Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 13C (Mathematical Biology)

This is an immigration--death process. New individuals enter independently of the current population at the constant rate $\lambda$, producing transitions $n\to n+1$. Each existing individual dies at rate $\beta$, so the total transition rate $n\to n-1$ is $\beta n$.

Multiply the master equation by $s^n$ and sum over $n\ge 0$. For the immigration terms,For the death terms, shifting the first sum givesTherefore the probability generating function obeys

The normalized ansatz intended in the question isfor which $\varphi (1,t)=1$. Substitution into the equation from part (ii) and cancellation of $(s-1)\varphi$ gives the linear ordinary differential equationThusThis generating function is that of a Poisson distribution with parameter $f(t)$. Equivalently, differentiating at $s=1$ givesConsequently

The additional transition is $n\to n+k$ at rate $ϵ(n)$. Its gain term at state $n$ comes from state $n-k$, so the new master equation iswhere the gain term is zero when $n<k$. In the generating-function equation the new contribution isFor a general state-dependent $ϵ$, this is not closed in $\varphi$. Even when it is closed, $s^k-1$ is not proportional to $s-1$ for $k>1$, so the Poisson ansatz used in part (a) is not preserved.

Let $m(t)=⟨n⟩={\sum }_{n\ge 0}n{p}_n(t)$. Multiplying the new master equation by $n$ and summing, each immigration event changes $n$ by $+1$, each death changes it by $-1$, and each extra event changes it by $+k$. Hence

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 $m(t)=m(0)+(\lambda +kb)t$.