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

TL;DR

The paper addresses finite-time blowup statistics in a super-Malthusian population by analyzing a simple stochastic birth process with rule $2A\to3A$. It combines exact backward-master-equation methods and Laplace-transform techniques to obtain the full blowup-time distribution and its moments, complemented by the optimal-fluctuation-method analysis of the short-time tail. The main results include the exact average blowup time $\langle T\rangle = \Theta(m)=\frac{2}{m-1}$, a closed-form representation of the distribution for $m=2$ with a purely exponential long-time tail $\mathcal{P}(T\to\infty)\sim e^{-T}$ and an essential-singularity short-time tail $\mathcal{P}(T)\sim e^{-\pi^2/(2T)}$, together with the optimal trajectory $n(t;T)=\frac{\pi}{T}\tan\left(\frac{\pi t}{2T}\right)$. The work highlights the tractability of the discrete model and its qualitative agreement with corresponding Langevin descriptions, while also clarifying differences in tail behavior and analytical complexity between the discrete and continuous formulations.

Abstract

Motivated by the paradigm of a super-Maltusian 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 $2A \to 3A$, where two identical individuals give birth to a third one. We determine exactly the average blowup time as well as the probability distribution $\mathcal{P}(T)$ of the blowup time and its moments. In particular, we show that the long-time tail $\mathcal{P}(T\to \infty)$ is purely exponential. The short-time tail $\mathcal{P}(T\to 0)$ exhibits an essential singularity at $T=0$, and it is dominated by a single (the most likely) population trajectory which we determine analytically.

Figures (3)

• Figure 1: The blowup time distribution $\mathcal{P}(T)$ for $m=2$, as described by Eqs. (\ref{['PTresult']}) and (\ref{['a(k)']}) (solid line). Dashed lines: the long-time asymptotic (\ref{['PlargeT']}) and the short-time asymptotic (\ref{['PsmallT']}).
• Figure 2: The phase portrait of the system in the limit of $T\to 0$. Shown are three phase trajectories, as described by Eq. (\ref{['p(n)']}) for $E=10^3$, $10^2$ and $10$ (from top to bottom). In view of Eq. (\ref{['TvsE']}), the blowup times in these cases are $0.070\dots$, $0.222\dots$ and $0.702\dots$, respectively.
• Figure 3: The optimal population trajectory conditioned on unusually fast blowup times $T=0.1$, $0.3$ and $0.5$, as described by Eq. (\ref{['nvst']}).