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 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.