Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 14E (Mathematical Biology)

This bistable reaction-diffusion equation models an invading population with an Allee effect: densities below $r$ decline, while those above it grow toward carrying capacity one.

The homogeneous equilibria are $0,r,1$. Since $f\prime (0)=-r<0$, $f\prime (1)=r-1<0$, and $f\prime (r)=r(1-r)>0$, the first and last are linearly stable and the middle one unstable.

Multiplying $Dn\prime \prime +f(n)=0$ by $n\prime$ and using the endpoint conditions gives the logistic frontIt has $n_0(0)=1/2$ and direct differentiation verifies the equation for $r=1/2$.

With $\xi =x-ct$, the equation is $Du\prime \prime +cu\prime +f(u)=0$. Since $f_r(u)-f_{1/2}(u)=u(1-u)(1/2-r)$ and $n_0\prime =-n_0(1-n_0)/\sqrt{2D}$, the same profile works with

The speed is positive for $r<1/2$, zero at $1/2$, and negative for $r>1/2$. The state with the larger basin-weighted reaction potential invades the other; raising the Allee threshold favours extinction.