Note: Here time is assumed to be discrete and indexed by the naturals $\mathbb N$.
A curious species of amoeba undergoes the following evolution rule - at each time step, with probability $0 < p < 1$ it splits into two copies of itself of size $\varepsilon S$, where $S$ is the original size of the amoeba, and $0 < \varepsilon < 1$ is a fixed number. With probability $q = 1-p$, it doubles in size instead.
Start with one amoeba of size $1$. Denoting by $S^{\max}_t$ the size of the largest amoeba at time $t$, for what values of the parameters $\varepsilon, p$ do we have
$$\lim_{t \to \infty} S^{\max}_t = \infty$$
almost surely?
