Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 6A (Mathematical Biology)

For $F(n)=\alpha n-\beta n^3$, the nonnegative equilibria areSincethe zero equilibrium is unstable and $K$ is asymptotically stable.

For $0<n<K$, $F(n)>0$, so solutions increase monotonically toward $K$. For $n>K$, $F(n)<0$, so they decrease monotonically toward $K$. The solution $n=0$ remains at zero. Thus a phase-line sketch has arrows away from $0$ and toward $K$; a time-series sketch has all positive curves tending to the same horizontal asymptote $K$. This is the cubic saturation population model.

With the same low-density rate and carrying capacity, the two per-capita growth rates arefor the cubic model andfor logistic growth. The logistic correction is linear in a small population, whereas the cubic correction is quadratic. Consequently the cubic trajectory initially follows pure exponential growth more closely and lies above the logistic trajectory before both approach $K$. Plotting measured per-capita growth against $n$ distinguishes a linear decline from a curve with zero slope at the origin.