Global Stability Thresholds in Two-Species Migration Models

Definitions and mathematical statement

Consider the source's two-species Lotka--Volterra competition or migration system with nonnegative spatially heterogeneous growth rate $m(x)$. Retain $c^*$ for its threshold, $\Sigma_c$ for its specified set of migration-parameter pairs, and $(\theta,0)$ for its semitrivial steady state. The system, spatial domain, boundary conditions, the equation for $\theta$, and the definitions of $c^*$ and $\Sigma_c$ are not contained in the supplied record.

The cleaned conjecture asks whether, for every admissible model and parameter choice with \[ c\in(c^*,1),\qquad (\mu,\nu)\in\Sigma_c, \] so that $(\theta,0)$ is linearly stable as stated in the source, that equilibrium is globally asymptotically stable for every $b\in(0,1)$. The universal quantifier over $b$ is inside the fixed admissible model setup; it does not assert a single uniform convergence rate over all choices of $b,c,\mu,\nu,m$. Both upper endpoints $b=1$ and $c=1$, and the lower endpoints $b=0$ and $c=c^*$, are excluded.

To make the global conclusion precise, let $\mathcal X_{\mathrm{adm}}$ be the intended initial-data class, $\|\cdot\|_{\mathcal X}$ the intended norm, and $(u(t),v(t))$ a solution. The attraction component would require global existence for each initial datum in that class and \[ \|u(t)-\theta\|_{\mathcal X}+\|v(t)\|_{\mathcal X}\longrightarrow0 \quad\text{as }t\to\infty. \] Asymptotic stability also includes Lyapunov stability relative to that phase space. The class and topology must be recovered before this is a complete assertion. In ordinary competition models the absent-species faces are invariant, so one cannot silently take all nonnegative initial data, including extinction of the species represented by $\theta$, as a basin attracted to a nonzero $\theta$. The relevant positivity or nonextinction requirement is not supplied.

The original question says that the conclusion of Theorem 6 holds with $b^*=1$; the verification field reports expanding this into the global-stability question and interval $0<b<1$. This does not add a conclusion at $b=1$, nor independently establish an optimality claim for $b>1$. Theorem 6 and any additional hypotheses remain absent. The literature triage checked 2026-08-21 calls the $b^*=1$ assertion conjectural and reports no authoritative proof or disproof located in its search. These are dated source claims, not independently checked status conclusions.

Short English statement

Under the source's parameter conditions giving linear stability, does the semitrivial state attract every admissible trajectory and remain stable for all $0<b<1$? The intended system, admissible initial data and stability topology are missing. The conjectural threshold $b^*=1$ refers to that open interval and does not supply endpoint stability.

Sources

• Local UnsolvedMath registry, permanent ID 30000268, source code catalogue entry, "Global Stability Thresholds in Two-Species Migration Models," including the original Theorem 6 threshold question and expanded cleaned statement.
• Topological and Variational Methods for Differential Equations (2006), Oberwolfach Reports, pp. 1601--1678, DOI 10.4171/owr/2005/29: https://doi.org/10.4171/owr/2005/29. This is the sole supplied literature reference, labelled "original problem report." The citation uses 2006, while the DOI and proposed-year metadata use 2005.
• Literature triage checked 2026-08-21; registry record updated 2026-09-16. The verification field reports direct-source expansion of the question. No external source has been visited or independently checked here.