# Professor Anne Skeldon

## Academic and research departments

Department of Mathematics, Centre for Mathematical and Computational Biology.### 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 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 currently lead the Mathematics of Life and Social Sciences research group and co-lead the 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.

### Supervision

## Postgraduate research supervision

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

- Michelle Grant
- Imran Usmani (co-supervised with Prof Derk-Jan Dijk)
- Rafal Zajac (co-supervised with Prof Derk-Jan Dijk)

Previous PhD students include:

Matthew Bailey, James Allen, Gary Chaffey, Anna Chuter, Naowarut Meejun, Nicholas Robertson, Diep Ho

### My teaching

In 2019/20 I am teaching

- MAT3046 Game Theory with Applications in Biology and Economics (co-taught with Prof Gianne Derks)
- MAT3043 Graphs and Networks

### My publications

### Publications

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

The multi- resolution Horn- Schunck method is employed here because it can cope with the sometimes fictitious ?large? displacements that fluid deformation seems to produce [2]. But some of the recovered divergences here have unrealistically ?large? magnitudes (relative to those near the injection location) where they ought to be comparatively ?small? [2].

Quantifying error in flow fields is difficult when the true solution is unknown. One can subjectively define uncertainty in their components, and observations, to follow Gaussian distributions which are updated using Kalman filtering [2]. Given a pair of synthetic images, posterior variances seem to get reduced most where angular errors are comparatively ?small? [2]. So posterior variances are used to infer errors when the true solution is unknown here because they are independent of it [2].

Unrealistic ?large? divergence magnitudes (relative to those near injection locations) still appear where they ought to be comparatively ?small? [2]. In line with previous research, one tries modelling flows induced by bioirrigating Arenicola marina as two- dimensional incompressible point sources. Only three parameters, namely the source strength, x- and y- coordinates, need estimating rather than flow components at each grid cell. A Markov chain Monte- Carlo method is employed for this task, instead of the Kalman filter, because the state being estimated is no longer proportional to observations. Although comparatively ?large? divergence magnitudes now only appear near locations of fluid injection, this approach seems computationally expensive on one?s Dell Optiplex 7010 computer.

Outflow appears to be induced at the sediment- water interface by a two- dimensional incompressible point source beneath it. One questions whether there should be a little inflow as well because when an organism burrows forwards, the volume that it previously occupied ought to refill with surrounding fluid. This could be accounted for here by considering an additional two- dimensional incompressible flow at the sediment- water interface, as well as a point source at the injection location. But more parameters would need estimating. In an attempt to reduce computation times, the simulations involving the Markov chain Monte- Carlo method are rerun using two iterative ensemble Kalman filters (respectively).