论文标题
带有种子库的随机Fisher-kpp方程，以及开/离分支的布朗运动
The stochastic Fisher-KPP Equation with seed bank and on/off-branching-coalescing Brownian motion
论文作者
论文摘要
我们引入了一类新的随机部分微分方程（SPDE），种子库对有益等位基因在空间人群中的传播进行建模，其中个体可以在活动状态和休眠状态之间切换。融合休眠和由此产生的种子库导致两种状态之间迁移的两种方程式耦合系统。我们首先讨论种子库SPDE的存在和独特性，并提供同等的延迟表示，可以清楚地解释种子库组件中的年龄结构。延迟表示在证明中也将至关重要。此外，我们表明，种子库SPDE产生了一类有趣的“开/关”时刻偶。特别是，在带有种子库的F-KPP方程式的特殊情况下，双重时刻由“开/分支布朗尼运动”给出。该系统与经典的布朗尼运动的不同之处在于，对于所有个人而言，动作和分支都可以“关闭”指数级的时间，然后再次“打开”。在这里，作为我们二元性的应用，我们表明，有益等位基因的传播在经典的F-KPP方程式中始于Heaviside Intial条件，它随着速度$ \ sqrt {2} $的拉动波动而演变，在相应的种子Bank F-KPP模型中却大大降低了。实际上，通过计算双重粒子在双重旋转的布朗尼运动中的位置，我们获得了一个上限，以实现$ \ sqrt {\ sqrt {\ sqrt {5} {5} -1} -1} -1} -1} \大约1.111 $在单位交换率下的传播速度。这表明，与人口遗传学和生态学的直观推理一致，种子库确实会减慢健身波并保留遗传变异性。
We introduce a new class of stochastic partial differential equations (SPDEs) with seed bank modeling the spread of a beneficial allele in a spatial population where individuals may switch between an active and a dormant state. Incorporating dormancy and the resulting seed bank leads to a two-type coupled system of equations with migration between both states. We first discuss existence and uniqueness of seed bank SPDEs and provide an equivalent delay representation that allows a clear interpretation of the age structure in the seed bank component. The delay representation will also be crucial in the proofs. Further, we show that the seed bank SPDEs give rise to an interesting class of "on/off" moment duals. In particular, in the special case of the F-KPP Equation with seed bank, the moment dual is given by an "on/off-branching Brownian motion". This system differs from a classical branching Brownian motion in the sense that independently for all individuals, motion and branching may be "switched off" for an exponential amount of time after which they get "switched on" again. Here, as an application of our duality, we show that the spread of a beneficial allele, which in the classical F-KPP Equation, started from a Heaviside intial condition, evolves as a pulled traveling wave with speed $\sqrt{2}$, is slowed down significantly in the corresponding seed bank F-KPP model. In fact, by computing bounds on the position of the rightmost particle in the dual on/off-branching Brownian motion, we obtain an upper bound for the speed of propagation of the beneficial allele given by $\sqrt{\sqrt{5}-1}\approx 1.111$ under unit switching rates. This shows that seed banks will indeed slow down fitness waves and preserve genetic variability, in line with intuitive reasoning from population genetics and ecology.
