Past exam of the mathematical physiology course of the University of Oxford / 2011 / 3 / a / Solution

For a constant solution $u(t)\equiv U$, its delayed value is also $U$. The equation becomes $0=U(1-U)$, so the only constant equilibria are $\boxed{U=0\text{ and }U=1}$. The delay differential equation requires the same constant history to realize each equilibrium.