Atto-Foxes and Other Minutiae

Abstract

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 $$10^{-18}$$ 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.

Special Collection: Celebrating J.D. Murray’s Contributions to Mathematical Biology

“…It is shown that the model is amenable to a multiple timescale analysis and the contribution of key cell processes to plaque lipid accumulation is explored in detail. Fowler (2021) extends the aforementioned work of Murray et al (1986) on rabid foxes to address the issue of extinction, which is a general problem faced by con-tinuous models of population dynamics, whose applicability is called into question at low numbers of individuals. A number of ways to address/resolve these problems is presented in this paper, and illustrated through application to foxes approaching extinction; oscillatory dynamics with extremely small minima, explored in the context of immune dynamics and microbial growth models; and finally frogspawn, where an age-structured model is proposed and analysed.…”

Permanence via invasion graphs: incorporating community assembly into modern coexistence theory

“…Specifically, these are communities where (i) species i is missing, and (ii) all other missing species have a negative invasion growth rate. communities can be found, approximately (see Remark 4 ), by simulating initial conditions supporting all species but species i for a sufficiently long time, removing “atto-foxes” (Sari and Lobry 2015 ; Fowler 2021 ), and seeing what species are left. This characterization ensures that each species i has at least one community associated with it.…”

A Rose by any Other Name: Towards a quantitative theory of neuroimmune interactions

“…This cut-off mechanism extends the notion of network activity being proportional to the concentration of population present, but prevents the existence of an exponentially-small population that suddenly resurges later in time. In population dynamics, this is sometimes referred to as the 'atto-fox problem' (Murray et al 1986;Mollison, 1991;Lobry and Sari, 2015), in which an infeasibly small population persists and can resurge to its original population size; an equivalent 'yocto-cell problem' (Fowler, 2021) has been coined for mathematical models at the cellular scale. Similar cut-off mechanisms could be incorporated for the immune system and feedback signal, but are omitted for brevity and simplicity.…”