突发繁殖下随机种群中的稀有事件
Rare events in stochastic populations under bursty reproduction
摘要：
最近，第一步是由作者对反应步长噪声的影响进行了系统的调查-不确定性的反应步长-随机种群的动态。这是通过调查突发流入的随机种群的开关动力学的影响。在这里，我们扩展这种形式主义占突发再现过程，并提高准确性的形式主义，包括subleading顺序校正。爆发性繁殖出现在各种情况下，其中值得注意的例子包括从受感染的细胞产生爆发性病毒，以及涉及不同数量后代的哺乳动物的繁殖。我们定量研究的主要问题是，爆发性繁殖如何影响人口的总体命运。我们考虑两个互补的情况：人口灭绝和人口生存;在前者的人口灭绝后，保持一个长期的亚稳态，而在后者的人口激增，尽管经历了一个确定性的漂移走向灭绝。在这两种模型中，繁殖都是从任意分布中抽样的突发性繁殖。在灭绝问题中，我们发现，爆发性繁殖扩大了准平稳分布的人口规模的亚稳态，这导致在指数下降的平均灭绝时间。在生存问题中，爆发性繁殖使种群的生存概率呈指数增长。接近分歧的限制，我们的分析结果大大简化，并仅取决于平均值和方差的突发大小分布。我们的形式主义证明了几个现实的分布，所有比较以及与数值蒙特卡罗模拟。
英文摘要：
Recently, a first step was made by the authors towards a systematic investigation of the effect of reaction-step-size noise - uncertainty in the step size of the reaction - on the dynamics of stochastic populations. This was done by investigating the effect of bursty influx on the switching dynamics of stochastic populations. Here we extend this formalism to account for bursty reproduction processes, and improve the accuracy of the formalism to include subleading-order corrections. Bursty reproduction appears in various contexts, where notable examples include bursty viral production from infected cells, and reproduction of mammals involving varying number of offspring. The main question we quantitatively address is how bursty reproduction affects the overall fate of the population. We consider two complementary scenarios: population extinction and population survival; in the former a population gets extinct after maintaining a long-lived metastable state, whereas in the latter a population proliferates despite undergoing a deterministic drift towards extinction. In both models reproduction occurs in bursts, sampled from an arbitrary distribution. In the extinction problem, we show that bursty reproduction broadens the quasi-stationary distribution of population sizes in the metastable state, which results in an exponential decrease of the mean time to extinction. In the survival problem, bursty reproduction yields an exponential increase in survival probability of the population. Close to the bifurcation limit our analytical results simplify considerably and are shown to depend solely on the mean and variance of the burst-size distribution. Our formalism is demonstrated on several realistic distributions which all compare well with numerical Monte-Carlo simulations.
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