# Professor David J.B. Lloyd

## Academic and research departments

Mathematics at the Interface Group, Centre for Mathematical and Computational Biology, Centre for Criminology, School of Mathematics and Physics.## About

### 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 with applications to criminology and space. I am the Society for Industrial and Applied Mathematics 2024 T. Brook Benjamin Prize winner, and 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.

### Areas of specialism

### University roles and responsibilities

- Co-founder and co-director of the Surrey Centre for Criminology

## 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), Jason Bramburger.

PhD Students: Tasos Rossides, Jacob Brooks, Daniel Hill

#### Mathematical criminology and data Science

Research here focuses on modelling of crime and social norms associated with crime (e.g. collective efficacy), data science and assimilation techniques, and network analysis.

Collaborators: Drs. Naratip Santitissadeekorn (Surrey), and Martin B. Short (Georgia Tech.), Giulia Berlusconi (Surrey - Sociology), Ian Brunton-Smith (Surrey - Sociology), and Martin B. Short (Georgia Tech. - Maths)

PhD Students: Laura Jones, Steve Falconer, Jessica Furber, Daniel Catlin

Industry: Surrey Police, APHA

#### Mathematics for Space Applications

Research here focuses on astrodynamics and satellite trajectory design in uncertain environments.

Collaborators: Nicola Baresi (Surrey - Space Eng)

PhD Student: Giacomo Acciarini

Industry: European Space Agency

### Indicators of esteem

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

Society for Industrial and Applied Mathematics 2024 T. Brooke Benjamin Prize winner

Fellow of the Institute of Mathematics and Its Applications 2024

Front Cover of Nonlinearity journal, January 2024

**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

- 2022-present: Daniel Catlin (co-supervised with Dr. Giulia Berlusconi (Sociology)). Project: Dynamic network analysis of criminal networks.
- 2022-present: Giacomo Acciarini (co-supervised with Dr. Nicola Baresi (Surrey Space Centre) and Dario Izzo (European Space Agency)). Project: Stochastic continuation for space trajectory design in uncertain environments.
- 2021-present: Jessica Furber (co-supervised with Drs. Stefan Klus (Maths) and Giovanni Lo Lacono (Vet School)). Project: Modelling Badger movement.
- 2020-present: Laura Jones (co-supervised with Prof. Ian Brunton-Smith (Sociology)

Project: Modelling collective efficacy and crime

- 2019-2023: 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

- 2017-2021: Daniel Hill (co-supervised with Dr. Matt Turner)

Project: *Localised Ferrofluid Patterns*

- 2016-2020: 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*

## Publications

Small bodies are ubiquitous in our Solar System, and they constitute a key element in understanding the origin of Earth and the emergence of life. Yet navigating a spacecraft around these bodies is very challenging, due to the difficulties of fully observing and characterizing these environments from the ground. These difficulties often translate into large uncertainties in the parameters that characterize these dynamical systems, ranging from uncertainties in the shape and mass distribution of the target bodies to those regarding the position and velocity of the spacecraft that navigates them. Small-body environments remain among the most perturbed and chaotic, making preliminary mission analysis particularly challenging. In particular, since the discovery of the first binary asteroids (Ida-Dactyl), binary systems have attracted much interest due to their considerable number (about 15% of the near-Earth orbit population) and their potential to reveal hints about the formation and evolution of our Solar System. Previous studies have modeled the dynamical environment of these systems using a perturbed version of the circular restricted three-body problem (CR3BP), where solar radiation pressure and irregularities of the gravity field of the body are accounted for. However, these analyses have predominantly focused on deterministic periodic orbits, which are then perturbed to examine sensitivity concerning initial conditions and/or parameter uncertainties. More recently, stochastic continuation approaches have been identified as a promising tool for integrating uncertainties into preliminary mission design for three-body systems. Unlike traditional iterative procedures, these techniques directly incorporate uncertainties, offering a more streamlined approach. By identifying natural regions of motion where spacecraft are statistically more likely to maintain periodic orbits, these methods offer a robust framework for bounded motion analysis. Expanding upon that, we extend the approach to small bodies, accounting for the irregularities in their gravity field. As a case study, we will apply these methodologies to the Hera spacecraft's mission to the Didymos & Dimorphos binary system, showing how these techniques can serve as a powerful preliminary mission design tool to identify safe regions of bounded motion around small bodies.

Numerical continuation techniques are powerful tools that have been extensively used to identify particular solutions of nonlinear dynamical systems and enable trajectory design in chaotic astrodynamics problems such as the Circular Restricted Three-Body Problem. However , the applicability of equilibrium points and periodic orbits may be questionable in real-world applications where the uncertainties of the initial conditions of the spacecraft and dynamical parameters of the problem (e.g., mass ratio parameter) are taken into consideration. Usually, the robustness of a candidate trajectory is tested via a two-step approach, whereby trajectories are first designed in a deterministic scenario, and then Monte Carlo methods are a posteriori used to check their robustness. While this strategy is ubiquitous in preliminary mission design, it can however lead to time-consuming and potentially not robust solutions, meaning that the found tra-jectories are not designed to account for uncertainties. Instead, the robustness of the determin-istic optimal solutions is usually ensured. Due to uncertain parameters and initial conditions, the spacecraft might not follow the reference periodic orbit owing to growing uncertainties that cause the satellite to deviate from its nominal path. Hence, it is crucial to keep track of the probability of finding the spacecraft in a given region. Building on previous work, we extend numerical continuation to moments of the distribution (i.e., stochastic continuation) by directly continuing moments of the probability density function of the spacecraft state. Only assuming normality of the initial conditions, and leveraging moment-generating functions, Isserlis' theorem, and the algebra of truncated polynomials, we propagate the distribution of the spacecraft state at consecutive surface of section crossings while retaining a symbolic map of the final moments of the distribution that depend on the initial mean and covariance matrix only. While the technique is only valid for initial Gaussian distributions, it does not assume that the distribution maintains its normality throughout the integration. The symbolic Poincaré map can then be directly used to evaluate the final moments of an initial distribution , as a function of the initial mean and covariance. This can therefore be used to accelerate the evolving step in the stochastic continuation procedure. The goal of the work is to offer a differential algebra-based general framework to continue 3D periodic orbits in the presence of uncertain dynamical systems. The proposed approach is compared against traditional Monte Carlo simulations to validate the uncertainty propagation approach and demonstrate the advantages of the proposed in terms of uncertainty propagation computational burden and access to higher-dimensional problems.

Many networks have event-driven dynamics (such as communication, social media and criminal networks), where the mean rate of the events occurring at a node in the network changes according to the occurrence of other events in the network. In particular, events associated with a node of the network could increase the rate of events at other nodes, depending on their influence relationship. Thus, it is of interest to use temporal data to uncover the directional, time-dependent, influence structure of a given network while also quantifying uncertainty even when knowledge of a physical network is lacking. Typically, methods for inferring the influence structure in networks require knowledge of a physical network or are only able to infer small network structures. In this paper, we model event-driven dynamics on a network by a multidimensional Hawkes process. We then develop a novel ensemble-based filtering approach for a time-series of count data (i.e., data that provides the number of events per unit time for each node in the network) that not only tracks the influence network structure over time but also approximates the uncertainty via ensemble spread. The method overcomes several deficiencies in existing methods such as existing methods for inferring multidimensional Hawkes processes are too slow to be practical for any network over ∼ 50 nodes, can only deal with timestamp data (i.e. data on just when events occur not the number of events at each node), and that we do not need a physical network to start with. Our method is massively parallelizable, allowing for its use to infer the influence structure of large networks (∼ 10, 000 nodes). We demonstrate our method for large networks using both synthetic and real-world email communication data.

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.

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.

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.

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.

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.

This paper explores the application of stochastic continuation methods in the context of mission analysis for spacecraft trajectories around libration points in the planar circular restricted three-body problem. Traditional deterministic approaches have limitations in accounting for uncertainties, requiring a two-step process involving Monte Carlo techniques for assessing the robustness of the deterministic design. This might lead to suboptimal solutions and to a long and time-consuming design process. Stochastic continuation methods, which extend numerical continuation techniques to moments of probability density functions, offer a promising alternative. This paper aims to pioneer the application of stochastic continuation procedures in mission analysis, incorporating and acknowledging the stochastic nature of spacecraft missions from the early design phases. By extending existing frameworks to handle fixed points of stroboscopic or Poincaré mappings, the study focuses on robustifying and enhancing trajectory design by considering uncertainties in the determination of periodic orbits. The proposed approach has the potential to discover new solutions that may remain hidden in deterministic analyses, offering improved mission design outcomes. Specifically, this work concentrates on the planar circular restricted three-body problem, assuming uncertainties in both initial conditions and the mass ratio parameter. Stochastic continuation is employed to identify equilibrium points and periodic orbits in this uncertain dynamical system. The generalization of steady states and periodic orbits in uncertain environments is discussed, demonstrating the effectiveness of stochastic continuation in identifying safe operational regions in uncertain astrodynamics problems.

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.

This paper investigates the existence of localised axisymmetric (radial) patterns on the surface of a ferrofluid in the presence of a uniform vertical magnetic field. We formally investigate all possible small-amplitude solutions which remain bounded close to the pattern's centre (the core region) and decay exponentially away from the pattern's centre (the far-field region). The results are presented for a finite-depth, infinite expanse of ferrofluid equipped with a linear magnetisation law. These patterns bifurcate at the Rosensweig instability, where the applied magnetic field strength reaches a critical threshold. Techniques for finding localised solutions to a non-autonomous PDE system are established; solutions are decomposed onto a basis which is independent of the radius, reducing the problem to an infinite set of nonlinear, non-autonomous ODEs. Using radial centre manifold theory, local manifolds of small-amplitude solutions are constructed in the core and far-field regions, respectively. Finally, using geometric blow-up coordinates, we match the core and far-field manifolds; any solution that lies on this intersection is a localised radial pattern. Three distinct classes of stationary radial solutions are found: spot A and spot B solutions, which are equipped with two different amplitude scaling laws and achieve their maximum amplitudes at the core, and ring solutions, which achieve their maximum amplitudes away from the core. These solutions correspond exactly to the classes of localised radial solutions found for the Swift-Hohenberg equation. Different values of the linear magnetisation and depth of the ferrofluid are investigated and parameter regions in which the various localised radial solutions emerge are identified. The approach taken in this paper outlines a route to rigorously establishing the existence of axisymmetric localised patterns in the future.

We formulate an effective numerical scheme that can readily, and accurately, calculate the dynamics of weakly interacting multi-pulse solutions of the quintic complex Ginzburg-Landau equation (QCGLE) in one space dimension. The scheme is based on a global centre-manifold reduction where one considers the solution of the QCGLE as the composition of individual pulses plus a remainder function, which is orthogonal to the adjoint eigenfunctions of the linearised operator about a single pulse. This centre-manifold projection overcomes the difficulties of other, more orthodox, numerical schemes, by yielding a fast-slow system describing 'slow' ordinary differential equations for the locations and phases of the individual pulses, and a 'fast' partial differential equation for the remainder function. With small parameter ϵ = e −λr d 0 where λr is a constant and d0 > 0 is the minimal pulse separation distance, we write the fast-slow system in terms of first-order and second-order correction terms only, a formulation which is solved more efficiently than the full system. This fast-slow system is integrated numerically using adaptive time-stepping. Results are presented here for two-and three-pulse interactions. For the two-pulse problem, cells of periodic behaviour, separated by an infinite set of heteroclinic orbits, are shown to 'split' under perturbation creating complex spiral behaviour. For the case of three pulse interaction a range of dynamics, including chaotic pulse interaction, are found. While results are presented for pulse interaction in the QCGLE, the numerical scheme can also be applied to a wider class of parabolic PDEs.

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.

Collective organisation of patterns into ring-like configurations has been well-studied when patterns are subject to either weak or semi-strong interactions. However, little is known numerically or analytically about their formation when the patterns are strongly interacting. We prove that approximate strongly interacting patterns can emerge in various ring-like dihedral configurations, bifurcating from quiescence near a Turing instability in generic two-component reaction-diffusion systems. The methods used are constructive and provide accurate initial conditions for numerical continuation methods to path-follow these ring-like patterns in parameter space. Our analysis is complemented by numerical investigations that illustrate our findings.

The primary focus is a sequential data assimilation method for count data modelled by an inhomogeneous Poisson process. In particular, a quadratic approximation technique similar to the extended Kalman filter is applied to develop a sub-optimal, discrete-time, filtering algorithm, called the extended Poisson-Kalman filter (ExPKF), where only the mean and covariance are sequentially updated using count data via the Poisson likelihood function. The performance of ExPKF is investigated in several synthetic experiments where the true solution is known. In numerical examples, ExPKF provides a good estimate of the “true” posterior mean, which can be well-approximated by the particle filter (PF) algorithm in the very large sample size limit. In addition, the experiments demonstrates that the ExPKF algorithm can be conveniently used to track parameter changes; on the other hand, a non-filtering framework such as a maximum likelihood estimation (MLE) would require a statistical test for change points or implement time-varying parameters. Finally, to demonstrate the model on real-world data, the ExPKF is used to approximate the uncertainty of urban crime intensity and parameters for self-exciting crime models. The Chicago Police Department’s CLEAR (Citizen Law Enforcement Analysis and Reporting) system data is used as a case study for both univariate and multivariate Hawkes models. An improved goodness of fit measured by the Kolomogrov-Smirnov (KS) statistics is achieved by the filtered intensity. The potential of using filtered intensity to improve police patrolling prioritisation is also tested. By comparing with the prioritisation based on MLE-derived intensity and historical frequency, the result suggests an insignificant difference between them. While the filter is developed and tested in the context of urban crime, it has the potential to make a contribution to data assimilation in other application areas.

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.

Analytically tracking patterns emerging from a small amplitude Turing instability to large amplitude remains a challenge as no general theory exists. In this paper, we consider a three component reaction-diffusion system with one of its components singularly perturbed, this component is known as the fast variable. We develop an analytical theory describing the periodic patterns emerging from a Turing instability using geometric singular perturbation theory. We show analytically that after the initial Turing instability, spatially periodic patterns evolve into a small amplitude spike in the fast variable whose amplitude grows as one moves away from onset. This is followed by a secondary transition where the spike in the fast variable widens, its periodic pattern develops two sharp transitions between two flat states and the amplitudes of the other variables grow. The final type of transition we uncover analytically is where the flat states of the fast variable develop structure in the periodic pattern. The analysis is illustrated and motivated by a numerical investigation. We conclude with a preliminary numerical investigation where we uncover more complicated periodic patterns and snaking-like behaviour that are driven by the three transitions analysed in this paper. This paper provides a crucial step towards understanding how periodic patterns transition from a Turing instability to large amplitude

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.

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.

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.

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.

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.

Data assimilation techniques, such as ensemble Kalman filtering, have been shown to be a highly effective and efficient way to combine noisy data with a mathematical model to track and forecast dynamical systems. However, when dealing with high-dimensional data, in many situations one does not have a model, so data assimilation techniques cannot be applied. In this paper, we use dynamic mode decomposition to generate a low-dimensional, linear model of a dynamical system directly from high-dimensional data, which is defined by temporal and spatial modes, that we can then use with data assimilation techniques such as the ensemble Kalman filter. We show how the dynamic mode decomposition can be combined with the ensemble Kalman filter (which we call the DMDEnKF) to iteratively update the current state and temporal modes as new data becomes available. We demonstrate that this approach is able to track time varying dynamical systems in synthetic examples, and experiment with the use of time-delay embeddings. We then apply the DMDEnKF to real world seasonal influenza-like illness data from the USA Centers for Disease Control and Prevention, and find that for short term forecasting, the DMDEnKF is comparable to the best mechanistic models in the ILINet competition.

Many complex systems occurring in the natural or social sciences or economics are frequently described on a microscopic level, e.g., by latticeor agent-based models. To analyze the states of such systems and their bifurcation structure on the level of macroscopic observables, one has to rely on equation-free methods like stochastic continuation. Here we investigate how to improve stochastic continuation techniques by adaptively choosing the parameters of the algorithm. This allows one to obtain bifurcation diagrams quite accurately, especially near bifurcation points. We introduce lifting techniques which generate microscopic states with a naturally grown structure, which can be crucial for a reliable evaluation of macroscopic quantities. We show how to calculate fixed points of fluctuating functions by employing suitable linear fits. This procedure offers a simple measure of the statistical error. We demonstrate these improvements by applying the approach in analyses of (i) the Ising model in two dimensions, (ii) an active Ising model, and (iii) a stochastic Swift-Hohenberg model. We conclude by discussing the abilities and remaining problems of the technique.

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.

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.

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.

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.

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.

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.

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.

We investigate the bifurcation structure of stationary localized patterns of the two dimensional Swift–Hohenberg equation on an infinitely long cylinder and on the plane. On cylinders, we find localized roll, square, and stripe patches that exhibit snaking and nonsnaking behavior on the same bifurcation branch. Some of these patterns snake between four saddle-node limits; in this case, recent analytical results predict the existence of a rich bifurcation structure to asymmetric solutions, and we trace out these branches and the PDE spectra along these branches. On the plane, we study the bifurcation structure of fully localized roll structures, which are often referred to as worms. In all the above cases, we use geometric ideas and spatial-dynamics techniques to explain the phenomena that we encounter.

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.

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.

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.

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.

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.

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.

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.

Fully localised patterns involving cellular hexagons or squares have been found experimentally and numerically in various continuum models. However, there is currently no mathematical theory for the emergence of these localised cellular patterns from a quiescent state. A key issue is that standard techniques for one-dimensional patterns have proven insufficient for understanding localisation in higher dimensions. In this work, we present a comprehensive approach to this problem by using techniques developed in the study of axisymmetric patterns. Our analysis covers localised patterns equipped with a wide range of dihedral symmetries, avoiding a restriction to solutions on a predetermined lattice. The context in this paper is a theory for the emergence of such patterns near a Turing instability for a general class of planar reaction-diffusion equations. Posing the reaction-diffusion system in polar coordinates, we carry out a finite-mode Fourier decomposition in the angular variable to yield a large system of coupled radial ordinary differential equations. We then utilise various radial spatial dynamics methods, such as invariant manifolds, rescaling charts, and normal form analysis, leading to an algebraic matching condition for localised patterns to exist in the finite-mode reduction. This algebraic matching condition is nontrivial, which we solve via a combination of by-hand calculations and Gröbner bases from polynomial algebra to reveal the existence of a plethora of localised dihedral patterns. These results capture the essence of the emergent localised hexagonal patterns witnessed in experiments. Moreover, we combine computer-assisted analysis and a Newton-Kantorovich procedure to prove the existence of localised patches with 6m-fold symmetry for arbitrarily large Fourier decompositions. This includes the localised hexagon patches that have been elusive to analytical treatment.

Stationary fronts connecting the trivial state and a cellular (distorted) hexagonal pattern in the Swift-Hohenberg equation with a quadratic-cubic nonlinearity are known to undergo a process of infinitely many folds as a parameter is varied, known as homoclinic snaking, where new hexagon cells are added to the core, leading to a region of infinitely-many, co-existing localized states. Outside the homoclinic snaking region, the hexagon fronts can invade the trivial state in a bursting fashion. In this paper, we use a far-field core decomposition to set up a numerical path-following routine to trace out the bifurcation diagrams of hexagon fronts for the two main orientations of cellular hexagon pattern with respect to the interface in the bistable region. We find for one orientation that the hexagon fronts can destabilize as the istorted hexagons are stretched in the transverse direction leading to defects occurring in the deposited cellular pattern. We then plot diagrams showing when the selected fronts for the two main orientations, aligned perpendicular to each other, are compatible leading to a hexagon wavenumber selection prediction for hexagon patches on the plane. Finally, we verify the compatibility criterion for hexagon patches in the Swift-Hohenberg equation with a quadratic-cubic nonlinearity and a large non-variational perturbation. The numerical algorithms presented in this paper can be adapted to general reaction-diffusion systems.