Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 13C / b / iii / Solution

For $ϵ(n)=an+b$, linearity of expectation gives $⟨ϵ(n)⟩=am+b$. The mean therefore satisfiesIf $\beta >ak$, the coefficient of $m$ is negative and the stable equilibrium isIf $\beta <ak$, the positive feedback from batch addition dominates deaths and $m(t)$ grows exponentially at rate $ak-\beta$. At the threshold $\beta =ak$, it grows linearly as $m(t)=m(0)+(\lambda +kb)t$.