 Solution

Linearizing at extinction gives $n_t=(r-\delta )n+D{n}_{xx}$. The principal eigenfunction for Neumann data at $0$ and Dirichlet data at $L$ is $\cos (\pi x/(2L))$, with growth rateA population is viable exactly when this is positive, requiring $r>\delta$ and $L>(\pi /2)\sqrt{D/(r-\delta )}$. Thus intrinsic growth must beat predation and the refuge must exceed the diffusion-dependent critical size.