Dr David Lloyd
Academic and research departments
Department of Mathematics, Centre for Mathematical and Computational Biology, Centre for Criminology.Biography
I am a Reader in the Department of Mathematics at the University of Surrey. My research interests are in localised pattern formation and mathematical modelling. I am also cofounder and director of the Surrey Centre for Criminology at the University of Surrey.
Please see my personal webpage for more details.
Research
Research interests
Localised Patterns
Spots and localised patches of cellular hexagons have been observed in a variety of experiments from magnetic fluids to vertical vibrated media. Research here is focused on understanding two and higherdimensional localised structures in pattern forming systems.
Collaborators: Drs. Daniele Avitabile (Surrey), John Burke (Boston) and Profs. Jurgen Knobloch (Ilmenau), Edgar Knobloch (UC, Berkeley), Bjorn Sandstede (Brown), Sergey Zelik (Surrey), Reinhard Richter (Bayreuth)
PhD Students: Tasos Rossides, Jacob Brooks, Daniel Hill
Mathematical Criminology
Analysis of models of burglary hotspots and Data Assimilation issues
Collaborators: Drs. Naratip Santitissadeekorn (Surrey), and Martin B. Short (Georgia Tech.)
PhD Student: Daniel Ennis
Indicators of esteem

Runnerup  DSWeb 2018 Software Contest: EMBER (Emergent and Macroscopic Behaviour ExtRaction)
AUTO Tutorial for localised patterns
With Bjorn Sandstede, this tutorial is part of the workshop The stability of coherent structures and patterns, (1112th June 2012). The course materials can be downloaded from:
Help on installing AUTO under various platforms can be found here.
EMBER (Emergent and Macroscopic Behaviour ExtRaction)
Stochastic continuation toolbox written in Java:
https://dsweb.siam.org/Software/emberemergentandmacroscopicbehaviourextraction
Runnerup  DSWeb 2018 Software Contest with Dr. Spencer Thomas (NPL) and Prof. Anne Skeldon (Surrey)
2D Localised Pattern Codes for the SwiftHohenberg equation
These codes (tgz) were created to produce all the figures in the paper:
Localized hexagon patterns in the planar SwiftHohenberg equation, DJB Lloyd, B Sandstede, D Avitabile and AR Champneys, SIAM J. Appl. Dyn. Sys. 7(3) 10491100, 2008. pdf.
and may be downloaded from the SIAM J. Appl. Dyn. Sys. webpage. The list of programs in localised_pattern_codes.tgz are given below:
Requirements: Matlab with optimization toolbox (tested on version 2007b) and AUTO07p.
(FSOLVE in the optimization toolbox is used to solve the BVPS. However, BVPS have been setup
so that any globalised Newton solver will work.)
To untar files use: tar xvzf localised_pattern_codes.tgz
Note: All subdirectories have README files to allow immediate running of all codes.
Matlab codes:
Matlab codes: 1D BVP solvers:
/1D_SH/solve_SH1D.m
 solves 1D quadratic/cubic SwiftHohenberg equation BVP on Half line. Finds a localised pulse and computes its stability with respect to perturbations on the full line. Uses Fourier differentiation matrices.
/1D_SH/solve_SH1Dfinite.m
 solves 1D quadratic/cubic SwiftHohenberg equation BVP on Half line. Finds a localised pulse and computes its stability with respect to perturbations on the full line. Uses finite differences and sparse matrices to speed up computations.
Matlab codes: 2D BVP solvers:
/BVPS/SH2DBVPFOUR_hex_10.m
 Solves 2D quadratic/cubic SwiftHohenberg equation BVP on the positive quadrant with Neumann BCS using Fourier differentiation matrices. Code finds a localised planar <10> hexagon pulse and plots the solution.
/BVPS/SH2DBVPFOUR_hex_11.m
 Solves 2D quadratic/cubic SwiftHohenberg equation BVP on the positive quadrant with Neumann BCS using Fourier differentiation matrices. Code finds a localised planar <11> hexagon pulse and plots the solution.
/BVPS/SH2DBVPFOUR_hexagon.m
 Solves 2D quadratic/cubic SwiftHohenberg equation BVP on the positive quadrant with Neumann BCS using Fourier differentiation matrices. Code finds a localised hexagon patch and plots the solution.
/BVPS/SH2DBVPFOUR_rhomboid.m
 Solves 2D quadratic/cubic SwiftHohenberg equation BVP on the positive quadrant with Neumann BCS using Fourier differentiation matrices. Code finds a localised rhomboid patch and plots the solution.
/Hexagon_Maxwell/Continue_Maxwell.m
 Gets initial data and continues Hexagon Maxwell curve in two parameters of the quadratic/cubic SwiftHohenberg equation. Calls compute_Maxwell.m and SH2DBVPFOUR.m
/radial_SH/solve_radial_SH.m
 solves quadratic/cubic radial SwiftHohenberg equation BVP on [0,L] with Neumann bcs at r=L. Uses L'Hopitals rule for r=0 boundary conditions. Finds a localised pulse and computes its stability with respect to perturbations on the half line. Uses finite differences and sparse matrices to speed up computations.
Matlab codes: 2D IVP solvers
/IVPS/swifthohen2DETD_hex.m
 solves quadratic/cubic SwiftHohenberg equation IVP with periodic BCs on [L,L]^2 computation is based on v = fft2(u) and firstorder exponential time stepping of Cox and Matthews (2002). Code computes figure 1(a) of "Localised Hexagon patterns in the planar SwiftHohenberg equation" by Lloyd, Sandstede, Avitabile and Champneys, SIADS 2008.
/IVPS/swifthohen2DETD_hexpatch.m
 solves quadratic/cubic SwiftHohenberg equation IVP with periodic BCs on [L,L]^2 computation is based on v = fft2(u) and firstorder exponential time stepping of Cox and Matthews (2002). Code finds a hexagon patch.
/IVPS/swifthohen2DETD_front10.m
 solves quadratic/cubic SwiftHohenberg equation IVP with periodic BCs on [L,L]^2 computation is based on v = fft2(u) and firstorder exponential time stepping of Cox and Matthews (2002). Code finds <10> hexagon pulse in the SwiftHohenberg equation.
/IVPS/swifthohen2DETD_front11.m
 solves quadratic/cubic SwiftHohenberg equation IVP with periodic BCs on [L,L]^2 computation is based on v = fft2(u) and firstorder exponential time stepping of Cox and Matthews (2002). Code finds <11> hexagon pulse in the SwiftHohenberg equation.
/IVPS/swifthohen2DETD_radial.m
 solves quadratic/cubic SwiftHohenberg equation IVP with periodic BCs on [L,L]^2 computation is based on v = fft2(u) and firstorder exponential time stepping of Cox and Matthews (2002). Code finds a localised ring in the SwiftHohenberg equation.
/IVPS/swifthohen2DETD_randompatch.m
 solves quadratic/cubic SwiftHohenberg equation IVP with periodic BCs on [L,L]^2 computation is based on v = fft2(u) and firstorder exponential time stepping of Cox and Matthews (2002). Code starts from a localised random patch.
AUTO codes:
Note: All codes are tested on AUTO07p. Initial data is supplied for immediate running. Conversion scripts and Matlab codes for data handling (procurement of initial data and postprocessing of AUTO output), are supplied. README files in each tar file for instruction on immediate running and data handling.
/Fourier_cont.tgz
 Code computes pulses on a finite cylinder of the quadratic/cubic SwiftHohenberg equation using a Fouriercosine projection in the circumference direction. Code computes <10> and <11> hexagon pulses on the half line/cylinder.
/periodicSH.tgz
 Continues periodic solutions, Maxwell curves and localised pulses of the 1D SwiftHohenberg equation with periodic boundary conditions.
/polarftSH.tgz
 Continues hexagon patches in the 2D quadratic/cubic SwiftHohenberg equation using a Fouriercosine projections in the angular direction as described in section 4.4 of "Localised Hexagon patterns in the planar SwiftHohenberg equation" by Lloyd, Sandstede, Avitabile and Champneys, SIADS 2008.
/radialodeSHC.tgz
 Continues radial localised pulses in the radial quadratic/cubic SwiftHohenberg equation.
/SH1Dstability.tgz
 Continues pulses in the 1D quadratic/cubic SwiftHohenberg equation and the leading eigenfunction.
/radialodeSH2hexeig.zip
 Continues radial pulses and hexagon eigenfunction in the quadratic/cubic SwiftHohenberg equation. Code traces out the hexagon pitchfork locus in the linear and quadratic bifurcation parameters.
/to_matlab_autox
 Converts AUTO output files b.foo and s.foo to matlab readable files. To use type $autox to_matlab.autox foo convertedfoo
Supervision
Postgraduate research supervision
 2019present: Steven Falconer (cosupervised with Drs. N. Santitissadeekorn (Maths), Nadia Smith and Spencer Thomas (National Physical Laboratory))
Project: Data analysis and modeling of the Royal College of GPs, Research and Surveillance data
 2019Present: Daniel Ennis (cosupervised with Dr. N. Santitissadeekorn (Maths), Prof. I. BruntonSmith (Sociology))
Project: Modeling and data analysis of crime
 2017Present: Daniel Hill (cosupervised with Dr. Matt Turner)
Project: Localised Ferrofluid Patterns
 2016Present: Jacob Brooks (cosupervised with Prof. Gianne Derks (Maths)).
Project: Nonlinear Wave Equations
 20122016: Craig Shenton (cosupervised with Prof. Angela Druckman (Centre for Environmental Strategy)).
Project: Food Security
 20112014: Tasos Rossides (cosupervised with Prof. Sergy Zelik (Maths)).
Thesis: Computing multilocalised structures for some parabolic PDE systems
 20102014: Gary Chaffey (cosupervised with Drs. Norman Kirkby (Civil Eng) and Anne Skeldon (Maths)).
Thesis: Modelling the Cell Cycle
 20102014: Jessica Rowden (cosupervised with Prof. Nigel Gilbert (Sociology)).
Thesis: Application of Two Mathematical Modelling Approaches for Real World Systems
 20072011. Jeremy Chamard
Thesis: Mountain Pass Algorithms and Applications
My publications
Publications
with application to urban crime data, Computational Statistics and Data Analysis 128 pp. 163183 Elsevier
? 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
realtime 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 stateparameter estimation for an inhomogeneous Poisson
process by deriving an approximating PoissonGamma ?Kalman? filter
that allows for uncertainty quantification. The ensemblebased implementation
of the filter is developed in a similar approach to the ensemble Kalman
filter, making the filter applicable to largescale 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 samplesize 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.
a spatially dependent scaling of the sineGordon potential term. The uncoupled inhomogeneous
sineGordon equation has stable stationary front solutions that persist in the coupled system.
Carrying out a numerical investigation it is found that these inhomogeneous sineGordon fronts
loose stability, provided the coupling between the two inhomogeneous sineGordon 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 ?hatlike? spatial inhomogeneity by introducing a fastslow structure and
using geometric singular perturbation theory.
We investigate stationary spatially localized hexagon patterns of the twodimensional (2D) SwiftHohenberg 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 oneparameter 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.
state) for general magnetic permeabilities. Using umerical continuation techniques we investigate the existence of localized hexagons in the YoungLaplace 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.
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 twodimensional SwiftHohenberg equation with either a quadraticcubic or cubicquintic
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 farfield wavenumber and a propagation
wavespeed whereas retreating parallel depinning fronts come in families where the wavespeed is
a function of the farfield wavenumber. Employing a farfield core decomposition, we propose
a boundary value problem for the invading depinning fronts which we numerically solve and
use pathfollowing 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 cubicquintic SwiftHohenberg equation
bifurcate off parallel invasion fronts and coexist 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 reactiondiffusion systems.