The development of deep-ocean anoxia in a comprehensive ocean phosphorus model.

We analyse a model of the phosphorus cycle in the ocean given by Slomp and Van Cappellen (Biogeosciences 4:155-171, 2007. 10.5194/bg-4-155-2007). This model contains four distinct oceanic boxes and includes relevant parts of the water, carbon and oxygen cycles. We show that the model can essentially be solved analytically, and its behaviour completely understood without recourse to numerical methods. In particular, we show that, in the model, the carbon and phosphorus concentrations in the different ocean reservoirs are all slaved to the concentration of soluble reactive phosphorus in the deep ocean, which relaxes to an equilibrium on a time scale of 180,000 y, and we show that the deep ocean is either oxic or anoxic, depending on a critical parameter which we can determine explicitly. Finally, we examine how the value of this critical parameter depends on the physical parameters contained in the model. The presented methodology is based on tools from applied mathematics and can be used to reduce the complexity of other large, biogeochemical models. Supplementary Information: The online version contains supplementary material available at 10.1007/s13137-023-00221-0.

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 [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.

Quasi-steady uptake and bacterial community assembly in a mathematical model of soil-phosphorus mobility.

We mathematically model the uptake of phosphorus by a soil community consisting of a plant and two bacterial groups: copiotrophs and oligotrophs. Four equilibrium states emerge, one for each of the species monopolising the resource and dominating the community and one with coexistence of all species. We show that the dynamics are controlled by the ratio of chemical adsorption to bacterial death permitting either oscillatory states or quasi-steady uptake. We show how a steady state can emerge which has soil and plant nutrient content unresponsive to increased fertilization. However, the additional fertilization supports the copiotrophs leading to community reassembly. Our results demonstrate the importance of time-series measurements in nutrient uptake experiments.

Regularization of the Ostwald supersaturation model for Liesegang bands.

In a previous paper, we analysed the Keller-Rubinow formulation of Ostwald's supersaturation theory for the formation of Liesegang rings or Liesegang bands, and found that the model is ill-posed, in the sense that after the termination of the first crystal front growth, secondary bands form, as in the experiment, but these are numerically found to be a single grid space wide, and thus an artefact of the numerical method. This ill-posedness is due to the discontinuity in the crystal growth rate, which itself reflects the supersaturation threshold inherent in the theory. Here we show that the ill-posedness can be resolved by the inclusion of a relaxation mechanism describing an impurity coverage fraction, which physically enables the transition in heterogeneous nucleation from precipitate-free impurity to precipitate-covered impurity.

Phase Transition in the Boltzmann-Vlasov Equation.

In this paper we revisit the problem of explaining phase transition by a study of a form of the Boltzmann equation, where inter-molecular attraction is included by means of a Vlasov term in the evolution equation for the one particle distribution function. We are able to show that for typical gas densities, a uniform state is unstable if the inter-molecular attraction is large enough. Our analysis relies strongly on the assumption, essential to the derivation of the Boltzmann equation, that ν ≪ 1 , where ν = d / l is the ratio of the molecular diameter to the mean inter-particle distance; in this case, for fluctuations on the scale of the molecular spacing, the collision term is small, and an explicit approximate solution is possible. We give reasons why we think the resulting approximation is valid, and in conclusion offer some possibilities for extension of the results to finite amplitude.

Correction to: The Dynamics of Ascaris lumbricoides Infections.

In the original article, the second author's name was incorrect in the metadata. The given name is T. Déirdre, and the family name is Hollingsworth.

Microbial dormancy and boom-and-bust population dynamics under starvation stress.

We propose a model for the growth of microbial populations in the presence of a rate-limiting nutrient which accounts for the switching of cells to a dormant phase at low densities in response to decreasing concentration of a putative biochemical signal. We then show that in conditions of nutrient starvation, self-sustained oscillations can occur, thus providing a natural explanation for such phenomena as plankton blooms. However, unlike results of previous studies, the microbial population minima do not become unrealistically small, being buffered during minima by an increased dormant phase population. We also show that this allows microbes to survive in extreme environments for very long periods, consistent with observation. The mechanism provides a natural vehicle for other such sporadic outbreaks, such as viral epidemics.

On the Keller-Rubinow model for Liesegang ring formation.

We study the model of Keller & Rubinow (Keller & Rubinow 1981 J. Chem. Phys74, 5000-5007. (doi:10.1063/1.441752)) describing the formation of Liesegang rings due to Ostwald's supersaturation mechanism. Keller and Rubinow provided an approximate solution both for the growth and equilibration of the first band, and also for the formation of secondary bands, based on a presumed asymptotic limit. However, they did not provide a parametric basis for the assumptions in their solution, nor did they provide any numerical corroboration, particularly of the secondary band formation. Here, we provide a different asymptotic solution, based on a specific parametric limit, and we show that the growth and subsequent cessation of the first band can be explained. We also show that the model is unable to explain the formation of finite width secondary bands, and we confirm this result by numerical computation. We conclude that the model is not fully posed, lacking a transition variable which can describe the hysteretic switch across the nucleation threshold.

Numerical simulations of drumlin formation.

We summarize the present form of the instability theory for drumlin formation, which describes the coupled subglacial flow of ice, water and sediment. This model has evolved over the last 20 years, and is now at the point where it can predict instabilities corresponding to ribbed moraine, drumlins and mega-scale glacial lineations, but efforts to provide numerical solutions of the model have been limited. The present summary adds some slight nuances to previously published versions of the theory, notably concerning the constitutive description of the subglacial water film and its flow. A new numerical method is devised to solve the model, and we show that it can be solved for realistic values of most of the parameters, with the exception of that corresponding to the water film thickness. We show that evolved bedforms can be three-dimensional and of the correct sizes, and we explore to some extent the variation of the solutions with the model's parameters.

A theoretical explanation of grain size distributions in explosive rock fragmentation.

We have measured grain size distributions of the results of laboratory decompression explosions of volcanic rock. The resulting distributions can be approximately represented by gamma distributions of weight per cent as a function of [Formula: see text], where d is the grain size in millimetres measured by sieving, with a superimposed long tail associated with the production of fines. We provide a description of the observations based on sequential fragmentation theory, which we develop for the particular case of 'self-similar' fragmentation kernels, and we show that the corresponding evolution equation for the distribution can be explicitly solved, yielding the long-time lognormal distribution associated with Kolmogorov's fragmentation theory. Particular features of the experimental data, notably time evolution, advection, truncation and fines production, are described and predicted within the constraints of a generalized, 'reductive' fragmentation model, and it is shown that the gamma distribution of coarse particles is a natural consequence of an assumed uniform fragmentation kernel. We further show that an explicit model for fines production during fracturing can lead to a second gamma distribution, and that the sum of the two provides a good fit to the observed data.

The development of biofilm architecture.

We extend the one-dimensional polymer solution theory of bacterial biofilm growth described by Winstanley et al. (2011 Proc. R. Soc. A467, 1449-1467 (doi:10.1098/rspa.2010.0327)) to deal with the problem of the growth of a patch of biofilm in more than one lateral dimension. The extension is non-trivial, as it requires consideration of the rheology of the polymer phase. We use a novel asymptotic technique to reduce the model to a free-boundary problem governed by the equations of Stokes flow with non-standard boundary conditions. We then consider the stability of laterally uniform biofilm growth, and show that the model predicts spatial instability; this is confirmed by a direct numerical solution of the governing equations. The instability results in cusp formation at the biofilm surface and provides an explanation for the common observation of patterned biofilm architectures.

The Dynamics of Ascaris lumbricoides Infections.

The Anderson-May model of human parasite infections and specifically that for the intestinal worm Ascaris lumbricoides is reconsidered, with a view to deriving the observed characteristic negative binomial distribution which is frequently found in human communities. The means to obtaining this result lies in reformulating the continuous Anderson-May model as a stochastic process involving two essential populations, the density of mature worms in the gut, and the density of mature eggs in the environment. The resulting partial differential equation for the generating function of the joint probability distribution of eggs and worms can be partially solved in the appropriate limit where the worm lifetime is much greater than that of the mature eggs in the environment. Allowing for a mean field nonlinearity, and for egg immigration from neighbouring communities, a negative binomial worm distribution can be predicted, whose parameters are determined by those in the continuous Anderson-May model; this result assumes no variability in predisposition to the infection.

Simple Approximations for Epidemics with Exponential and Fixed Infectious Periods.

Analytical approximations have generated many insights into the dynamics of epidemics, but there is only one well-known approximation which describes the dynamics of the whole epidemic. In addition, most of the well-known approximations for different aspects of the dynamics are for the classic susceptible-infected-recovered model, in which the infectious period is exponentially distributed. Whilst this assumption is useful, it is somewhat unrealistic. Equally reasonable assumptions are that the infectious period is finite and fixed or that there is a distribution of infectious periods centred round a nonzero mean. We investigate the effect of these different assumptions on the dynamics of the epidemic by deriving approximations to the whole epidemic curve. We show how the well-known sech-squared approximation for the infective population in 'weak' epidemics (where the basic reproduction rate R0 ≈ 1) can be extended to the case of an arbitrary distribution of infectious periods having finite second moment, including as examples fixed and gamma-distributed infectious periods. Further, we show how to approximate the time course of a 'strong' epidemic, where R0 ≫ 1, demonstrating the importance of estimating the infectious period distribution early in an epidemic.

A 2D channel-clogging biofilm model.

We present a model of biofilm growth in a long channel where the biomass is assumed to have the rheology of a viscous polymer solution. We examine the competition between growth and erosion-like surface detachment due to the flow. A particular focus of our investigation is the effect of the biofilm growth on the fluid flow in the pores, and the issue of whether biomass can grow sufficiently to shut off fluid flow through the pores, thus clogging the pore space. Net biofilm growth is coupled along the pore length via flow rate and nutrient transport in the pore flow. Our 2D model extends existing results on stability of 1D steady state biofilm thicknesses to show that, in the case of flows driven by a fixed pressure drop, full clogging of the pore can indeed happen in certain cases dependent on the functional form of the detachment term.

An instability theory for the formation of ribbed moraine, drumlins and mega-scale glacial lineations.

We present a theory for the coupled flow of ice, subglacial water and subglacial sediment, which is designed to represent the processes which occur at the bed of an ice sheet. The ice is assumed to flow as a Newtonian viscous fluid, the water can flow between the till and the ice as a thin film, which may thicken to form streams or cavities, and the till is assumed to be transported, either through shearing by the ice, squeezing by pressure gradients in the till, or by fluvial sediment transport processes in streams or cavities. In previous studies, it was shown that the dependence of ice sliding velocity on effective pressure provided a mechanism for the generation of bedforms resembling ribbed moraine, while the dependence of fluvial sediment transport on water film depth provides a mechanism for the generation of bedforms resembling mega-scale glacial lineations. Here, we combine these two processes in a single model, and show that, depending largely on the granulometry of the till, instability can occur in a range of types which range from ribbed moraine through three-dimensional drumlins to mega-scale glacial lineations.

Subglacial swamps.

The existence of both water and sediment at the bed of ice streams is well documented, but there is a lack of fundamental understanding about the mechanisms of ice, water and sediment interaction. We pose a model to describe subglacial water flow below ice sheets, in the presence of a deformable sediment layer. Water flows in a rough-bedded film; the ice is supported by larger clasts, but there is a millimetric water layer submerging the smaller particles. Partial differential equations describing the water film are derived from a description of the dynamics of ice, water and mobile sediment. We assume that sediment transport is possible, either as fluvial bedload, but more significantly by ice-driven shearing and by internal squeezing. This provides an instability mechanism for rivulet formation; in the model, downstream sediment transport is compensated by lateral squeezing of till towards the incipient streams. We show that the model predicts the formation of shallow, swamp-like streams, with a typical depth of the order of centimetres. The swamps are stable features, typically with a width of the order of tens to hundreds of metres.

Modelling spatial oscillations in soil borehole bacteria.

Spatial oscillations in groundwater contaminant concentrations can be successfully explained by consideration of a competitive microbial community in conditions of poor nutrient supply, in which the effects of spatial diffusion of the nutrient sources are included. In previous work we showed that the microbial competition itself allowed oscillations to occur, and, in common with other reaction-diffusion systems, the addition of spatial diffusion transforms these temporal oscillations into travelling waves, sometimes chaotic. We therefore suggest that irregular chemical profiles sometimes found in contaminant plume borehole profiles may be a consequence of this competition.

Subglacial hydrology and the formation of ice streams.

Antarctic ice streams are associated with pressurized subglacial meltwater but the role this water plays in the dynamics of the streams is not known. To address this, we present a model of subglacial water flow below ice sheets, and particularly below ice streams. The base-level flow is fed by subglacial melting and is presumed to take the form of a rough-bedded film, in which the ice is supported by larger clasts, but there is a millimetric water film which submerges the smaller particles. A model for the film is given by two coupled partial differential equations, representing mass conservation of water and ice closure. We assume that there is no sediment transport and solve for water film depth and effective pressure. This is coupled to a vertically integrated, higher order model for ice-sheet dynamics. If there is a sufficiently small amount of meltwater produced (e.g. if ice flux is low), the distributed film and ice sheet are stable, whereas for larger amounts of melt the ice-water system can become unstable, and ice streams form spontaneously as a consequence. We show that this can be explained in terms of a multi-valued sliding law, which arises from a simplified, one-dimensional analysis of the coupled model.

Oscillations in soil bacterial redox reactions.

Spatial oscillations in soil contaminant concentration profiles are sometimes observed, but rarely commented on, or are attributed to noisy data. In this paper we consider a possible mechanism for the occurrence of oscillatory reactant profiles within contaminant plumes. The bioremediative reactions which occur are effected by bacteria, whose rôle is normally conceived of as being passive. Here we argue that competition, for example between heterotrophic and fermentative bacteria, can occur in the form of an activator-inhibitor system, thus promoting oscillations. We describe a simple model for the competition between two such microbial populations, and we show that in normal oligotrophic groundwater conditions, oscillatory behaviour is easily obtained. When such competition occurs in a dispersive porous medium, travelling waves can be generated, which provide a possible explanation for the observed soil column oscillations.

Biomat development in soil treatment units for on-site wastewater treatment.

We provide a simple mathematical model of the bioremediation of contaminated wastewater leaching into the subsoil below a septic tank percolation system. The model comprises a description of the percolation system's flows, together with equations describing the growth of biomass and the uptake of an organic contaminant concentration. By first rendering the model dimensionless, it can be partially solved, to provide simple insights into the processes which control the efficacy of the system. In particular, we provide quantitative insight into the effect of a near surface biomat on subsoil permeability; this can lead to trench ponding, and thus propagation of effluent further down the trench. Using the computed vadose zone flow field, the model can be simply extended to include reactive transport of other contaminants of interest.