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) \,= \fracα{2}Δu +β\,u(1-u) + \sqrt{γ\,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 ($β=0$), quantifying the effect of spatial diffusion on fixation time. We explicitly express the fixation rate in terms of the migration rate $α$ for all $α\in (0,\infty)$, finding in particular that the fixation rate is given by $γ[1-\fracγ{12α}+\mathcal{O}(\frac{γ^2}{α^2})]$ for fast migration and $π^2α[1-\frac{8α}γ+\mathcal{O}(\frac{α^2}{γ^2})]$ 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.

Figures (3)

• Figure 1: A trajectory of the 2-type branching-coalescing Brownain motions (2-type BCBM) starting with one green particle and one red particle, where $\tau_{\partial}$ is the first time when all red particles die off.
• Figure 2: (Left) the eigenvalue $\lambda=e^{-4\alpha \theta_{\ast}^2}$ in \ref{['eq:principal eigenvalue no selection']} against $\alpha$ for fixed values of $\gamma$. The eigenvalue $\lambda$ tends to $e^{-\gamma}$ as $\alpha\uparrow\infty$. (Right) the fixation rate $\kappa=-\ln \lambda=4\alpha \theta_{\ast}^2$ against $\alpha$ for fixed values of $\gamma$. It increases to $\gamma$ (the fixation rate of the well-mixed case) as $\alpha\uparrow\infty$ and it decreases to 0 as $\alpha \downarrow 0$.
• Figure 3: The key idea of our proofs exploits the duality (Proposition \ref{['prop:killDual']}) between the killed BCBM and the killed stochastic FKPP through Propositions \ref{['prop:duality relationship for quasi-stationarity']}- \ref{['prop:uniqueness QSD for FKPP']}.

Theorems & Definitions (75)

• Remark 1.1
• Theorem 2.1
• Remark 2.2: Feller property versus Feller semigroup
• Theorem 2.3: Right eigenpair and fixation time
• Remark 2.4: Discontinuous initial conditions
• Remark 2.5: Fixation time
• Remark 2.6: Convergence/non-convergence in total variation
• Definition 3.1: $2$-type branching coalescing Brownian motions
• Definition 3.2
• Lemma 3.3