Uniform subcritical reproduction conjecture for global stability

Let $u_*\equiv 0$ denote the trivial steady state of the nonlinear problem $$, let $\mathcal{Y}$ be the parameter space, and let $\mathcal{R}:\mathcal{Y}\to\mathbb{R}$ be the net reproduction function. Assume that $E(0)={\bf 0}$ and that there exists an $\varepsilon>0$ such that, for every $\bf p\in\mathcal{Y}$, $\mathcal{R}(\bf p)<1-\varepsilon$. Uniform subcritical reproduction conjecture. The trivial steady state $u_*\equiv 0$ is globally asymptotically stable. This conjecture would extend the threshold interpretation of the net reproduction function beyond the linearised setting, since it requires no linearisation of the nonlinear operator; its resolution is not supplied here.