Net reproduction threshold conjecture for local stability of the trivial steady state

Let $u_*\equiv 0$ denote the trivial steady state of the nonlinear problem $$. Assume that its linearisation exists, and that $E(0)={\bf 0}$. Net reproduction threshold conjecture. The trivial steady state $u_*\equiv 0$ is locally asymptotically stable if $\mathcal{R}({\bf 0})<1$, and unstable if $\mathcal{R}({\bf 0})>1$. This conjecture links the stability of the zero population state to the net reproduction function and is motivated by the fact that the linearisation at the trivial steady state is the corresponding parametrised linear problem; its resolution is not supplied here.