-lotka-volterra-d-stable-global-attraction.md},
note = {Entry 489 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
No: for n = 4, x* = (1,1,1,1) and an explicit rational D-stable matrix A, the system has a nonconstant periodic solution with all coordinates in (1/8, 4), which does not converge to x* (and so for every n >= 4)..
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, 490-lotka-volterra-counterexample/PROOF.md. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00694, "Global attraction in D-stable Lotka–Volterra systems", clopen.solutions.