David Lloyd

Dr David Lloyd


Reader in Mathematics
PhD (Bristol), MEng (Bristol), PGCAP (Surrey)

Biography

Research

Research interests

Indicators of esteem

  • Runner-up - DSWeb 2018 Software Contest: EMBER (Emergent and Macroscopic Behaviour ExtRaction)

Supervision

Postgraduate research supervision

My publications

Publications

Knobloch J, Lloyd David, Sandstede B, Wagenknecht T (2010) Isolas of 2-Pulse Solutions in Homoclinic Snaking Scenarios, Journal of Dynamics and Differential Equations 23 (1) pp. 93-114 Springer
Homoclinic snaking refers to the bifurcation structure of symmetric localised roll patterns that are often found to lie on two sinusoidal ?snaking? bifurcation curves, which are connected by an infinite number of ?rung? segments along which asymmetric localised rolls of various widths exist. The envelopes of all these structures have a unique maximum and we refer to them as symmetric or asymmetric 1-pulses. In this paper, the existence of stationary 1D patterns of symmetric 2-pulses that consist of two well-separated 1-pulses is established. Corroborating earlier numerical evidence, it is shown that symmetric 2-pulses exist along isolas in parameter space that are formed by parts of the snaking curves and the rungs mentioned above.
Rossides T, Lloyd David, Zelik S (2015) Computing Interacting Multi-fronts in One Dimensional Real Ginzburg Landau Equations, JOURNAL OF SCIENTIFIC COMPUTING 63 (3) pp. 799-819 SPRINGER/PLENUM PUBLISHERS
We develop an efficient and robust numerical scheme to compute multi-fronts in one-dimensional real Ginzburg?Landau equations that range from well-separated to strongly interacting and colliding. The scheme is based on the global centre-manifold reduction where one considers an initial sum of fronts plus a remainder function (not necessarily small) and applying a suitable projection based on the neutral eigenmodes of each front. Such a scheme efficiently captures the weakly interacting tails of the fronts. Furthermore, as the fronts become strongly interacting, we show how they may be added to the remainder function to accurately compute through collisions. We then present results of our numerical scheme applied to various real Ginzburg Landau equations where we observe colliding fronts, travelling fronts and fronts converging to bound states. Finally, we discuss how this numerical scheme can be extended to general PDE systems and other multi-localised structures.
Knight CJK, Penn AS, Lloyd DJB (2014) Linear and sigmoidal fuzzy cognitive maps: An analysis of fixed points, Applied Soft Computing Journal 15 pp. 193-202
Fuzzy cognitive mapping is commonly used as a participatory modelling technique whereby stakeholders create a semi-quantitative model of a system of interest. This model is often turned into an iterative map, which should (ideally) have a unique stable fixed point. Several methods of doing this have been used in the literature but little attention has been paid to differences in output such different approaches produce, or whether there is indeed a unique stable fixed point. In this paper, we seek to highlight and address some of these issues. In particular we state conditions under which the ordering of the variables at stable fixed points of the linear fuzzy cognitive map (iterated to) is unique. Also, we state a condition (and an explicit bound on a parameter) under which a sigmoidal fuzzy cognitive map is guaranteed to have a unique fixed point, which is stable. These generic results suggest ways to refine the methodology of fuzzy cognitive mapping. We highlight how they were used in an ongoing case study of the shift towards a bio-based economy in the Humber region of the UK. © 2013 Elsevier B.V. All rights reserved.
Wadee M. Khurram, Lloyd David J.B., Bassom Andrew P. (2016) On the interaction of uni-directional and bi-directional buckling of a plate supported by an elastic foundation, Proceedings Royal Society London A 472 (2188) ROYAL SOCIETY
A thin flat rectangular plate supported on its edges and subjected to in-plane loading exhibits stable post-buckling behaviour. However, the introduction of a nonlinear (softening) elastic foundation may cause the response to become unstable. Here the post-buckling of such a structure is investigated and several important phenomena are identified, including the transition of patterns from stripes to spots and back again. The interaction between these forms is of importance for understanding the possible post-buckling behaviours of this structural system. In addition, both periodic and some localized responses are found to exist as the dimensions of the plate are increased and this becomes relevant when the characteristic wavelengths of the buckle pattern are small compared to the size of the plate. Potential application of the model range from macroscopic industrial manufacturing of structural elements to the understanding of micro- and nano-scale deformations in materials.
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.
Chamard J, Otta J, Lloyd David (2011) Computation of Minimum Energy Paths for Quasi-Linear Problems, Journal of Scientific Computing 49 (2) pp. 180-194 Springer
We investigate minimum energy paths of the quasi-linear problem with the p-Laplacian operator and a double-well potential. We adapt the String method of E, Ren, and Vanden-Eijnden (J. Chem. Phys. 126, 2007) to locate saddle-type solutions. In one-dimension, the String method is shown to find a minimum energy path that can align along one-dimensional ?ridges? of saddle-continua. We then apply the same method to locate saddle solutions and transition paths of the two-dimensional quasi-linear problem. The method developed is applicable to a general class of quasi-linear PDEs.
Lloyd David, O'Farrell H (2013) On localised hotspots of an urban crime model, Physica D: Nonlinear Phenomena 253 pp. 23-39
We investigate stationary, spatially localised crime hotspots on the real line and the plane of an urban crime model of Short et al. [M. Short, M. DÓrsogna, A statistical model of criminal behavior, Mathematical Models and Methods in Applied Sciences 18 (2008) 1249-1267]. Extending the weakly nonlinear analysis of Short et al., we show in one-dimension that localised hotspots should bifurcate off the background spatially homogeneous state at a Turing instability provided the bifurcation is subcritical. Using path-following techniques, we continue these hotspots and show that the bifurcating pulses can undergo the process of homoclinic snaking near the singular limit. We analyse the singular limit to explain the existence of spike solutions and compare the analytical results with the numerical computations. In two-dimensions, we show that localised radial spots should also bifurcate off the spatially homogeneous background state. Localised planar hexagon fronts and hexagon patches are found and depending on the proximity to the singular limit these solutions either undergo homoclinic snaking or act like "multi-spot" solutions. Finally, we discuss applications of these localised patterns in the urban crime context and the full agent-based model. © 2013 Elsevier B.V. All rights reserved.
Moschoyiannis S, Elia N, Penn AS, Lloyd DJB, Knight C (2016) A web-based tool for identifying strategic intervention points in complex systems, In: Proc. Games for the Synthesis of Complex Systems (CASSTING?16 @ ETAPS 2016), EPTCS 220 Proc. Games for the Synthesis of Complex Systems (CASSTING'16 @ ETAPS 2016), EPTCS 220 pp. 39-52
Steering a complex system towards a desired outcome is a challenging task. The lack of clarity on the system?s exact architecture and the often scarce scientific data upon which to base the op- erationalisation of the dynamic rules that underpin the interactions between participant entities are two contributing factors. We describe an analytical approach that builds on Fuzzy Cognitive Map- ping (FCM) to address the latter and represent the system as a complex network. We apply results from network controllability to address the former and determine minimal control configurations - subsets of factors, or system levers, which comprise points for strategic intervention in steering the system. We have implemented the combination of these techniques in an analytical tool that runs in the browser, and generates all minimal control configurations of a complex network. We demonstrate our approach by reporting on our experience of working alongside industrial, local-government, and NGO stakeholders in the Humber region, UK. Our results are applied to the decision-making process involved in the transition of the region to a bio-based economy.
Lloyd David, Penn A, knight C, Chalkias G, Velenturf A (2016) Extending Participatory Fuzzy Cognitive Mapping with a Control Nodes Methodology: a case study of the development bio-based economy in the Humber region, UK, In: Gray S, Paolisso M, Jordan R, Gray S (eds.), Environmental Modeling with Stakeholders Springer International Publishing
Fuzzy Cognitive Mapping (FCM) is a widely used participatory modeling methodology in which stakeholders collaboratively develop a cognitive map (a weighted, directed graph), representing the perceived causal structure of their sys- tem. FCM can be an extremely useful tool to enable stakeholders to collaborative- ly represent and consolidate their understanding of the structure of their system. Analysis of an FCM using tools from network theory enables the calculation of ?control configurations? for the system; subsets of system factors which if con- trolled could be used to drive the system to any given state. We have developed a technique that allows us to calculate all possible, minimally-sized control configu- rations of a stakeholder-generated FCM within a workshop context. In order to evaluate our results in terms of real world ?controllability,? stakeholders score all factors on the basis of their ability to influence them, allowing us to rank the con- figurations by their potential local controllability. This provides a starting point for discussions about effective policy, or other interventions from the specific perspective of regional actors and decision makers. We describe this methodology and report on a participatory process in which it was tested: the construction of an FCM focusing on the development of a bio-based economy in the Humber region (UK) by key stakeholders from local companies and organizations. Results and stakeholder responses are discussed in the context of our case study, but also, more generally, in the context of the use of participatory modeling for decision making in complex socio-ecological-economic systems.
Avitabile D, Lloyd DJB, Burke J, Knobloch E, Sandstede B (2010) To Snake or Not to Snake in the Planar Swift-Hohenberg Equation, SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 9 (3) pp. 704-733 SIAM PUBLICATIONS
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.
Lloyd David, Rankin J, Avitabile D, Baladron J, Faye G (2014) Continuation of Localised Coherent Structures in Nonlocal Neural Field Equations, SIAM Journal on Scientific Computing 36 (1) pp. B70-B93
Faye G, Rankin J, Lloyd David (2013) Localized radial bumps of a neural field equation on the Euclidean plane and the Poincaré disc, Nonlinearity 26 (2) pp. 437-478
We analyse radially symmetric localized bump solutions of an integro-differential neural field equation posed in Euclidean and hyperbolic geometry. The connectivity function and the nonlinear firing rate function are chosen such that radial spatial dynamics can be considered. Using integral transforms, we derive a partial differential equation for the neural field equation in both geometries and then prove the existence of small amplitude radially symmetric spots bifurcating from the trivial state. Numerical continuation is then used to path follow the spots and their bifurcations away from onset in parameter space. It is found that the radial bumps in Euclidean geometry are linearly stable in a larger parameter region than bumps in the hyperbolic geometry. We also find and path follow localized structures that bifurcate from branches of radially symmetric solutions with D -symmetry and D-symmetry in the Euclidean and hyperbolic cases, respectively. Finally, we discuss the applications of our results in the context of neural field models of short term memory and edges and textures selectivity in a hypercolumn of the visual cortex. © 2013 IOP Publishing Ltd & London Mathematical Society.
Dalah E, Lloyd David, Bradley D, Nisbet A (2009) Computational simulation of tumour hypoxia as applied to radiation therapy applications, IFMBE Proceedings 25 (1) pp. 64-66
It has long been appreciated that hypoxia plays a significant role in tumour resistance to radiotherapy treatment, chemotherapy treatment and also in surgery. For present interests, it is noted that tumour radio-sensitivity increases with the increase of oxygen concentration across tumour regions. A theoretical representation of oxygen distribution in 2D vascular architecture using a reaction diffusion model enables relationships between tissue diffusivity, tissue metabolism, anatomical structure of blood vessels and oxygen gradients to be characterized quantitatively. We present a refinement to the work of Kelly and Brady (2006) and demonstrate the significant effect of the role of the venules supply on the microcirculation process at the intracellular level. With our representation of the two latter forces, the model is being developed to simulate the uptake of various PET reagents, such as 64Cu-ATSM, to demonstrate their potential use in radiation therapy treatment planning as an indicator of tumour hypoxic regions.
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.
Lloyd David, Champneys AR, Wilson RE (2005) Robust heteroclinic cycles in the one-dimensional complex Ginzburg-Landau equation, PHYSICA D-NONLINEAR PHENOMENA 204 3-4 pp. {240-268} ELSEVIER SCIENCE BV
Numerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the one-dimensional complex Ginzburg-Landau equation (CGL) on the unit, spatially periodic domain. These cycles connect different spatially and temporally inhomogeneous time-periodic solutions as t -> infinity. A careful analysis of the connections is made using a projection onto five complex Fourier modes. It is shown first that the time-periodic solutions can be treated as (relative) equilibria after consideration of the symmetries of the CGL. Second, the cycles are shown to be robust since the individual heteroclinic connections exist in invariant subspaces. Thirdly, after constructing appropriate Poincare maps around the cycle, a criteria for temporal stability is established, which is shown numerically to hold in specific parameter regions where the cycles are found to be of Shil?nikov type. This criterion is also applied to a much higher-mode Fourier truncation where similar results are found. In regions where instability of the cycles occurs, either Shil?nikov-Hopf or blow-out bifurcations are observed, with numerical evidence of competing attractors. Implications for observed spatio-temporal intermittency in situations modelled by the CGL are discussed. (c) 2005 Elsevier B.V. All rights reserved.
Rowden J, Lloyd David, Gilbert N (2014) A model of political voting behaviours across different countries, Physica A: Statistical Mechanics and its Applications 413 pp. 609-625
This paper analyses, models mathematically, and compares national voting behaviours across seven democratic countries that have a long term election history, focusing on re-election rates, leaders' reputation with voters and the importance of friends' and family influence. Based on the data, we build a Markov model to test and explore national voting behaviour, showing voters are only influenced by the most recent past election. The seven countries can be divided into those in which there is a high probability that leaders will be re-elected and those in which incumbents have relatively less success. A simple stochastic phenomenological dynamical model of electoral districts in which voters may be influenced by social neighbours, political parties and political leaders is then created to explore differences in voter behaviours in the countries. This model supports the thesis that an unsuccessful leader has a greater negative influence on individual voters than a successful leader, while also highlighting that increasing the influence on voters of social neighbours leads to a decrease in the average re-election rate of leaders, but raises the average amount of time the dominant party is in charge.© 2014 Elsevier B.V. All rights reserved.
Lloyd David, Santitissadeekorn N, Short MB (2015) Exploring data assimilation and forecasting issues for an urban crime model, European Journal of Applied Mathematics Cambridge University Press
Lloyd DJB, Santitissadeekorn N, Short MB Exploring data assimilation and forecasting issues for an urban crime model, CAMBRIDGE UNIV PRESS
Penn AS, Knight CJ, Lloyd David, Avitabile D, Kok K, Schiller F, Woodward A, Druckman Angela, Basson L (2013) Participatory development and analysis of a fuzzy cognitive map of the establishment of a bio-based economy in the Humber region., PLoS One 8 (11) pp. e78319-?
Fuzzy Cognitive Mapping (FCM) is a widely used participatory modelling methodology in which stakeholders collaboratively develop a 'cognitive map' (a weighted, directed graph), representing the perceived causal structure of their system. This can be directly transformed by a workshop facilitator into simple mathematical models to be interrogated by participants by the end of the session. Such simple models provide thinking tools which can be used for discussion and exploration of complex issues, as well as sense checking the implications of suggested causal links. They increase stakeholder motivation and understanding of whole systems approaches, but cannot be separated from an intersubjective participatory context. Standard FCM methodologies make simplifying assumptions, which may strongly influence results, presenting particular challenges and opportunities. We report on a participatory process, involving local companies and organisations, focussing on the development of a bio-based economy in the Humber region. The initial cognitive map generated consisted of factors considered key for the development of the regional bio-based economy and their directional, weighted, causal interconnections. A verification and scenario generation procedure, to check the structure of the map and suggest modifications, was carried out with a second session. Participants agreed on updates to the original map and described two alternate potential causal structures. In a novel analysis all map structures were tested using two standard methodologies usually used independently: linear and sigmoidal FCMs, demonstrating some significantly different results alongside some broad similarities. We suggest a development of FCM methodology involving a sensitivity analysis with different mappings and discuss the use of this technique in the context of our case study. Using the results and analysis of our process, we discuss the limitations and benefits of the FCM methodology in this case and in general. We conclude by proposing an extended FCM methodology, including multiple functional mappings within one participant-constructed graph.
Lloyd David, Champneys AR (2005) Efficient numerical continuation and stability analysis of spatiotemporal quadratic optical solitons, SIAM JOURNAL ON SCIENTIFIC COMPUTING 27 (3) pp. 759-773 Society for Industrial and Applied Mathematics (SIAM)
A numerical method is set out which efficiently computes stationary (z-independent) two- and three-dimensional spatiotemporal solitons in second-harmonic-generating media. The method relies on a Chebyshev decomposition with an infinite mapping, bunching the collocation points near the soliton core. Known results for the type-I interaction are extended and a stability boundary is found by two- parameter continuation as defined by the Vakhitov-Kolokolov criteria. The validity of this criterion is demonstrated in (2+1) dimensions by simulation and direct calculation of the linear spectrum. The method has wider applicability for general soliton-bearing equations in (2+1) and (3+1) dimensions.
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.
Beck M, Knobloch J, Lloyd David, Sandstede B, Wagenknecht T (2009) Snakes, ladders, and isolas of localized patterns, SIAM Journal on Mathematical Analysis 41 (3) pp. 936-972
Stable localized roll structures have been observed in many physical problems and model equations, notably in the 1D Swift?Hohenberg equation. Reflection-symmetric localized rolls are often found to lie on two ?snaking? solution branches, so that the spatial width of the localized rolls increases when moving along each branch. Recent numerical results by Burke and Knobloch indicate that the two branches are connected by infinitely many ?ladder? branches of asymmetric localized rolls. In this paper, these phenomena are investigated analytically. It is shown that both snaking of symmetric pulses and the ladder structure of asymmetric states can be predicted completely from the bifurcation structure of fronts that connect the trivial state to rolls. It is also shown that isolas of asymmetric states may exist, and it is argued that the results presented here apply to 2D stationary states that are localized in one spatial direction.
Lloyd David, Sandstede B, Avitabile D, Champneys AR (2008) Localized hexagon patterns of the planar Swift-Hohenberg equation, SIAM Journal on Applied Dynamical Systems 7 (3) pp. 1049-1100 SIAM PUBLICATIONS
We investigate stationary spatially localized hexagon patterns of the two-dimensional (2D) Swift? Hohenberg equation in the parameter region where the trivial state and regular hexagon patterns are both stable. Using numerical continuation techniques, we trace out the existence regions of fully localized hexagon patches and of planar pulses which consist of a strip filled with hexagons that is embedded in the trivial state. We find that these patterns exhibit snaking: for each parameter value in the snaking region, an infinite number of patterns exist that are connected in parameter space and whose width increases without bound. Our computations also indicate a relation between the limits of the snaking regions of planar hexagon pulses with different orientations and of the fully localized hexagon patches. To investigate which hexagons among the one-parameter family of hexagons are selected in a hexagon pulse or front, we derive a conserved quantity of the spatial dynamical system that describes planar patterns which are periodic in the transverse direction and use it to calculate the Maxwell curves along which the selected hexagons have the same energy as the trivial state. We find that the Maxwell curve lies within the snaking region, as expected from heuristic arguments.
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.
The international trade in commodities forms a complex network of economic interdependencies. This network now plays a central role in promoting global economic development and security. However, significant asymmetries have been noted in terms of access to this network, and in the unequal distribution of the benefits and risks accrued from the system as a whole. Understanding the statistical properties and dynamics of the trade network have therefore, become important tools for investigating a multitude of real-world policy concerns relevant to economics, public policy, and international development. This thesis focuses on investigating three of these issues---market growth, price inequality, and supply risks. The first of these projects focuses on modelling the growth of commodity markets, and the resulting effect on network topology. The second, looks at how asymmetries in network can lead to varying prices for the same good, and explores the implications for developing more equitable market structures. The final project contributes to our understanding of how export restrictions affect the network structure of trade and how these risks can undermine global food security. Throughout, a network science approach is employed, whereby trade is modelled as a graph-like structure, with the topology of trade being the primary focus of analysis. To support this approach, we introduce several theoretical models, and apply simulations on both real-world, and artificially produced trade network data. The outcome of this research improves on our ability to identify and target key participants within a market, and predict policies that favour more stable and equitable structures that better facilitate trade.
Groves M, Lloyd D, Stylianou A (2017) Pattern formation on the free surface of a ferrofluid: spatial dynamics and homoclinic bifurcation, Physica D: Nonlinear Phenomena 350 pp. 1-12 Elsevier
We establish the existence of spatially localised one-dimensional free surfaces of a ferrofluid near onset of the Rosensweig instability, assuming a general (nonlinear) magnetisation law. It is shown that the ferrohydrostatic equations can be derived from a variational principle that allows one to formulate them as an (infinite-dimensional) spatial Hamiltonian system in which the unbounded free-surface direction plays the role of time. A centremanifold reduction technique converts the problem for small solutions near onset to an equivalent Hamiltonian system with finitely many degrees of freedom. Normal-form theory yields the existence of homoclinic solutions to the reduced system, which correspond to spatially localised solutions of the ferrohydrostatic equations.
Lloyd David, Scheel A (2017) Continuation and Bifurcation of Grain Boundaries in the Swift-Hohenberg Equation, SIAM Journal on Applied Dynamical Systems 16 (1) pp. 252-293 Society for Industrial and Applied Mathematics (SIAM)
We study grain boundaries between striped phases in the prototypical Swift-Hohenberg equation. We propose an analytical and numerical far- field-core decomposition that allows us to study existence and bifurcations of grain boundaries analytically and numerically using continuation techniques. This decomposition overcomes problems with computing grain boundaries in a large doubly periodic box with phase conditions. Using the spatially conserved quantities of the time-independent Swift-Hohenberg equation, we show that symmetric grain boundaries must select the marginally zig-zag stable stripes. We find that as the angle between the stripes is decreased, the symmetric grain boundary undergoes a parity-breaking pitchfork bifurcation where dislocations at the grain boundary split into disclination pairs. A plethora of asymmetric grain boundaries (with different angles of the far- field stripes either side of the boundary) is found and investigated. The energy of the grain boundaries is then mapped out. We find that when the angle between the stripes is greater than a critical angle, the symmetric grain boundary is energetically preferred while when the angle is less than the critical angle, the grain boundaries where stripes on one side are parallel to the interface are energetically preferred. Finally, we propose a classification of grain boundaries that allows us to predict various non-standard asymmetric grain boundaries.
Lloyd D, Sandstede B (2009) Localized radial solutions of the Swift-Hohenberg equation, NONLINEARITY 22 (2) pp. 485-524 IOP PUBLISHING LTD
Santitissadeekorn Naratip, Short M, Lloyd David (2018) Sequential data assimilation for 1D self-exciting processes
with application to urban crime data,
Computational Statistics and Data Analysis 128 pp. 163-183 Elsevier
A number of models ? such as the Hawkes process and log Gaussian Cox process
? have been used to understand how crime rates evolve in time and/or
space. Within the context of these models and actual crime data, parameters
are often estimated using maximum likelihood estimation (MLE) on batch
data, but this approach has several limitations such as limited tracking in
real-time and uncertainty quantification. For practical purposes, it would be
desirable to move beyond batch data estimation to sequential data assimilation.
A novel and general Bayesian sequential data assimilation algorithm is
developed for joint state-parameter estimation for an inhomogeneous Poisson
process by deriving an approximating Poisson-Gamma ?Kalman? filter
that allows for uncertainty quantification. The ensemble-based implementation
of the filter is developed in a similar approach to the ensemble Kalman
filter, making the filter applicable to large-scale real world applications unlike
nonlinear filters such as the particle filter. The filter has the advantage
that it is independent of the underlying model for the process intensity,
and can therefore be used for many different crime models, as well as other
application domains. The performance of the filter is demonstrated on synthetic
data and real Los Angeles gang crime data and compared against a
very large sample-size particle filter, showing its effectiveness in practice. In addition the forecast skill of the Hawkes model is investigated for a forecast
system using the Receiver Operating Characteristic (ROC) to provide a useful
indicator for when predictive policing software for a crime type is likely
to be useful. The ROC and Brier scores are used to compare and analyse
the forecast skill of sequential data assimilation and MLE. It is found that
sequential data assimilation produces improved probabilistic forecasts over
the MLE.
Brooks Jacob, Derks Gianne, Lloyd David J.B. (2019) Existence of stationary fronts in a system of two coupled wave equations with spatial inhomogeneity, Nonlinearity London Mathematical Society
We investigate the existence of stationary fronts in a coupled system of two sine-Gordon equations with a smooth, ?hat-like? spatial inhomogeneity. The spatial inhomogeneity corresponds to
a spatially dependent scaling of the sine-Gordon potential term. The uncoupled inhomogeneous
sine-Gordon equation has stable stationary front solutions that persist in the coupled system.
Carrying out a numerical investigation it is found that these inhomogeneous sine-Gordon fronts
loose stability, provided the coupling between the two inhomogeneous sine-Gordon equations is
strong enough, with new stable fronts bifurcating. In order to analytically study the bifurcating
fronts, we first approximate the smooth spatial inhomogeneity by a piecewise constant function.
With this approximation, we prove analytically the existence of a pitchfork bifurcation. To
complete the argument, we prove that transverse fronts for a piecewise constant inhomogeneity
persist for the smooth ?hat-like? spatial inhomogeneity by introducing a fast-slow structure and
using geometric singular perturbation theory.
Lloyd David, Sandstede B, Avitabile D, Champneys AR (2008) Localized hexagon patterns of the planar Swift-Hohenberg equation, SIAM Journal on Applied Dynamical Systems 7 (3) pp. 1049-1100

We investigate stationary spatially localized hexagon patterns of the two-dimensional (2D) Swift-Hohenberg equation in the parameter region where the trivial state and regular hexagon patterns are both stable. Using numerical continuation techniques, we trace out the existence regions of fully localized hexagon patches and of planar pulses which consist of a strip filled with hexagons that is embedded in the trivial state. We find that these patterns exhibit snaking: for each parameter value in the snaking region, an infinite number of patterns exist that are connected in parameter space and whose width increases without bound. Our computations also indicate a relation between the limits of the snaking regions of planar hexagon pulses with different orientations and of the fully localized hexagon patches. To investigate which hexagons among the one-parameter family of hexagons are selected in a hexagon pulse or front, we derive a conserved quantity of the spatial dynamical system that describes planar patterns which are periodic in the transverse direction and use it to calculate the Maxwell curves along which the selected hexagons have the same energy as the trivial state. We find that the Maxwell curve lies within the snaking region, as expected from heuristic arguments.

Griffin BA, Paynton SE, Lloyd David, Gabe SM, Mateos AR, Lovegrove JA (2007) Plasma long-chain n-3 PUFA status in patients receiving long-term home parenteral nutrition, PROCEEDINGS OF THE NUTRITION SOCIETY 66 pp. 92A-92A CAMBRIDGE UNIV PRESS
Lloyd David, Gollwitzer C, Rehberg I, Richter R (2015) Homoclinic snaking near the surface instability of a polarizable fluid, Journal of Fluid Mechanics 783 pp. 283-305 Cambridge University Press
We report on localized patches of cellular hexagons observed on the surface of a magnetic fluid in a vertical magnetic field. These patches are spontaneously generated by jumping into the neighborhood of the unstable branch of the domain covering hexagons of the Rosensweig instability upon which the patches equilibrate and stabilise. They are found to co-exist in intervals of the applied magnetic field strength parameter around this branch. We formulate a general energy functional for the system and a corresponding Hamiltonian that provides a pattern selection principle allowing us to compute Maxwell points (where the energy of a single hexagon cell lies in the same Hamiltonian level set as the flat
state) for general magnetic permeabilities. Using umerical continuation techniques we investigate the existence of localized hexagons in the Young-Laplace equation coupled to the Maxwell equations. We find cellular hexagons possess a Maxwell point providing an energetic explanation for the multitude of measured hexagon patches. Furthermore,it is found that planar hexagon fronts and hexagon patches undergo homoclinic snaking corroborating the experimentally detected intervals. Besides making a contribution to the specific area of ferrofluids, our work paves the ground for a deeper understanding of homoclinic snaking of 2D localized patches of cellular patterns in many physical systems.
Guest S, Catmur C, Lloyd David, Spence C (2002) Audiotactile interactions in roughness perception, EXPERIMENTAL BRAIN RESEARCH 146 (2) pp. 161-171 SPRINGER-VERLAG
Williamson S, Healey RC, Lloyd David, Tevaarwerk JL (1997) Rotor cage anomalies and unbalanced magnetic pull in single-phase induction motors, IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS 33 (6) pp. 1553-1562
Chaffey GS, Lloyd David, Skeldon AC, Kirkby NF (2014) The Effect of the G 1 - S transition Checkpoint on an Age Structured Cell Cycle Model., PLoS One 9 (1) pp. 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 [Formula: see text]-[Formula: see text] 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 [Formula: see text]-[Formula: see text] checkpoint. By considering the probability of cells transitioning at the [Formula: see text]-[Formula: see text] 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.
Knobloch J, Lloyd David, Sandstede B, Wagenknecht T (2011) Isolas of 2-pulse solutions in homoclinic snaking scenarios, Journal of Dynamics and Differential Equations 23 (1) pp. 93-114
Homoclinic snaking refers to the bifurcation structure of symmetric localised roll patterns that are often found to lie on two sinusoidal "snaking" bifurcation curves, which are connected by an infinite number of "rung" segments along which asymmetric localised rolls of various widths exist. The envelopes of all these structures have a unique maximum and we refer to them as symmetric or asymmetric 1-pulses. In this paper, the existence of stationary 1D patterns of symmetric 2-pulses that consist of two well-separated 1-pulses is established. Corroborating earlier numerical evidence, it is shown that symmetric 2-pulses exist along isolas in parameter space that are formed by parts of the snaking curves and the rungs mentioned above. © Springer Science+Business Media, LLC 2010.
Lloyd David, Paynton SE, Bassett P, Mateos AR, Lovegrove JA, Gabe SM, Griffin BA (2008) Assessment of long chain n-3 polyunsaturated fatty acid status and clinical outcome in adults receiving home parenteral nutrition, CLINICAL NUTRITION 27 (6) pp. 822-831 CHURCHILL LIVINGSTONE
Williamson S, Healey RC, Lloyd David (1996) Rotor cage anomalies and unbalanced magnetic pull in single-phase induction motors .1. Analysis, IAS '96 - CONFERENCE RECORD OF THE 1996 IEEE INDUSTRY APPLICATIONS CONFERENCE, THIRTY-FIRST IAS ANNUAL MEETING, VOLS 1-4 pp. 558-565
Lloyd David J.B. (2019) Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation, SIAM Journal on Applied Dynamical Systems Society for Industrial and Applied Mathematics (SIAM)
It is well-known that stationary localised patterns involving a periodic stripe core can undergo
a process that is known as 'homoclinic snaking' where patterns are added to the stripe
core as a bifurcation parameter is varied. The parameter region where homoclinic snaking takes
place usually occupies a small region in the bistability region between the stripes and quiescent
state. Outside the homoclinic snaking region, the localised patterns invade or retreat where
stripes are either added or removed from the core forming depinning fronts. It remains an
open problem to carry out a numerical bifurcation analysis of depinning fronts. In this paper,
we carry out numerically bifurcation analysis of depinning of fronts near the homoclinic
snaking region, involving a spatial stripe cellular pattern embedded in a quiescent state, in
the two-dimensional Swift-Hohenberg equation with either a quadratic-cubic or cubic-quintic
nonlinearity. We focus on depinning fronts involving stripes that are orientated either parallel,
oblique and perpendicular to the front interface, and almost planar depinning fronts. We show
that invading parallel depinning fronts select both a far-field wavenumber and a propagation
wavespeed whereas retreating parallel depinning fronts come in families where the wavespeed is
a function of the far-field wavenumber. Employing a far-field core decomposition, we propose
a boundary value problem for the invading depinning fronts which we numerically solve and
use path-following routines to trace out bifurcation diagrams. We then carry out a thorough
numerical investigation of the parallel, oblique, perpendicular stripe, and almost planar invasion
fronts. We find that almost planar invasion fronts in the cubic-quintic Swift-Hohenberg equation
bifurcate off parallel invasion fronts and co-exist close to the homoclinic snaking region.
Sufficiently far from the 1D homoclinic snaking region, no almost planar invasion fronts exist
and we find that parallel invasion stripe fronts may regain transverse stability if they propagate
above a critical speed. Finally, we show that depinning fronts shed light on the time simulations
of fully localised patches of stripes on the plane. The numerical algorithms detailed have wider
application to general modulated fronts and reaction-diffusion systems.

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