Fluctuations of blowup time in a simple model of a super-Malthusian catastrophe

Motivated by the paradigm of a super-Malthusian population catastrophe, we study a simple stochastic population model which exhibits a finite-time blowup of the population size and is strongly affected by intrinsic noise. We focus on the fluctuations of the blowup time T in the asexual binary reproduction model 2 A → 3 A , where two identical individuals give birth to a third one. We determine exactly the average blowup time, as well as the probability distribution P ( T ) of the blowup time and its moments. In particular, we show that the long-time tail P ( T → ∞ ) is purely exponential. The short-time tail P ( T → 0 ) exhibits an essential singularity at T = 0, and it is dominated by a single (most likely) population trajectory which we determine analytically.