Atto-Foxes and Other Minutiae

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 10(-18) (atto-) per square kilometre. The second is howthe 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 fewindividuals 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.

Automated summary

This paper addresses the issue of extinction in continuous population dynamics models with small numbers of individuals. It discusses extinction in the context of a stochastic logistic model, provides examples of 'small number dynamics,' such as the 'atto-fox' and 'frogspawn' problems, and proposes modeling solutions to address these issues.