Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 6C (Mathematical Biology)

Conservation requires zero flux: $n_x(0,t)=n_x(L,t)=0$. Indeed $d{\int }_0^{L}n,dx/dt=D[n_x{]}_0^L=0$.

Add exponential growth: $n_t=D{n}_{xx}+\alpha n$.

With hostile Dirichlet boundaries, expand in $\sin (k\pi x/L)$. The $k$th amplitude grows at rate $\alpha -D(k\pi /L{)}^2$. Persistence requires the principal rate to be nonnegative, hence $\alpha \ge D{\pi }^2/L^2$; strict inequality gives growth.