论文标题
在随机环境中具有种子银行的空间种群:iii。与单型平衡的收敛
Spatial populations with seed-banks in random environment: III. Convergence towards mono-type equilibrium
论文作者
论文摘要
我们认为在丹·霍兰德(Den Hollander)和南丹(Nandan)(2021)中引入的种子银行的空间不均匀模型(2021)。包括$ Active $和$休眠$个人的种群以$ \ Mathbb {Z}^d,〜d \ geq 1 $标记的殖民地结构。人口大小来自形成随机环境的千古,翻译不变的,均匀的椭圆形场。个人携带两种类型之一:$ \ heartsuit $,$ \ scadesuit $。休眠个体位于所谓的种子银行中。活跃的个体通过根据对称的迁移内核从活性种群中选择父母,从自己菌落的种子银行交换类型。在Den Hollander和Nandan(2021)中,通过使用双重(相互作用的聚合粒子系统),我们表明空间系统在$ clustering $(单型型平衡)和$ coexiscastence $(多类型平衡)之间表现出二分法。在本文中,我们确定了$固定$环境的聚类制度中每个单型平衡的吸引力领域。我们还表明,当迁移内核为$ recurrent $时,对于A.E.实现环境的实现,最初具有$一致的$类型分布的系统薄弱地收集到单型平衡,其中键入键入的概率 - $ \ heartsuit $配置不取决于环境。固定概率的公式是根据类型的退火平均值 - $ \ heartsuit $ $ \ heartsuit $在目标菌落的两个种群大小比率偏见的处于休眠和积极种群的密度。 为了证明证明,我们使用dolgopyat和Goldsheid(2019)在带状条上引入的二元性和环境。马尔可夫操作员的光谱分析产生了与单粒子双重二元相关的环境过程静止的弱收敛,我们使用二元性将其转移到空间系统中。
We consider the spatially inhomogeneous Moran model with seed-banks introduced in den Hollander and Nandan (2021). Populations comprising $active$ and $dormant$ individuals are structured in colonies labelled by $\mathbb{Z}^d,~d\geq 1$. The population sizes are drawn from an ergodic, translation-invariant, uniformly elliptic field that form a random environment. Individuals carry one of two types: $\heartsuit$, $\spadesuit$. Dormant individual resides in what is called a seed-bank. Active individuals exchange type from seed-bank of their own colony and resample type by choosing parent from the active populations according to a symmetric migration kernel. In den Hollander and Nandan (2021) by using a dual (an interacting coalescing particle system), we showed that the spatial system exhibits a dichotomy between $clustering$ (mono-type equilibrium) and $coexistence$ (multi-type equilibrium). In this paper we identify the domain of attraction for each mono-type equilibrium in the clustering regime for a $fixed$ environment. We also show that when the migration kernel is $recurrent$, for a.e. realization of the environment, the system with an initially $consistent$ type distribution converges weakly to a mono-type equilibrium in which the fixation probability to type-$\heartsuit$ configuration does not depend on the environment. A formula for the fixation probability is given in terms of an annealed average of type-$\heartsuit$ densities in dormant and active population biased by ratio of the two population sizes at the target colony. For the proofs, we use duality and environment seen by particle introduced in Dolgopyat and Goldsheid (2019) for RWRE on a strip. A spectral analysis of Markov operator yields quenched weak convergence of the environment process associated with single-particle dual to a reversible ergodic distribution which we transfer to the spatial system by using duality.