Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 6C (Mathematical Biology)

The positive fixed point satisfies $x_{\ast }=x_{\ast }{e}^{r(1-x_{\ast })}$, hence $x_{\ast }=1$. Put $x_n=1+u_n$ and retain first-order terms:Thuswith characteristic polynomial ${\lambda }^2-\lambda +r$.

The Jury stability criterion for ${\lambda }^2-\lambda +r$ gives both roots inside the unit disc exactly whenThe decay is monotone when both roots are real and positive. Since the discriminant is $1-4r$, this occurs for

For $1/4<r<1$, the roots are a complex-conjugate pair with modulus $\sqrt{r}<1$. The perturbation therefore changes sign or phase while its envelope decays:

At the stability boundary $r_{\ast }=1$, the characteristic roots areThey lie on the unit circle and satisfy ${\lambda }^6=1$, with no smaller positive common period. Hence