Multiscale analysis: Fisher-Wright diffusions with rare mutations and selection, logistic branching system
Multiscale analysis: Fisher-Wright diffusions with rare mutations and selection, logistic branching system
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多尺度分析：具有罕见突变和选择的 Fisher-Wright 扩散、逻辑分支系统
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
A. Greven
中科院分区：
文献类型：
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作者：
D. Dawson;A. Greven
我们研究了两种类型的随机过程，第一个平均场空间系统的相互作用的Fisher-Wright扩散与劣势和优势的类型与罕见的突变（劣势的优势）和选择和第二个平均场空间系统的超临界分支随机游动的额外死亡率，这是二次在当地的粒子数。前者描述了一个标准的两型人口下的选择，突变和后者模型描述了一个人口稀缺的资源造成额外的死亡，在高的本地人口密度。地理空间由1，...，N建模。第一个过程开始于初始状态，只有劣质类型存在或可交换的配置，第二个过程具有单个初始颗粒。这种材料是一个特殊情况下的理论发展和描述的结果第7节在那里。我们研究的行为在两个时间窗口，第一次之间的时间0和T和第二个大的时间后，在费舍尔-赖特模型的罕见的突变体成功，分别在分支随机行走的粒子人口达到一个积极的空间强度。证明了当N → ∞时，两个模型的第二阶段渐近地在时间α− 1 logN之后开始，如果N是地理空间的大小，N − 1是稀有突变率，α ∈（0，∞）依赖于其他参数.我们将两个时间窗口中的极限动力学确定为N → ∞，并将两个模型分别确定为非线性马尔可夫动力学（McKean-Vlasov动力学），即该动力学从时间− ∞开始的相应随机进入律。最后，我们解释说，这两个过程只是同一枚硬币的两面，一个事实产生于一种新形式的二元性，特别是粒子模型产生的家谱的费舍尔-赖特扩散与选择和突变。我们讨论了这种对偶关系的扩展到一个多型模型与两个以上的类型。
We study two types of stochastic processes, first a mean-field spatial system of interacting Fisher–Wright diffusions with an inferior and an advantageous type with rare mutation (inferior to advantageous) and selection and second a mean-field spatial system of supercritical branching random walks with an additional death rate, which is quadratic in the local number of particles. The former describes a standard two-type population under selection, mutation and the latter model describes a population under scarce resources causing additional death at high local population intensity. Geographic space is modelled by 1, …, N. The first process starts in an initial state with only the inferior type present or an exchangeable configuration and the second one with a single initial particle. This material is a special case of the theory developed in and describes the results of Section 7 therein. We study the behaviour in two time windows, first between time 0 and T and second after a large time when in the Fisher–Wright model the rare mutants succeed, respectively, in the branching random walk the particle population reaches a positive spatial intensity. It is shown that asymptotically as N → ∞ the second phase for both models sets in after time α− 1logN, if N is the size of geographic space and N − 1 the rare mutation rate and α ∈ (0, ∞) depends on the other parameters. We identify the limit dynamics as N → ∞ in both time windows and for both models as a nonlinear Markov dynamic (McKean–Vlasov dynamic), respectively, a corresponding random entrance law from time − ∞ of this dynamic. Finally, we explain that the two processes are just two sides of the very same coin, a fact arising from a new form of duality, in particular the particle model generates the genealogy of the Fisher–Wright diffusions with selection and mutation. We discuss the extension of this duality in relation to a multitype model with more than two types.
影响因子：
2
作者：
A. Greven;P. Pfaffelhuber;A. Winter
通讯作者：
A. Winter