# Professor Anne Skeldon

## Academic and research departments

Department of Mathematics, Centre for Mathematical and Computational Biology.## About

### Biography

I am a mathematician with a keen interest in applying mathematics in the real world. My background is in dynamical systems and bifurcation theory.

Much of my early work was in pattern formation and fluid mechanics, particularly the Faraday wave problem. More recently I have shifted towards biology/sociology. I was a co-investigator on the 6 year complexity science project, Evolution and Resilience of Industrial Ecosystems. I am part of the Mathematics of Life and Social Sciences research group. I co-lead the cross-faculty Centre for Mathematical and Computational Biology

My current research interests are largely centred on sleep and circadian rhythms and data science. I collaborate with researchers at the Surrey Sleep Research Centre developing and analysing mathematical models of sleep-wake regulation. Our work recently featured in the UK parliamentary debate ``School should start at 10am because teenagers are too tired''. I have a particular interest in the impact of our light environment on sleep, including the impact of permanent DST.

## Research

## Supervision

### Postgraduate research supervision

Please contact me if you are interested in PhD projects. My current PhD students are:

- Rachel Bernasconi
- Daniel Knott
- Elaheh Kalantari (Lead supervisor: Samaneh Kouchaki, other co-supervisor Derk-Jan Dijk)

### Completed postgraduate research projects I have supervised

- Danielle Kurtin
- Imran Usmani
- Michelle Grant
- Matthew Bailey
- James Allen
- Gary Chaffey
- Anna Chuter
- Naowarut Meejun
- Nicholas Robertson
- Diep Ho

## Teaching

In 2023/24 I am teaching

- MAT3046 Game Theory with Applications in Biology and Economics
- MAT3040 Mathematical Ecology and Epidemiology

Recently I have also taught

- MAT3043 Graphs and Networks

## Publications

Changes in our climate and environment make it ever more important to understand the processes involved in Earth systems, such as the carbon cycle. There are many models that attempt to describe and predict the behaviour of carbon stocks and stores but, despite their complexity, significant uncertainties remain. We consider the qualitative behaviour of one of the simplest carbon cycle models, the Data Assimilation Linked Ecosystem Carbon (DALEC) model, which is a simple vegetation model of processes involved in the carbon cycle of forests, and consider in detail the dynamical structure of the model. Our analysis shows that the dynamics of both evergreen and deciduous forests in DALEC are dependent on a few key parameters and it is possible to find a limit point where there is stable sustainable behaviour on one side but unsustainable conditions on the other side. The fact that typical parameter values reside close to this limit point highlights the difficulty of predicting even the correct trend without sufficient data and has implications for the use of data assimilation methods.

Tumours that are low in oxygen (hypoxic) tend to be more aggressive and respond less well to treatment. Knowing the spatial distribution of oxygen within a tumour could therefore play an important role in treatment planning, enabling treatment to be targeted in such a way that higher doses of radiation are given to the more radioresistant tissue. Mapping the spatial distribution of oxygen in vivo is difficult. Radioactive tracers that are sensitive to different levels of oxygen are under development and in the early stages of clinical use. The concentration of these tracer chemicals can be detected via positron emission tomography resulting in a time dependent concentration profile known as a tissue activity curve (TAC). Pharmaco-kinetic models have then been used to deduce oxygen concentration from TACs. Some such models have included the fact that the spatial distribution of oxygen is often highly inhomogeneous and some have not. We show that the oxygen distribution has little impact on the form of a TAC; it is only the mean oxygen concentration that matters. This has significant consequences both in terms of the computational power needed, and in the amount of information that can be deduced from TACs.

Three-wave interactions form the basis of our understanding of many pattern forming systems because they encapsulate the most basic nonlinear interactions. In problems with two comparable length scales, it is possible for two waves of the shorter wavelength to interact with one wave of the longer, as well as for two waves of the longer wavelength to interact with one wave of the shorter. Consideration of both types of three-wave interactions can generically explain the presence of complex patterns and spatio-temporal chaos. Two length scales arise naturally in the Faraday wave experiment with multi-frequency forcing, and our results enable some previously unexplained experimental observations of spatio-temporal chaos to be interpreted in a new light. Our predictions are illustrated with numerical simulations of a model partial differential equation.

When a layer of fluid is oscillated up and down with a sufficiently large amplitude, patterns form on the surface, a phenomenon first observed by Faraday. A wide variety of such patterns have been observed from regular squares and hexagons to superlattice and quasipatterns and more exotic patterns such as oscillons. Previous work has investigated the mechanisms of pattern selection using the tools of symmetry and bifurcation theory. The hypotheses produced by these generic arguments have been tested against an equation derived by Zhang and Vinals in the weakly viscous and large depth limit. However, in contrast, many of the experiments use shallow viscous layers of fluid to counteract the presence of high frequency weakly damped modes that can make patterns hard to observe. Here we develop a weakly nonlinear analysis of the full Navier-Stokes equations for the two-frequency excitation Faraday experiment. The problem is formulated for general depth, although results are presented only for the infinite depth limit. We focus on a few particular cases where detailed experimental results exist and compare our analytical results with the experimental observations. Good agreement with the experimental results is found.

Cities are constantly evolving and so are the living conditions within and between them. Rapid urbanization and the ever-growing need for housing have turned large areas of many cities into concrete landscapes that lack greenery. Green infrastructure can support human health, provide socio-economic and environmental benefits, and bring color to an otherwise grey urban landscape. Sometimes, benefits come with downsides in relation to its impact on air quality and human health, requiring suitable data and guidelines to implement effective greening strategies. Air pollution and human health, as well as green infrastructure and human health, are often studied together. Linking green infrastructure with air quality and human health together is a unique aspect of this article. A holistic understanding of these links is key to enabling policymakers and urban planners to make informed decisions. By critically evaluating the link between green infrastructure and human health via air pollution mitigation, we also discuss if our existing understanding of such interventions is enabling their uptake in practice. Both the natural science and epidemiology approach the topic of green infrastructure and human health very differently. The pathways linking health benefits to pollution reduction by urban vegetation remain unclear and that the mode of green infrastructure deployment is critical to avoid unintended consequences. Strategic deployment of green infrastructure may reduce downwind pollution exposure. However, the development of bespoke design guidelines is vital to promote and optimize greening benefits and measuring green infrastructure’s socio-economic and health benefits are key for their uptake. Greening cities to mitigate pollution effects is on the rise and these needs to be matched by scientific evidence and appropriate guidelines. We conclude that urban vegetation can facilitate broad health benefits, but there is little empirical evidence linking these benefits to air pollution reduction by urban vegetation, and appreciable efforts are needed to establish the underlying policies, design and engineering guidelines governing its deployment.

Optimising policy choices to steer social/economic systems efficiently towards desirable outcomes is challenging. The inter-dependent nature of many elements of society and the economy means that policies designed to promote one particular aspect often have secondary, unintended, effects. In order to make rational decisions, methodologies and tools to assist the development of intuition in this complex world are needed. One approach is the use of agent-based models. These have the ability to capture essential features and interactions and predict outcomes in a way that is not readily achievable through either equations or words alone. In this paper we illustrate how agent-based models can be used in a policy setting by using an example drawn from the biowaste industry. This example describes the growth of in-vessel composting and anaerobic digestion to reduce food waste going to landfill in response to policies in the form of taxes and financial incentives. The fundamentally dynamic nature of an agent-based modelling approach is used to demonstrate that policy outcomes depend not just on current policy levels but also on the historical path taken.

Recent experiments (Kudrolli, Pier and Gollub, 1998) on two-frequency parametrically excited surface waves exhibit an intriguing "superlattice" wave pattern near a codimension-two bifurcation point where both subharmonic and harmonic waves onset simultaneously, but with different spatial wavenumbers. The superlattice pattern is synchronous with the forcing, spatially periodic on a large hexagonal lattice, and exhibits small-scale triangular structure. Similar patterns have been shown to exist as primary solution branches of a generic 12-dimensional $D_6____dot{+}T^2$-equivariant bifurcation problem, and may be stable if the nonlinear coefficients of the bifurcation problem satisfy certain inequalities (Silber and Proctor, 1998). Here we use the spatial and temporal symmetries of the problem to argue that weakly damped harmonic waves may be critical to understanding the stabilization of this pattern in the Faraday system. We illustrate this mechanism by considering the equations developed by Zhang and Vinals (1997, J. Fluid Mech. 336) for small amplitude, weakly damped surface waves on a semi-infinite fluid layer. We compute the relevant nonlinear coefficients in the bifurcation equations describing the onset of patterns for excitation frequency ratios of 2/3 and 6/7. For the 2/3 case, we show that there is a fundamental difference in the pattern selection problems for subharmonic and harmonic instabilities near the codimension-two point. Also, we find that the 6/7 case is significantly different from the 2/3 case due to the presence of additional weakly damped harmonic modes. These additional harmonic modes can result in a stabilization of the superpatterns.

Agent based models (ABM)s are increasingly used in social science, economics, mathematics, biology and computer science to describe time dependent systems in circumstances where a description in terms of equations is difficult. Yet few tools are currently available for the systematic analysis of ABM behaviour. Numerical continuation and bifurcation analysis is a well established tool for the study of deterministic systems. Recently, equation-free (EF) methods have been developed to extend numerical continuation techniques to systems where the dynamics are described at a microscopic scale and continuation of a macroscopic property of the system is considered. To date, the practical use of EF methods has been limited by; 1) the over-head of application-specific implementation; 2) the laborious configuration of problem-specific parameters; and 3) large ensemble sizes (potentially) leading to computationally restrictive run-times. In this paper we address these issues with our tool for the EF continuation of stochastic systems, which includes algorithms to systematically configuration problem specific parameters and enhance robustness to noise. Our tool is generic and can be applied to any `black-box' simulator and determine the essential EF parameters prior to EF analysis. Robustness is significantly improved using our convergence-constraint with a corrector-repeat method (C3R) method. This algorithm automatically detects outliers based on the dynamics of the underlying system enabling both an order of magnitude reduction in ensemble size and continuation of systems at much higher levels of noise than classical approaches. We demonstrate our method with application to several ABM models, revealing parameter dependence, bifurcation and stability analysis of these complex systems giving a deep understanding of the dynamical behaviour of the models in a way that is not otherwise easily obtainable. In each case we demonstrate our systematic parameter determination stage for configuring the system specific EF parameters.

Adequate synchronisation of endogenous circadian rhythms to external time is beneficial for human health [1]. But how circadian time (biological time) and the numbers on the clock (clock time) are related is tricky to understand, as many of us experience when we change from standard time (ST) to daylight saving time (DST) and during jet-lag. How confused we can be is also exemplified by two bills currently making their way through the Californian state legislature. Senate Bill SB-328 Pupil Attendance: School Start Time [2] prohibits middle and high schools from starting earlier than 8:30 in the morning. Senate Bill AB-807 Daylight Saving Time [3] would result in a switch to permanent DST. Similar debates on school start times and DST are happening throughout North America and Europe. Here we explain why a switch to permanent DST could negate any beneficial effects of delaying school start times.

Numerous empirical studies show that when people play social dilemma games in the laboratory they often cooperate conditionally, and the frequency of conditional cooperators differs between communities. However, this has not yet been fully explained by social dilemma models in structured populations. Here we model a population as a two-layer multiplex network, where the two layers represent economic and social interactions respectively. Players play a conditional public goods game on the economic layer, their donations to the public good dependent on the donations of their neighbours, and player strategies evolve through a combination of payoff comparison and social influence. We find that both conditional cooperation and social influence lead to increased cooperation in the public goods game, with social influence being the dominant factor. Cooperation is more prevalent both because conditional cooperators are less easily exploited by free-riders than unconditional cooperators, and also because social influence tends to preserve strategies over time. Interestingly the choice of social imitation rule does not appear to be important: it is rather the separation of strategy imitation from payoff comparison that matters. Our results highlight the importance of social influence in maintaining cooperative behaviour across populations, and suggest that social behaviour is more important than economic incentives for the maintenance of cooperation.

We investigate how the dynamics of a mathematical model of a nephron depend on the precise form of the delay in the tubuloglomerular feed- back loop. Although qualitative behavioral similarities emerge for di®erent orders of delay, we ¯nd that signi¯cant quantitative di®erences occur. With- out more knowledge of the form of the delay, this places restrictions on how reasonable it is to expect close quantitative agreement between the mathemat- ical model and experimental data.

We consider the symmetry-breaking steady state bifurcation of a spatially-uniform equilibrium solution of E(2)-equivariant PDEs. We restrict the space of solutions to those that are doubly-periodic with respect to a square or hexagonal lattice, and consider the bifurcation problem restricted to a finite-dimensional center manifold. For the square lattice we assume that the kernel of the linear operator, at the bifurcation point, consists of 4 complex Fourier modes, with wave vectors K_1=(a,b), K_2=(-b,a), K_3=(b,a), and K_4=(-a,b), where a>b>0 are integers. For the hexagonal lattice, we assume that the kernel of the linear operator consists of 6 complex Fourier modes, also parameterized by an integer pair (a,b). We derive normal forms for the bifurcation problems, which we use to compute the linear, orbital stability of those solution branches guaranteed to exist by the equivariant branching lemma. These solutions consist of rolls, squares, hexagons, a countable set of rhombs, and a countable set of planforms that are superpositions of all of the Fourier modes in the kernel. Since rolls and squares (hexagons) are common to all of the bifurcation problems posed on square (hexagonal) lattices, this framework can be used to determine their stability relative to a countable set of perturbations by varying a and b. For the hexagonal lattice, we analyze the degenerate bifurcation problem obtained by setting the coefficient of the quadratic term to zero. The unfolding of the degenerate bifurcation problem reveals a new class of secondary bifurcations on the hexagons and rhombs solution branches.

For more than thirty years the `two process model' has played a central role in the under- standing of sleep/wake regulation. This ostensibly simple model is an interesting example of a nonsmooth dynamical system whose rich dynamical structure has been relatively un- explored. The two process model can be framed as a one-dimensional map of the circle which, for some parameter regimes, has gaps. We show how border collision bifurcations that arise naturally in maps with gaps extend and supplement the Arnold tongue saddle- node bifurcation set that is a feature of continuous circle maps. The novel picture that results shows how the periodic solutions that are created by saddle-node bifurcations in continuous maps transition to periodic solutions created by period-adding bifurcations as seen in maps with gaps.

Self-organization of matter is essential for natural pattern formation, chemical synthesis, as well as modern material science. Here we show that isovolumetric reactions of a single organometallic precursor allow symmetry breaking events from iron nuclei to the creation of different symmetric carbon structures: microspheres, nanotubes, and mirrored spiraling microcones. A mathematical model, based on mass conservation and chemical composition, quantitatively explains the shape growth. The genesis of such could have significant implications for material design.

In part I, the Two-Process model for sleep-wake regulation was discussed and it was shown that it could usefully be represented as a one-dimensional map with discontinuities. Here we discuss some recent, more physiological, models of sleep wake dynamics. We describe how their fast-slow structure means that one can expect them to inherit many of the dynamical features of the Two-Process model.

Sleep is essential for the maintenance of human life, yet many features of sleep are poorly understood and mathematical models are an important tool for probing proposed biological mechanisms. The most well-known mathematical model of sleep regulation, the two-process model, models the sleep-wake cycle by two oscillators: a circadian oscillator and a homeostatic oscillator. An alternative, more recent, model considers the reciprocal interaction of sleep promoting neurons and the ascending arousal system regulated by homeostatic and circadian processes. Here we show there are fundamental similarities between these two models. The implications are illustrated with two important sleep-wake phenomena. Firstly, we show that in the two-process model, transitions between different numbers of daily sleep episodes can be classified as grazing bifurcations. This provides the theoretical underpinning for numerical results showing that the sleep patterns of many mammals can be explained by the reciprocal interaction model. Secondly, we show that when sleep deprivation disrupts the sleep-wake cycle, ostensibly different measures of sleepiness in the two models are closely related. The demonstration of the mathematical similarities of the two models is important because not only does it it allow some features of the two-process model to be interpreted physiologically but it also means that knowledge gained from the study of the two-process model can be used to inform understanding of the behaviour of the mutual inhibition model. This is important because the mutual inhibition model and its extensions are increasingly being used as a tool to understand a diverse range of sleep-wake phenomena sucah as the design of optimal shift-patterns, yet the values it uses for the parameters associated with the circadian and homeostatic processes are very different from those that have been experimentally measured in the context of the two-process model

Sleep-wake regulation is an example of a system with multiple timescales, with switching between sleep and wake states occurring in minutes but the states of wake or sleep usually existing for some hours. Here we discuss some general features of models of sleep-wake regulation. We show that some typical models of sleep-wake regulation can be reduced to one-dimensional maps with discontinuities, and show that this reduction is useful in understanding some of the dynamical behaviour seen in sleep-wake models.

Motivated by experimental observations of exotic standing wave patterns in the two-frequency Faraday experiment, we investigate the role of normal form symmetries in the pattern selection problem. With forcing frequency components in ratio m/n, where m and n are co-prime integers, there is the possibility that both harmonic and subharmonic waves may lose stability simultaneously, each with a different wavenumber. We focus on this situation and compare the case where the harmonic waves have a longer wavelength than the subharmonic waves with the case where the harmonic waves have a shorter wavelength. We show that in the former case a normal form transformation can be used to remove all quadratic terms from the amplitude equations governing the relevant resonant triad interactions. Thus the role of resonant triads in the pattern selection problem is greatly diminished in this situation. We verify our general results within the example of one-dimensional surface wave solutions of the Zhang-Vinals model of the two-frequency Faraday problem. In one-dimension, a 1:2 spatial resonance takes the place of a resonant triad in our investigation. We find that when the bifurcating modes are in this spatial resonance, it dramatically effects the bifurcation to subharmonic waves in the case of forcing frequencies are in ratio 1/2; this is consistent with the results of Zhang and Vinals. In sharp contrast, we find that when the forcing frequencies are in ratio 2/3, the bifurcation to (sub)harmonic waves is insensitive to the presence of another spatially-resonant bifurcating mode.

Knowledge of how a population of cancerous cells progress through the cell cycle is vital if the population is to be treated effectively, as treatment outcome is dependent on the phase distributions of the population. Estimates on the phase distribution may be obtained experimentally however the errors present in these estimates may effect treatment efficacy and planning. If mathematical models are to be used to make accurate, quantitative predictions concerning treatments, whose efficacy is phase dependent, knowledge of the phase distribution is crucial. In this paper it is shown that two different transition rates at the G1-S checkpoint provide a good fit to a growth curve obtained experimentally. However, the different transition functions predict a different phase distribution for the population, but both lying within the bounds of experimental error. Since treatment outcome is effected by the phase distribution of the population this difference may be critical in treatment planning. Using an age-structured population balance approach the cell cycle is modelled with particular emphasis on the G1-S checkpoint. By considering the probability of cells transitioning at the G1-S checkpoint, different transition functions are obtained. A suitable finite difference scheme for the numerical simulation of the model is derived and shown to be stable. The model is then fitted using the different probability transition functions to experimental data and the effects of the different probability transition functions on the model's results are discussed.

In this paper we investigate the dynamics associated with a degenerate codimension-two Takens-Bogdanov bifurcation which arises in a recently derived model for self-exciting dynamo action introduced by Hide et al. [R. Hide, A.C. Skeldon, D.J. Acheson, A study of two novel self-exciting single-disk homopolar dynamos: theory, Proc. R. Soc. Lond. A 452 (1996) 1369-1395]. The general unfolding of such a codimension-three bifurcation has already been discussed in an abstract setting by Li and Rousseau [Codimension-2 symmetric homoclinic bifurcations and application to 1:2 resonance, Can J. Math. 42 (1990) 191-212].Here we describe the unfolding scenario in the context of the dynamo problem. In particular we compare the behaviour predicted by the normal form analysis with a bifurcation study of the full dynamo equations in the neighbourhood of the codimension-three point.

The Faraday problem is an important pattern-forming system that provides some middle ground between systems where the initial instability involves just a single mode and in which complexity then results from mode interactions or secondary bifurcations, and cases where a system is highly turbulent and many spatial and temporal modes are excited. It has been a rich source of novel patterns and of theoretical work aimed at understanding how and why such patterns occur. Yet it is particularly challenging to tie theory to experiment: the experiments are difficult to perform; the parameter regime of interest (large box, moderate viscosity) along with the technical difficulties of solving the free boundary Navier--Stokes equations make numerical solution of the problem hard; and the fact that the instabilities result in an entire circle of unstable wavevectors presents considerable theoretical difficulties. In principle, weakly nonlinear theory should be able to predict which patterns are stable near pattern onset. In this paper we present the first quantitative comparison between weakly nonlinear theory of the full Navier--Stokes equations and (previously published) experimental results for the Faraday problem with multiple frequency forcing. We confirm that three-wave interactions sit at the heart of why complex patterns are stablised but also highlight some discrepancies between theory and experiment. These suggest the need for further experimental and theoretical work to fully investigate the issues of pattern bistability and the role of bicritical/tricritical points in determining bifurcation structure.

Timing of the human sleep-wake cycle is determined by social constraints, biological processes (sleep homeostasis and circadian rhythmicity) and environmental factors, particularly natural and electrical light exposure. To what extent seasonal changes in the light-dark cycle affect sleep timing and how this varies between weekdays and weekends has not been firmly established. We examined sleep and activity patterns during weekdays and weekends in late autumn (standard time, ST) and late spring (daylight saving time, DST), and expressed their timing in relation to three environmental reference points: clock-time, solar noon (SN), which occurs one clock hour later during DST than ST, and the midpoint of accumulated light exposure (50%-LE). Observed sleep timing data were compared to simulated data from a mathematical model for the effects of light on the circadian and homeostatic regulation of sleep. A total of 715 days of sleep timing and light exposure were recorded in 19 undergraduates in a repeated-measures observational study. During each three-week assessment, light and activity were monitored, and self-reported bed and wake times were collected. Light exposure was higher in spring than in autumn. 50%-LE did not vary across season, but occurred later on weekends compared to weekdays. Relative to clock-time, bedtime, wake-time, mid-sleep, and midpoint of activity were later on weekends but did not differ across seasons. Relative to SN, sleep and activity measures were earlier in spring than in autumn. Relative to 50%-LE, only wake-time and mid-sleep were later on weekends, with no seasonal differences. Individual differences in mid-sleep did not correlate with SN but correlated with 50%-LE. Individuals with different habitual bedtimes responded similarly to seasonal changes. Model simulations showed that light exposure patterns are sufficient to explain sleep timing in spring but less so in autumn. The findings indicate that during autumn and spring, the timing of sleep associates with actual light exposure rather than sun time as indexed by SN.

We consider the transition from a spatially uniform state to a steady, spatially- periodic pattern in a partial differential equation describing long-wavelength convec- tion [1]. This both extends existing work on the study of rolls, squares and hexagons and demonstrates how recent generic results for the stability of spatially-periodic patterns may be applied in practice. We find that squares, even if stable to roll perturbations, are often unstable when a wider class of perturbations is considered. We also find scenarios where transitions from hexagons to rectangles can occur. In some cases we find that, near onset, more exotic spatially-periodic planforms are preferred over the usual rolls, squares and hexagons.

Why do we go to sleep late and struggle to wake up on time? Historically, light-dark cycles were dictated by the solar day, but now humans can extend light exposure by switching on artificial lights. We use a mathematical model incorporating effects of light, circadian rhythmicity and sleep homeostasis to provide a quantitative theoretical framework to understand effects of modern patterns of light consumption on the human circadian system. The model shows that without artificial light humans wake-up at dawn. Artificial light delays circadian rhythmicity and preferred sleep timing and compromises synchronisation to the solar day when wake-times are not enforced. When wake-times are enforced by social constraints, such as work or school, artificial light induces a mismatch between sleep timing and circadian rhythmicity (‘social jet-lag’). The model implies that developmental changes in sleep homeostasis and circadian amplitude make adolescents particularly sensitive to effects of light consumption. The model predicts that ameliorating social jet-lag is more effectively achieved by reducing evening light consumption than by delaying social constraints, particularly in individuals with slow circadian clocks or when imposed wake-times occur after sunrise. These theory-informed predictions may aid design of interventions to prevent and treat circadian rhythm-sleep disorders and social jet-lag.

Circle maps frequently arise in mathematical models of physical or biological systems. Motivated by Cherry flows and `threshold' systems such as integrate and fire neuronal models, models of cardiac arrhythmias, and models of sleep/wake regulation, we consider how structural transitions in circle maps occur. In particular we describe how maps evolve near the creation of a discontinuity. We show that the natural way to create discontinuities in the maps associated with both threshold systems and Cherry flows results in a square root singularity in the map as the discontinuity is approached from either one or both sides. We analyse the generic properties of maps with gaps displaying this local behaviour, showing how border collisions and saddle-node bifurcations are interspersed. This highlights how the Arnold tongue picture for tongues bordered by saddle-node bifurcations is amended once gaps are present. For the threshold models we also show that a loss of injectivity naturally results in the creation of multiple gaps giving rise to a novel codimension two bifurcation.

Since its inception four decades ago the two-process model introduced by Borbély has provided the conceptual framework to explain sleep wake regulation across many species, including humans. At its core, high level notions of circadian and homeostatic processes are modelled with a low dimensional description in the form of a one dimensional nonautonomous and nonsmooth flow, with the rate of change of homeostatic sleep pressure switching at specific times. These events in time can be described by an implicit map from one switching time to another and have given rise to an elegant mathematical description of periodic orbits and their instabilities using the theory of iterated maps. In this paper we show that an equivalent description can be obtained from a direct analysis of the underlying nonsmooth flow. We further show how to construct the Lyapunov exponent of the nonsmooth flow and use this to uncover a more detailed picture of the Arnol'd tongue structure of the model. Given the growing interest in studying networks of sleepers, where interactions may occur continuously throughout the day-night cycle and not just at event times, we advocate for the future use of techniques from nonsmooth dynamical systems in studying networks of the two-process model.

The Acute Medical Unit (AMU) provides care for unscheduled hospital admissions. Seven-day Consultant presence and morning AMU discharges have been advocated to improve hospital bed management. To determine whether a later time of daily peak AMU occupancy correlates with measures of hospital stress; whether seven-day Consultant presence, for COVID-19, abolished weekly periodicity of discharges. Retrospective cohort analysis. Anonymised AMU admission and discharge times were retrieved from the Profile Information Management System (PIMS), at a large, urban hospital from 14th April 2014-31st December 2018 and 20th March-2nd May 2020 (COVID-19 peak). Minute-by-minute admission and discharge times were combined to construct a running total of AMU bed occupancy. Fourier transforms were used to determine periodicity. We tested association between i) average AMU occupancy and ii) time of peak AMU occupancy, with measures of hospital stress (total medical bed occupancy and 'medical outliers' on non-medical wards). Daily, weekly and seasonal patterns of AMU bed occupancy were evident. Timing of AMU peak occupancy was unrelated to each measure of hospital stress: total medical inpatients (Spearman's rho, rs=0.04, P = 0.24); number of medical outliers (rs=-0.06, P = 0.05). During COVID-19, daily bed occupancy was similar, with continuation of greater Friday and Monday discharges than the weekend. Timing of peak AMU occupancy did not alter with hospital stress. Efforts to increase morning AMU discharges are likely to have little effect on hospital performance. Seven-day Consultant presence did not abolish weekly periodicity of discharges - other factors influence weekend discharges.

Sleep and circadian rhythm dysfunction is prevalent in schizophrenia, is associated with distress and poorer clinical status, yet remains an under-recognized therapeutic target. The development of new therapies requires the identification of the primary drivers of these abnormalities. Understanding of the regulation of sleep–wake timing is now sufficiently advanced for mathematical model-based analyses to identify the relative contribution of endogenous circadian processes, behavioral or environmental influences on sleep-wake disturbance and guide the development of personalized treatments. Here, we have elucidated factors underlying disturbed sleep-wake timing by applying a predictive mathematical model for the interaction of light and the circadian and homeostatic regulation of sleep to actigraphy, light, and melatonin profiles from 20 schizophrenia patients and 21 age-matched healthy unemployed controls, and designed interventions which restored sleep-circadian function. Compared to controls, those with schizophrenia slept longer, had more variable sleep timing, and received significantly fewer hours of bright light (light > 500 lux), which was associated with greater variance in sleep timing. Combining the model with the objective data revealed that non 24-h sleep could be best explained by reduced light exposure rather than differences in intrinsic circadian period. Modeling implied that late sleep offset and non 24-h sleep timing in schizophrenia can be normalized by changes in environmental light–dark profiles, without imposing major lifestyle changes. Aberrant timing and intensity of light exposure patterns are likely causal factors in sleep timing disturbances in schizophrenia. Implementing our new model-data framework in clinical practice could deliver personalized and acceptable light–dark interventions that normalize sleep-wake timing.

A recent elegant study published in this journal (Zerbini, Winnebeck & Merrow, J. Pineal Res, e12723, 2021) reported data on weekly and seasonal changes in circadian timing as assessed by the melatonin rhythm in dim light in a population which was exposed to a change from standard time to day light saving time. The authors highlight a one hour earlier timing of melatonin onset in summer compared to winter and a 20 minute delay on work-free days compared to work days. The variations in the timing of the melatonin rhythm are reported in standard time and the authors imply that the data are consistent with synchronisation to midday and that ‘we know that humans entrain to suntime’. Here we show that their extensive data are most parsimoniously explained by entrainment to local clock time and associated light exposure rather than ‘sun time’, indexed by either dawn or midday.

Study Objectives Assess the validity of a subjective measure of sleepiness as an indicator of sleep drive by quantifying associations between intra-individual variation in evening sleepiness and bedtime, sleep duration, and next morning and subsequent evening sleepiness, in young adults. Methods Sleep timing and sleepiness were assessed in 19 students in late autumn and late spring on a total of 771 days. Karolinska Sleepiness Scales (KSS) were completed at half-hourly intervals at fixed clock times starting four hours prior to participants’ habitual bedtime, and in the morning. Associations between sleepiness and sleep timing were evaluated by mixed model and non-parametric approaches and simulated with a mathematical model for the homeostatic and circadian regulation of sleepiness. Results Intra-individual variation in evening sleepiness was very large, covering four or five points on the 9-point KSS scale, and was significantly associated with subsequent sleep timing. On average, a one point higher KSS value was followed by 20 min earlier bedtime, which led to 11 min longer sleep, which correlated with lower sleepiness next morning and following evening. Associations between sleepiness and sleep timing were stronger in early compared to late sleepers. Model simulations indicated that the directions of associations between sleepiness and sleep timing are in accordance with their homeostatic and circadian regulation, even though much of the variance in evening sleepiness and details of its time course remain unexplained by the model. Conclusion Subjective sleepiness is a valid indicator of the drive for sleep which, if acted upon, can reduce insufficient sleep.

The field of industrial ecology applies ecosystem theory to industrial production, human consumption and societies. This article presents a case study of the development of the bio-based economy in the area surrounding the Humber estuary in the North-East of England. The study developed an agent-based model to simulate the evolution of the industrial system. We explain how the qualitative research process led to the development of a toy model that has successively been specified

Sleep is essential for the maintenance of human life, yet many features of sleep are poorly understood and mathematical models are an important tool for probing proposed biological mechanisms. The most well-known mathematical model of sleep regulation, the two-process model, models the sleep-wake cycle by two oscillators: a circadian oscillator and a homeostatic oscillator. An alternative, more recent, model considers the reciprocal interaction of sleep promoting neurons and the ascending arousal system regulated by homeostatic and circadian processes. Here we show there are fundamental similarities between these two models. The implications are illustrated with two important sleep-wake phenomena. Firstly, we show that in the two-process model, transitions between different numbers of daily sleep episodes can be classified as grazing bifurcations. This provides the theoretical underpinning for numerical results showing that the sleep patterns of many mammals can be explained by the reciprocal interaction model. Secondly, we show that when sleep deprivation disrupts the sleep-wake cycle, ostensibly different measures of sleepiness in the two models are closely related. The demonstration of the mathematical similarities of the two models is important because it brings a deeper insight to sleep-wake dynamics and allows some features of the two-process model to be interpreted physiologically.

We consider the transition from a spatially uniform state to a steady, spatially-periodic pattern in a partial differential equation describing long-wavelength convection. This both extends existing work on the study of rolls, squares and hexagons and demonstrates how recent generic results for the stability of spatially-periodic patterns may be applied in practice. We find that squares, even if stable to roll perturbations, are often unstable when a wider class of perturbations is considered. We also find scenarios where transitions from hexagons to rectangles can occur. In some cases we find that, near onset, more exotic spatially-periodic planforms are preferred over the usual rolls, squares and hexagons.

Java based analysis tool that is able to extract systematic behaviour of complex and stochastic systems directly from 'black box' simulations. Performing rigorous analysis such as; parameter dependence, sensitivity analysis, bifurcations or regime shifts, tipping or limit point identification, statistical properties (variance and underlying distributions), path dependence and stability analysis. This tool extracts insight directly from a simulator, e.g. micro-level or agent-based model, without the need to understand any of the underlying algorithms involved.