MACSI, University of Limerick, Limerick, Ireland.
OCIAM, University of Oxford, Oxford, UK.
Bull Math Biol. 2021 Aug 31;83(10):104. doi: 10.1007/s11538-021-00936-x.
This paper addresses the problem of extinction in continuous models of population dynamics associated with small numbers of individuals. We begin with an extended discussion of extinction in the particular case of a stochastic logistic model, and how it relates to the corresponding continuous model. Two examples of 'small number dynamics' are then considered. The first is what Mollison calls the 'atto-fox' problem (in a model of fox rabies), referring to the problematic theoretical occurrence of a predicted rabid fox density of [Formula: see text] (atto-) per square kilometre. The second is how the production of large numbers of eggs by an individual can reliably lead to the eventual survival of a handful of adults, as it would seem that extinction then becomes a likely possibility. We describe the occurrence of the atto-fox problem in other contexts, such as the microbial 'yocto-cell' problem, and we suggest that the modelling resolution is to allow for the existence of a reservoir for the extinctively challenged individuals. This is functionally similar to the concept of a 'refuge' in predator-prey systems and represents a state for the individuals in which they are immune from destruction. For what I call the 'frogspawn' problem, where only a few individuals survive to adulthood from a large number of eggs, we provide a simple explanation based on a Holling type 3 response and elaborate it by means of a suitable nonlinear age-structured model.
本文解决了与个体数量较少相关的连续人口动态模型中的灭绝问题。我们首先对随机 logistic 模型中灭绝的特殊情况进行了扩展讨论，以及它与相应的连续模型的关系。然后考虑了两个“小种群动力学”的例子。第一个是 Mollison 所谓的“atto-fox”问题（在狐狸狂犬病模型中），指的是理论上预测的狂犬病狐狸密度为[公式：见文本]（atto-）每平方公里的问题。第二个是个体产生大量卵子如何可靠地导致少数成虫最终存活，因为灭绝似乎成为了一种可能。我们描述了 atto-fox 问题在其他情况下的发生，例如微生物“yocto-cell”问题，我们建议允许灭绝挑战个体存在一个储层来解决模型问题。这在功能上类似于捕食者-猎物系统中的“避难所”概念，代表了个体的一种状态，使它们免受破坏。对于我所谓的“frogspawn”问题，即大量卵子中只有少数个体存活到成年，我们提供了一个基于 Holling 类型 3 反应的简单解释，并通过一个合适的非线性年龄结构模型进行了详细说明。
