Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 6E (Mathematical Biology)

The terms are logistic birth and crowding, $rn-r{n}^2/K$, linear predation mortality $-\delta n$, and diffusion $D{n}_{xx}$. The coast is reflecting, so $n_x(0)=0$; lethal habitat beyond $L$ gives the absorbing condition $n(L)=0$.

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.