# Professor David Lloyd

## Academic and research departments

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

I am a Professor in the Department of Mathematics at the University of Surrey. My research interests are in localised pattern formation and mathematical modelling. I am also co-founder 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 higher-dimensional 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: Laura Jones

### Indicators of esteem

Runner-up - 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, (11-12th 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/ember-emergent-and-macroscopic-behaviour-extraction

Runner-up - DSWeb 2018 Software Contest with Dr. Spencer Thomas (NPL) and Prof. Anne Skeldon (Surrey)

**2D Localised Pattern Codes for the Swift-Hohenberg equation**

These codes (tgz) were created to produce all the figures in the paper:

*Localized hexagon patterns in the planar Swift-Hohenberg equation*, DJB Lloyd, B Sandstede, D Avitabile and AR Champneys, SIAM J. Appl. Dyn. Sys. 7(3) 1049-1100, 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 set-up

so that any globalised Newton solver will work.)

To untar files use: tar xvzf localised_pattern_codes.tgz

Note: All sub-directories 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 Swift-Hohenberg 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 Swift-Hohenberg 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 Swift-Hohenberg 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 Swift-Hohenberg 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 Swift-Hohenberg 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 Swift-Hohenberg 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 Swift-Hohenberg equation. Calls compute_Maxwell.m and SH2DBVPFOUR.m

/radial_SH/solve_radial_SH.m

- solves quadratic/cubic radial Swift-Hohenberg 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 Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code computes figure 1(a) of "Localised Hexagon patterns in the planar Swift-Hohenberg equation" by Lloyd, Sandstede, Avitabile and Champneys, SIADS 2008.

/IVPS/swifthohen2DETD_hexpatch.m

- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code finds a hexagon patch.

/IVPS/swifthohen2DETD_front10.m

- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code finds <10> hexagon pulse in the Swift-Hohenberg equation.

/IVPS/swifthohen2DETD_front11.m

- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code finds <11> hexagon pulse in the Swift-Hohenberg equation.

/IVPS/swifthohen2DETD_radial.m

- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code finds a localised ring in the Swift-Hohenberg equation.

/IVPS/swifthohen2DETD_randompatch.m

- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order 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 post-processing 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 Swift-Hohenberg equation using a Fourier-cosine 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 Swift-Hohenberg equation with periodic boundary conditions.

/polarftSH.tgz

- Continues hexagon patches in the 2D quadratic/cubic Swift-Hohenberg equation using a Fourier-cosine projections in the angular direction as described in section 4.4 of "Localised Hexagon patterns in the planar Swift-Hohenberg equation" by Lloyd, Sandstede, Avitabile and Champneys, SIADS 2008.

/radialodeSHC.tgz

- Continues radial localised pulses in the radial quadratic/cubic Swift-Hohenberg equation.

/SH1Dstability.tgz

- Continues pulses in the 1D quadratic/cubic Swift-Hohenberg equation and the leading eigenfunction.

/radialodeSH2hexeig.zip

- Continues radial pulses and hexagon eigenfunction in the quadratic/cubic Swift-Hohenberg 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

- 2020-present: Laura Jones (co-supervised with Prof. Ian Brunton-Smith (Sociology)

Project: Modelling social norms and crime

- 2019-present: Steven Falconer (co-supervised 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

- 2017-Present: Daniel Hill (co-supervised with Dr. Matt Turner)

Project: *Localised Ferrofluid Patterns*

- 2016-Present: Jacob Brooks (co-supervised with Prof. Gianne Derks (Maths)).

Project: *Nonlinear Wave Equations*

- 2012-2016: Craig Shenton (co-supervised with Prof. Angela Druckman (Centre for Environmental Strategy)).

Project: *Food Security *

- 2011-2014: Tasos Rossides (co-supervised with Prof. Sergy Zelik (Maths)).

Thesis: *Computing multi-localised structures for some parabolic PDE systems*

- 2010-2014: Gary Chaffey (co-supervised with Drs. Norman Kirkby (Civil Eng) and Anne Skeldon (Maths)).

Thesis: *Modelling the Cell Cycle*

- 2010-2014: Jessica Rowden (co-supervised with Prof. Nigel Gilbert (Sociology)).

Thesis: *Application of Two Mathematical Modelling Approaches for Real World Systems*

- 2007-2011. Jeremy Chamard

Thesis: *Mountain Pass Algorithms and Applications*