Differential Equations | Part IA, 2002

Solve the differential equation

dydt=ry(1ay)\frac{d y}{d t}=r y(1-a y)

for the general initial condition y=y0y=y_{0} at t=0t=0, where r,ar, a, and y0y_{0} are positive constants. Deduce that the equilibria at y=a1y=a^{-1} and y=0y=0 are stable and unstable, respectively.

By using the approximate finite-difference formula

dydt=yn+1ynδt\frac{d y}{d t}=\frac{y_{n+1}-y_{n}}{\delta t}

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

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

where λ=1+rδt>1\lambda=1+r \delta t>1 and where un=λ1a(λ1)ynu_{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 λ>1\lambda>1 ) and that the other is stable or unstable according as λ<3\lambda<3 or λ>3\lambda>3. Show that this last instability is oscillatory with period 2δt2 \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 λ=3.2\lambda=3.2.

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