### Professor Anne Skeldon

Professor

Department of Mathematics, Centre for Mathematical and Computational Biology.

### Publications

Silber M, Skeldon AC (1999) Parametrically excited surface waves: Two-frequency forcing, normal form symmetries, and pattern selection, Physical Review E 59 (5) pp. 5446-5456 American Physical Society
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.
Schiller F, Skeldon A, Balke T, Grant M, Penn AS, Basson L, Jensen P, Gilbert N, Kalkan OD, Woodward A, Kaminski B, Koloch G (2014) Defining Relevance and Finding Rules: An Agent-Based Model of Biomass Use in the Humber Area, ADVANCES IN SOCIAL SIMULATION 229 pp. 373-384 SPRINGER-VERLAG BERLIN
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
SKELDON AC, MULLIN T (1992) MODE INTERACTION IN A DOUBLE PENDULUM, PHYSICS LETTERS A 166 (3-4) pp. 224-229 ELSEVIER SCIENCE BV
Dijk D-J, Skeldon AC (2015) Biological rhythms: human sleep before the industrial era, NATURE 527 (7577) pp. 176-177 NATURE PUBLISHING GROUP
Daniels PG, Ho D, Skeldon AC (2003) Solutions for nonlinear convection in the presence of a lateral boundary, PHYSICA D-NONLINEAR PHENOMENA 178 (1-2) pp. 83-102 ELSEVIER SCIENCE BV
SKELDON AC (1994) DYNAMICS OF A PARAMETRICALLY EXCITED DOUBLE PENDULUM, PHYSICA D 75 (4) pp. 541-558 ELSEVIER SCIENCE BV
Shiozawa H, Skeldon Anne, Lloyd David, Stolojan V, Cox DC, Silva SR (2011) Spontaneous emergence of long-range shape symmetry, Nano Letters 11 (1) pp. 160-163
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.
SKELDON A, MULLIN T, POTTINGER D (1989) THE 5-DIMENSIONAL PENDULUM PICTURE SHOW, NEW SCIENTIST 124 (1689) pp. 46-47 NEW SCIENTIST PUBL EXPEDITING INC
Skeldon Anne, Chaffey G, Lloyd David, Mohan V, Bradley DA, Nisbet A (2012) Modelling and detecting tumour oxygenation levels., PLoS One 7 (6) pp. e38597-?
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.
Skeldon AC, Purvey I (2005) The effect of different forms for the delay in a model of the nephron, Mathematical Biosciences and Engineering 2 (1) pp. 97-109 AMER INST MATHEMATICAL SCIENCES
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.
Dionne B, Silber M, Skeldon AC (1998) Stability Results for Steady, Spatially--Periodic Planforms, Nonlinearity 10 (2) 321 Institute of Physics
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.
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.
Skeldon AC, Riley DS, Cliffe KA (1996) Convection in a low Prandtl number fluid, JOURNAL OF CRYSTAL GROWTH 162 (1-2) pp. 95-106 ELSEVIER SCIENCE BV
POTTINGER D, TODD S, RODRIGUES I, MULLIN T, SKELDON A (1992) PHASE PORTRAITS FOR PARAMETRICALLY EXCITED PENDULA - AN EXERCISE IN MULTIDIMENSIONAL DATA VISUALIZATION, COMPUTERS & GRAPHICS 16 (3) pp. 331-337 PERGAMON-ELSEVIER SCIENCE LTD
Skeldon AC, Moroz IM (1998) On a codimension-three bifurcation arising in a simple dynamo model, Physica D-Nonlinear Phenomena 117 (1-4) pp. 117-127 ELSEVIER SCIENCE BV
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.
Skeldon AC, Cliffe KA, Riley DS (1997) Grid design for the computation of a hexagon-roll interaction using a finite element method, JOURNAL OF COMPUTATIONAL PHYSICS 133 (1) pp. 18-26 ACADEMIC PRESS INC JNL-COMP SUBSCRIPTIONS
Daniels PG, Ho D, Skeldon AC (2008) Non-linear wavelength selection for pattern-forming systems in channels, IMA JOURNAL OF APPLIED MATHEMATICS 73 (6) pp. 873-901 OXFORD UNIV PRESS
Skeldon AC, Silber M (1998) New stability results for patterns in a model of long-wavelength convection, Physica D-Nonlinear Phenomena 122 (1-4) pp. 117-133 ELSEVIER SCIENCE BV
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.
Skeldon AC, Porter J (2011) Scaling properties of weakly nonlinear coefficients in the Faraday problem, PHYSICAL REVIEW E 84 (1) ARTN 016209 AMER PHYSICAL SOC
Braunsfurth MG, Skeldon AC, Juel A, Mullin T, Riley DS (1997) Free convection in liquid gallium, JOURNAL OF FLUID MECHANICS 342 pp. 295-314 CAMBRIDGE UNIV PRESS
Silber M, Topaz CM, Skeldon AC (2000) Two-frequency forced Faraday waves: Weakly damped modes and pattern selection, Physica D: Nonlinear Phenomena 143 (1-4) pp. 205-225 Elsevier
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.
Skeldon AC, Guidoboni G (2007) Pattern selection for Faraday waves in an incompressible viscous fluid, SIAM Journal on Applied Mathematics 67 (4) pp. 1064-1100
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 Viñals 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. © 2007 Society for Industrial and Applied Mathematics.
Chuter AM, Aston PJ, Skeldon AC, Roulstone I (2014) A dynamical systems analysis of the data assimilation linked ecosystem carbon (DALEC) models, Chaos 25 (3) 036401
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.
Meejun N, Skeldon AC, Tuzun U, O'Sullivan C (2008) Wavelet analysis of DEM simulations of samples under biaxial compression, GRANULAR MATTER 10 (5) pp. 389-398 SPRINGER
Skeldon A, Dijk D-J, Derks G (2014) Changes in sleep across the lifespan: using mathematical models to explore hypotheses to explain sleep timing, JOURNAL OF SLEEP RESEARCH 23 pp. 169-170 WILEY-BLACKWELL
Skeldon AC (1995) On long-wave morphological instabilities in directional solidification, European Journal of Applied Mathematics 6 pp. 639-652
Skeldon AC, Rucklidge AM (2015) Can weakly nonlinear theory explain Faraday wave patterns near onset?, Journal of Fluid Mechanics 777 pp. 604-632
© 2015 Cambridge University Press.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 stabilised, 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.
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.
Hide R, Skeldon AC, Acheson DJ (1996) A study of two novel self-exciting single-disk homopolar dynamos: Theory, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 452 (1949) pp. 1369-1395 ROYAL SOC
Lloyd D, Chaffey GS, Skeldon AC, Kirkby NF (2014) The effect of the G1 - S transition checkpoint on an age structured cell cycle model, PLoS One 9 (1) 83477
Skeldon Anne, Derks Gianne (2017) Nonsmooth maps and the fast-slow dynamics of sleep-wake regulation: Part 1, Research Perspectives CRM Barcelona (Trends in Mathematics series) pp. 167-170 Springer
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.
Skeldon A, Phillips A, Dijk D (2017) The effects of self-selected light-dark cycles and social constraints on human sleep and circadian timing: a modeling approach, Scientific Reports 7 45158 Nature Publishing Group
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.
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
Skeldon Anne, Derks Gianne, Booth V (2017) Nonsmooth maps and the fast-slow dynamics of sleep-wake regulation: Part 2, Research Perspectives CRM Barcelona (Trends in Mathematics series) pp. 171-175 Springer
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.
Chaffey Gary, Lloyd David, Skeldon Anne, Kirkby Norman (2014) The effect of the G1-S transition checkpoint on an age structured cell cycle model., PLoS One 9 (1) e83477 Public Library of Science
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.
Rucklidge A, Silber M, Skeldon A (2012) Three-wave interactions and spatio-temporal chaos, Physical Review Letters 108 (7) 075450
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.
Cooperation is acting in the interests of one?s social group, often at a cost to yourself. When the level of cooperation is observed in the laboratory, people cooperate more often, and at higher levels than are predicted by standard theories. In this thesis I find two novel ways in which cooperation on multilayered populations is increased. These models contribute to an understanding of how people cooperate in real-world social situations, and help us to explain why people cooperate as much as they are observed to do.
In each study I model the tension between the individual and the group using the public goods game. This game is played on a structured population defined by a multilayered network. Each layer represents a different sphere of influence on the player?s decision to cooperate or defect.
The first model studies the effect of a player choosing whether to cooperate or defect on either all layers simultaneously (synchronously) or on one layer at a time (asynchronously). Updating asynchronously leads to increased cooperation across a number of different parameter regimes. This demonstrates a new way in which cooperation can be increased in a system with multiple influences, and also helps to understand exactly why cooperation is increased in multilayered systems.
Inspired by empirical examples, the second model adds to the standard model of the public goods game on networks in two ways. The first is to include conditional cooperators, and the second is the addition of a layer of social influence. This combination of economic and social influence has not been considered in previous models of the public goods game, and I find that this additional layer of influence results in high levels of cooperation. In the final chapter, I study these dynamics on more realistic network structures, with results echoing empirical findings under certain parameters.
Bailey Matthew, Derks Gerardina, Skeldon Anne (2018) Circle maps with gaps: understanding the dynamics of the two process model for sleep-wake regulation, European Journal of Applied Mathematics. Special issue 29 (5) pp. 845-868 Cambridge University Press
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.
Skeldon A.C., Schiller F., Yang A., Balke-Visser T., Penn A., Gilbert N. (2018) Agent-based modelling to predict policy outcomes: a food waste recycling example, Environmental Science and Policy 87 pp. 85-91 Elsevier

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.

Allen James M, Skeldon Anne C, Hoyle Rebecca B (2018) Social influence preserves cooperative strategies in the conditional cooperator public goods game on a multiplex network, Physical Review E 98 062305 pp. 062305-1 American Physical Society
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.
Sleep is essential for most living things to function. Many features of sleep are not yet understood however, mathematical models are playing an important role in developing our understanding of many of the physiological properties of sleep. We introduce the most well-known model of sleep regulation, the two process model which proposes that sleep-wake cycles can be modelled by the
interaction between two oscillators. This ostensibly simple model is an interesting example of a nonsmooth dynamical system whose rich dynamical structure has been relatively unexplored. A key aim of this work is to further understand how transitions between monophasic (one sleep a day) and polyphasic (many sleeps a day) sleep occur in the two process model.

The two process model can be framed as a one-dimensional map of the circle which, for some parameter regimes, has gaps. As is a feature of continuous circle maps the bifurcation set consists of saddle-node Arnold tongues. We show that border collision bifurcations that arise naturally in maps with gaps extend and supplement these tongues. We see 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. With this deeper understanding of the dynamics and bifurcation structure of the two process model we use modifiedý versions of the model to explain two experimental data sets.

An ultradian rhythm is a recurrent period or cycle which repeats multiple times across the day. We consider the sleep wake patterns of a the common vole, Microtus Arvalis, which has ultradian rest activity and feeding patterns. By deriving parameters for the two process model from EEG data and sleep/ wake onset times we are able to simulate with high accuracy the key features of spontaneous sleep-wake patterns in the voles. However, to explain phenomena seen in sleep deprivation experiments we include a high amplitude ultradian oscillation alongside the circadian, the results allow us to give some physiological insight into the internal mechanisms which drive sleep/wake onset times in the common vole.

Across the human lifespan there are many changes in the physiological properties of sleep, sleep timing and sleep duration. In adolescence sleep timing is delayed and there is a reduction in slow wave sleep which continues into old age as sleep timing gradually becomes earlier. Using a modified two process model which incorporates a van der Pol oscillator driven by external light signals into the circadian process we show that changes in sleep timing and duration across the lifespan can be explained by varying parameters. Model simulation show that these changes can be understood by a simultaneous reduction in the amplitude of the circadian oscillator and the upper asymptote of the homeostatic sleep pressure.

Skeldon Anne, Dijk Derk-Jan (2019) School start times and daylight saving time confuse Californian lawmakers, Current Biology 29 (8) pp. R278-R279 Elsevier
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.
Thomas SA, Lloyd David, Skeldon AC (2016) Equation-free analysis of agent-based models and systematic parameter determination, Physica A: Statistical Mechanics and its Applications 464 pp. 27-53 ELSEVIER SCIENCE BV
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.
We here consider modelling tissue growth and mechanotransduction, utilising a continuum approach based on the theory of elasticity. Our models are valid for both unhealthy and healthy tissues and assume a spherical symmetry is present. In principal, there are a number of tissue types which can be modelled using our framework, however, we choose to focus on three main tissue paradigms; epithelial cysts, ovarian follicles and avascular tumour spheroids, which all share common geometric features of a central core, surrounded by a proliferating rim of cells. In all cases, the tissues are embedded in a constraining outer gel.

We show that growth within the tissue leads to the build-up of internal stress, and we find potential mechanotransductive mechanisms occurring as a result of this growth. These mechanisms are deemed switching points and we show that they can occur both as a result of the internal stress and of the associated strains. We also find that, for systems with a deformable central core, this core can expand and shrink passively as a result of growth within the surrounding material in parameter regimes that we identify.

We also relax the assumption that the inner core is non-growing and consider that both phases of the tissue undergo growth. This leads to a competition of growth and we show that switching still occurs, but is now dependent upon the growth in both regions. This is considered specifically in the case of ovarian follicles, where we further observe that the cuboidalisation of cells can be produced as a consequence of the mechanics within the system. We next consider the addition of the effect of contractility upon the linear model using two techniques of implementation.

The necessity of the use of nonlinear elasticity is then tested, from which we show that for most parameters within the realms of soft biological tissues, the linear approximation to the nonlinear model is of a sufficient likeness to warrant the use of the linear scheme. We find that the ratio of the Young's modulus of the tissue and the surrounding medium is key in determining the effectiveness of the linear model.

These studies consistently highlight the importance of the mechanical properties of the tissue and surrounding extracellular matrix, specifically stressing the significance of the Young's modulus upon tissue growth dynamics.

Shochat Tamar, Santhi Nayantara, Herer Paula, Flavell Sapphira A., Skeldon Anne C., Dijk Derk-Jan (2019) Human Sleep Timing in Late Autumn and Late Spring Associates with Light Exposure Rather than Sun Time in College Students, Frontiers in Neuroscience Frontiers Media
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.
Kumar Prashant, Druckman Angela, Gallagher John, Gatersleben Birgitta, Allison Sarah, Eisenman Theodore S., Hoang Uy, Hama Sarkawt, Tiwari Arvind, Sharma Ashish, Abhijith K V, Adlakha Deepti, McNabola Aonghus, Astell-Burt Thomas, Feng Xiaoqi, Skeldon Anne, de Lusignan Simon, Morawska Lidia (2019) The Nexus between Air Pollution, Green Infrastructure and Human Health, Environment International Elsevier
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.