Quasi-stationary behavior of the stochastic FKPP equation on the circle

Abstract

We consider the stochastic Fisher-Kolmogorov-Petrovsky-Piscunov (FKPP) equation on the circle $\mathbb{S}$, \begin{equation*} \partial_t u(t,x) \,= \frac{\alpha}{2}\Delta u +\beta\,u(1-u) + \sqrt{\gamma\,u(1-u)}\,\dot{W}, \qquad (t,x)\in(0,\infty)\times \mathbb{S}, \end{equation*} where $\dot{W}$ is space-time white noise. While any solution will eventually be absorbed at one of two states, the constant 1 and the constant 0 on the circle, essentially nothing had been established about the absorption time (also called the fixation time in population genetics), or about the long-time behavior prior to absorption. We establish the existence and uniqueness of the quasi-stationary distribution (QSD) for the solution of the stochastic FKPP. Moreover, we show that the solution conditioned on not being absorbed at time $t$ converges to this unique QSD as $t\to\infty$, for any initial distribution, and characterize the leading-order asymptotics for the tail distribution of the fixation time. We obtain explicit calculations in the neutral case ($\beta=0$), quantifying the effect of spatial diffusion on fixation time. We explicitly express the fixation rate in terms of the migration rate $\alpha$ for all $\alpha\in (0,\infty)$, finding in particular that the fixation rate is given by $\gamma[1-\frac{\gamma}{12\alpha}+\mathcal{O}(\frac{\gamma^2}{\alpha^2})]$ for fast migration and $\pi^2\alpha[1-\frac{8\alpha}{\gamma}+\mathcal{O}(\frac{\alpha^2}{\gamma^2})]$ for slow migration. Our proof relies on the observation that the absorbed (or killed) stochastic FKPP is dual to a system of $2$-type branching-coalescing Brownian motions killed when one type dies off, and on leveraging the relationship between these two killed processes.