Extend upper semicontinuity and openness to nonnegative Allee parameters

Extend the upper semicontinuity of the maximal equilibrium map and the openness of the set of Allee parameters yielding population elimination from strictly positive Allee parameters to the full nonnegative parameter domain \(\mathbb{R}^n_+\), including cases in which some components of the Allee parameter vanish.

Background

The paper defines $\mathbf{E}_+(a)$ as the componentwise maximal equilibrium of the $n$-patch wild-population system, where $a\in\mathbb{R}^n_+$ is the vector of patch-specific Allee parameters. The main regularity theorem establishes upper semicontinuity of $a\mapsto\mathbf{E}_+(a)$, openness of the elimination set $\{a\gg0_n:\mathbf{E}_+(a)=0_{3n}\}$, and discontinuity at boundary points, but only under strict positivity of every component of $a$.

The authors explicitly identify extending these properties to all nonnegative Allee-parameter vectors as unresolved. The stated obstacle is that the fixed-point map used in the proof is not continuous at points where some $a_j=0$ and the corresponding non-adult component $L_j=0$, so the existing argument does not apply in those cases.