2.II.6D

Solve the differential equation

$\frac{d y}{d t}=r y(1-a y)$

for the general initial condition $y=y_{0}$ at $t=0$, where $r, a$, and $y_{0}$ are positive constants. Deduce that the equilibria at $y=a^{-1}$ and $y=0$ are stable and unstable, respectively.

By using the approximate finite-difference formula

$\frac{d y}{d t}=\frac{y_{n+1}-y_{n}}{\delta t}$

for the derivative of $y$ at $t=n \delta t$, where $\delta t$ is a positive constant and $y_{n}=y(n \delta t)$, show that the differential equation when thus approximated becomes the difference equation

$u_{n+1}=\lambda\left(1-u_{n}\right) u_{n},$

where $\lambda=1+r \delta t>1$ and where $u_{n}=\lambda^{-1} a(\lambda-1) y_{n}$. Find the two equilibria and, by linearizing the equation about them or otherwise, show that one is always unstable (given that $\lambda>1$ ) and that the other is stable or unstable according as $\lambda<3$ or $\lambda>3$. Show that this last instability is oscillatory with period $2 \delta t$. Why does this last instability have no counterpart for the differential equation? Show graphically how this instability can equilibrate to a periodic, finite-amplitude oscillation when $\lambda=3.2$.

*Typos? Please submit corrections to this page on GitHub.*