Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 13A (Mathematical Biology)

Open ends impose $C(0,t)=C(L,t)=0$. By separation of variables,Expanding the initial constant gives $b_n=2C_0(1-(-1{)}^n)/(n\pi )$, hence

The diffusive flux is $j=-D{C}_x$. The total outward flux through both ends is thereforeIt diverges as $t\downarrow 0$: the initially uniform profile is incompatible with the suddenly imposed zero boundary values, creating an idealized infinite initial gradient.

At late times the first sine mode dominates, with growth exponentThusand the population grows precisely when $\mu >{\mu }_c$. Faster diffusion carries organisms to the lethal ends more quickly and raises the threshold; a longer channel lowers the principal diffusion eigenvalue and therefore lowers the threshold.

Immediately after $t=0$, thin boundary layers enforce zero concentration at both ends while the interior remains close to $C_0$. The profile is symmetric about $L/2$. Higher sine modes then decay rapidly and the shape approachesFor $\mu <{\mu }_c$ successive late-time curves have the same sine shape and decreasing height; for $\mu >{\mu }_c$ they have that shape and increasing height. Those features determine the two requested sketches.

The boundary conditions are now $C(0,t)=0$ and $C_x(L,t)=0$. Their wave numbers areThe principal growth rate is $\mu -D{\pi }^2/(4L^2)$, so long-term growth occurs whenClosing one end reduces the critical growth rate by a factor of four because there is only one absorbing escape boundary.

Each spatial mode acquires the factorThe oscillatory factor $e^{{\mu }_1\sin t}$ is bounded and does not affect the long-term exponential rate. With two open ends, growth therefore occurs exactly whenOnly the time average of this spatially uniform fluctuating growth rate controls the asymptotic threshold.