m.sc. in mathematical modelling and scienti c …. in mathematical modelling and scienti c computing...
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M.Sc. in Mathematical Modelling and Scientific Computing
Dissertation Projects
March 2016
Contents
1 Numerical Analysis Projects 3
1.1 Iterative Solution Methods for Toeplitz Linear Systems . . . . . . . . . . . 3
1.2 All-At-Once Methods for Initial Value Problems . . . . . . . . . . . . . . 3
1.3 Chebfun Dissertation Topics . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 High Order Finite Volume Element Methods . . . . . . . . . . . . . . . . 5
1.5 Coupling Brinkman and Darcy Flows in Axisymmetric Domains . . . . . 6
1.6 Data Science, Compressed Sensing or Matrix Completion . . . . . . . . . 7
1.7 Multilevel Radial Basis Function Approximation of PDEs . . . . . . . . . 8
1.8 Combining Rough Paths Theory with Machine Learning and its Applica-tion in Chinese Handwriting Recognition . . . . . . . . . . . . . . . . . . . 8
1.9 Energy Profiling Using Integer Programming . . . . . . . . . . . . . . . . 10
2 Biological and Medical Application Projects 12
2.1 Multi-Species Interaction within Porous Living Tissue . . . . . . . . . . . 12
2.2 Modelling Syringomyelia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 High Speed Detachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Two-Phase Flow Models for Active Cell Motion . . . . . . . . . . . . . . . 16
2.5 Learning Strategies for an Iterated Game with an Unknown Asymmetry . 16
3 Physical Application Projects 18
3.1 Collective Behaviour: From Robot Experiments to Mathematical Models . 18
3.2 Algorithmic Classification of Writing Styles via Time-Series Analysis ofPunctuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 A Model for Thermoelastic Contact Oscillations . . . . . . . . . . . . . . . 20
3.4 Homogenized Boundary Conditions for Faraday Cages . . . . . . . . . . . 21
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3.5 Fluid System Models for Drip Coffee Makers . . . . . . . . . . . . . . . . 22
3.6 Wave Propagation in Granular Crystals . . . . . . . . . . . . . . . . . . . 23
3.7 Stability of Initially Slow Viscous Two-Phase Jets Driven by Gravity . . . 26
3.8 Interaction of Multiple Incoherent Wave-Packets and Optical Turbulence . 28
3.9 Nonlinear Waves along Interacting nearly Parallel Vortex Filaments . . . 30
4 Networks 33
4.1 Continuous-Time Analysis of Temporal Networks . . . . . . . . . . . . . . 33
4.2 Context-Dependent Metabolic Networks: Structure and Dynamics . . . . 34
4.3 Optimal Control of Contagions on Networks . . . . . . . . . . . . . . . . . 35
4.4 Predator-Prey-Subsidy Population Dynamics on Networks . . . . . . . . . 38
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1 Numerical Analysis Projects
Numerical Linear Algebra and Numerical Methods for Partial Differ-ential Equations
Supervisor: Prof. Andy WathenContact: [email protected]
I’m happy to supervise generally projects in the area of Numerical Linear Algebra andNumerical Methods for Partial Differential Equations, so please come and talk with meabout any ideas if you wish. Specific suggestions this year are listed below.
1.1 Iterative Solution Methods for Toeplitz Linear Systems
Supervisor: Prof. Andy WathenContact: [email protected]
A Toeplitz matrix is a matrix with constant diagonals, so the entries of its first row andcolumn for example completely determine it. For real symmetric Toeplitz methods, wayback in 1986, Gil Strang suggested the use of circulant matrices as preconditioners foriterative methods like Conjugate Gradients for Toeplitz matrices. Recently, Jen Pestanaand myself have introduced another simple trick which allows Strang’s approach to beapplied for non-symmetric Toeplitz matrices.
This project would be to consider Strang’s proposal and our more recent work andexplain some of the theory as well as compute a series of examples to demonstrate thecapability of these approaches.
References
[1] Strang, G. (1986) A proposal for Toeplitz matrix calculations, Studies in AppliedMaths, 74, 171–176.
[2] Pestana, J. and Wathen, A. (2015) A preconditioned MINRES method for nonsym-metric Toeplitz matrices, SIAM J. Matrix Anal. Appl., 36, 273–288.
1.2 All-At-Once Methods for Initial Value Problems
Supervisor: Prof. Andy WathenContact: [email protected]
The idea of time-stepping is ubiquitous across numerical methods for time-dependentODEs and PDEs. In different contexts there has been recent interest in so-called all-at-once methods where a system of equations is written down and used to determine thesolution at all timesteps. Time-stepping turns out to be a simple block forward substitu-tion in this context. Other solution methods can however be employed and the entirelysequential nature of time-stepping can be replaced, for example, by iterative solution
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methods for these equations. In some situations, such iterations can be guaranteed toconverge in many fewer steps than the total number of timesteps; such methods are ofinterest therefore in the context of parallel computing, though they are interesting inand of themselves as rather non-standard methodology.
This project would be to consider, describe and investigate all-at-once approaches forODEs and possibly for some simple time-dependent PDEs like the heat equation. Thisis really a project about the numerical linear algebra of discretisations rather than aproject about discretisation methods themselves, so please don’t be misled!
Reference
[1] McDonald, E., Hon S., Pestana, J. and Wathen, A. (2015) Preconditioning for non-symmetry and time-dependence, to appear in Proceedings of the 23rd Domain Decom-position Conference. (I can provide a copy to anyone potentially interested)
1.3 Chebfun Dissertation Topics
Supervisor: Prof. Nick Trefethen in collaboration with other members of the Cheb-fun teamContact: [email protected]
Chebfun is an open-source algorithms and software project for numerical computationwith 1D, 2D and 3D functions, based on the idea of overloading Matlab’s vectors andmatrices to functions and operators. Chebfun can do almost anything in 1D (integration,optimization, rootfinding, differential equations,. . . ) and quite a bit in 2D and 3D too.See www.chebfun.org, especially the Users Guide and the large collection of Examples.
A number of M.Sc. dissertations related to Chebfun have been written over the years.There are many possibilities and we can tailor the project to the student’s interests andexpertise in areas including differential equations, approximation theory, quadrature,rootfinding, linear algebra, complex analysis, optimization, and modelling (especially ofvirtually anything to do with ODEs).
Here are two specific possibilities.
(1) O.D.E. algorithms near boundary and interior layers A big topic in appliedmathematics, and ODEs, is the behaviour of boundary and interior layers, both linearand nonlinear. For example, a starting question is often to determine how their thicknessscales as a parameter epsilon approaches zero. Chebfun provides a beautiful laboratory inwhich to explore such matters, and it also raises basic algorithmic questions. Chebfun’srepresentations of functions are by high-order polynomials, not adaptive, but it canincorporate breakpoints between different polynomials. Sometimes great benefits canbe gained by introducing such breakpoints. What is the mathematics of this? Can theprocess be automated? See e.g. http://www.chebfun.org/examples/ode-linear/
Breakpoints.html.
(2) Reaction-diffusion equations A beautiful (and very Oxford) topic is the appear-
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ance of stripes, spots, and other structure in solutions of nonlinear reaction-diffusionequations. Recently, through the work Niall Bootland and Hadrien Montanelli, Cheb-fun has acquired powerful algorithms in this area based on exponential integrators —spin, spin2, spin3. It would be very interesting to pick a scientifically interesting prob-lem in this domain, like the Grey-Scott equations, and take advantage of this newtool to explore its behavior in new directions. (See https://people.maths.ox.ac.
uk/trefethen/pdectb/reaction2.pdf.)
The Chebfun team consists of about 10 people, and M.Sc. students doing projects inthis area would be welcome to participate in our weekly team meetings.
1.4 High Order Finite Volume Element Methods
Supervisor: Dr Ricardo Ruiz BaierContact: [email protected]
Scope: Despite numerous advances in handling the complexity of flow equations equa-tions, most numerical techniques used by practitioners still lack essential features toreliably couple flow and transport processes. In addition to efficiency, it is crucial thatthe schemes are accurate and robust under various ranges of model parameters and ge-ometry configurations. Moreover, discrete mass conservation is key to avoid artificialsinks or sources. Unfortunately, up to date there is no ultimate numerical tool ableto resolve all these issues at once: Some methods are easy to implement and can bereadily parallelized while others are more suitable for unstructured meshes and compli-cated geometries, or mass conservative by construction, or allow the natural derivationof error estimates, etc. Consequently, to resolve multiphysics problems, one must resortto schemes that combine, at least some of, these properties.
This project deals in particular with mixed finite finite volume element (FVE) formula-tions for coupled flow and transport problems (see the review in [2]). They feature localmass conservativity, flexibility for choosing accurate numerical fluxes, smaller controlvolumes, and suitability for error analysis in the L2-norm, however their analysis hasbeen restricted so far to low order approximations. Other hybrid methods for flow andtransport include [1,3]. All of these schemes exploit intrinsic joint properties of finiteelement (FE) and finite volume (FV)-based formulations. The envisaged schemes willshow robustness and applicability in a wide range of problems, whilst being suitable forestablishing well-posedness and deriving error estimates.
Goals of the project. Theoretical (derivation of error estimates for an extension ofthe classical Taylor-Hood method) and/or computational (implementation of the FVEmethod in 2D. A working code for pure diffusion will be provided).
Remarks. The main directions of the project can be discussed and modified accordinglyto the skills or preferences of the student.
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References
[1] Bergamaschi, L., Mantica, S., and Manzini, G. (1998) A mixed finite element-finitevolume formulation of the black-oil model, SIAM J. Sci. Comput., 20, 970–997.
[2] Cui, M. and Ye, X. (2010) Unified analysis of finite volume methods for the Stokesequations, SIAM J. Numer. Anal., 48, 824–839.
[3] Ohlberger, M. (1997) Convergence of a mixed finite element-finite volume method forthe two phase flow in porous media, East-West J. Numer. Math., 5, 183–210.
1.5 Coupling Brinkman and Darcy Flows in Axisymmetric Domains
Supervisor: Dr Ricardo Ruiz BaierContact: [email protected]
Scope: Let us consider an incompressible fluid within a bounded three-dimensionalporous domain Ω. A clearly identified interface Σ exists in the medium, separating tworegions where the flow passes from viscous to a non-viscous regime. Such a configurationsuggests to express momentum and mass conservation in each of these regions involvingBrinkman and Darcy equations for the viscous and non-viscous subdomains (ΩB, ΩD),respectively. We assume that ΩB consists of an array of low concentration fixed particles,whereas ΩD is a classical porous medium constituted by connected porous matrices. Thesystem can be written as [1]: Find the velocity u and pressure field p satisfying
σu− ν∆u +∇p = fB in ΩB,
µu +∇p = fD in ΩD,
div u = 0 in Ω,
(u|ΩB− u|ΩD
) · n = 0 on Σ,
p|ΩB− p|ΩD
= 0 on Σ,
u · n = 0 on Γ.
The numerical analysis of similar coupled problems using Stokes description of viscousflow has been extensively studied in recent years. Fully mixed, mortar-based, Nitsche,weak Galerkin, discontinuous Galerkin, and other methods have been successfully ap-plied, also covering the nonlinear case, but the specific form of the model we are interestedin, still represents an open problem.
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ΩB
ΩD
ΓB
Σ
ΓD
ΩB
ΣΓ0
Ω Ω
ΩD
Three dimensional and axisymmetric domains Ω, Ω split into two different porous regions with
inverse permeabilities σ and µ.
Goals of the project. The task would be to get acquainted with the formulation andfinite element analysis using classical theory of saddle point problems, to implement thecoupling using a finite element library such as FreeFem++ or FEniCS, to verify theproperties of the numerical scheme (convergence, stability), and to use the method athand to study certain cases of interest in applications to groundwater infiltration andliving tissue perfusion. A partial result is recently available [2].
The project will allow the student to learn (or to refine his/her expertise in) fluid dynam-ics in porous media, theory of PDEs involving saddle-point problems, implementationof large scale solvers, and finite element discretization techniques.
Remarks. The main directions of the project can be discussed and modified accordinglyto the skills or preferences of the student.
References
[1] Alvarez, M., Gatica G. N., and Ruiz-Baier, R. Analysis of a vorticity-based fully-mixedformulation for the 3D Brinkman-Darcy problem. (Submitted 2015).
[2] Anaya, V., Mora, D., Reales, C. and Ruiz-Baier, R. Solvability analysis for theBrinkman-Darcy coupling in axisymmetric domains. (In preparation.)
1.6 Data Science, Compressed Sensing or Matrix Completion
Supervisor: Prof. Jared TannerContact: [email protected]
I would be open to supervising a project in the area of data science, compressed sensingor matrix completion. If you are interested in a project in this area, then please get intouch!
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1.7 Multilevel Radial Basis Function Approximation of PDEs
Supervisor: Dr Kathryn GillowContact: [email protected]
Ref. [1] describes how to use compactly supported radial basis functions (RBFs) withina multiscale setting in order to solve partial differential equations. Each level usescompactly supported radial basis functions of smaller scale on an increasingly fine meshin order to find correction terms to the solution on the original coarse mesh. A symmetriccollocation method is utilised. The idea of this project is to extend the work in order tosee how well such a method can approximate linear functionals of the solution to partialdifferential equations.
The project would begin by reviewing work already undertaken in multilevel RBF meth-ods both for interpolation and for solving PDEs. We would then extend this to look atfunctionals which might take the form of point values, an integral of the solution overthe domain, or a flux along part of the domain boundary. The project would involveboth reviewing existing theory and computing in Matlab.
Reference
[1] Farrell, P., and Wendland, H. (2013) RBF multiscale collocation for second orderelliptic boundary value problems, SIAM J. Numer. Anal., 51(4), 2403–2425.
1.8 Combining Rough Paths Theory with Machine Learning and itsApplication in Chinese Handwriting Recognition
Supervisors: Prof. Terry Lyons and Dr Hao NiContact: [email protected] and [email protected]
Background In the past few years, deep learning has become one of the most popularmachine learning methods and has achieved some notable success in speech and imagerecognition [1]. Rough paths theory provides a general mathematical framework to de-scribe high oscillatory interacting systems driven by complex data streams [2]. Recently,it was found to provide a novel feature set for data streams the signature of a stream [3].Dr. Ben Graham won the first place in task 3 of online Chinese handwriting recognitionat the 12th International Conference on Document Analysis and Recognition (ICDAR2013) using the signature feature set and convolutional neural network (CNN), for itssuperior accuracy ([4], [5]). This result has been further improved by combining withdomain specific knowledge by the team from South China University of Technology [7].
Requirements For all projects, candidates should have a strong background in pro-gramming in C++ as well as a solid mathematical background in algebra and numericalanalysis. For the third project, experience in Linux system would be desirable.
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Project Description This project aims to find “better features” for online Chinesehandwriting classification problem in terms of the accuracy and efficiency of the algo-rithm. We propose the use of truncated signatures over dyadic sub-time intervals asthe new feature set. Instead of using CNN, as in [4], [5] and [7], we combine this novelfeature set with fully connected neural network. We will benchmark our results withstate-of-art methods. In this project we aim to further develop the existing package1 toimprove the efficiency of computing the signature feature sets, and utilize this packagealongside the standard deep learning platform, Caffe, to implement our method.
Potential projects are:
1. Implement efficient algorithms to solve large-scale matrix multiplication problemsusing matrix template library (mtl4). Apply them to compute the tensor productof two truncated signatures.
2. Write an abstract class of path that takes data streams of different kinds as inputvariables. Write a function which takes a path and the level of dyadic partition,and the level of truncated signatures as inputs, and outputs the collection of thetruncated signature of the path over sub-time intervals.
3. Implement, using Caffe, a fully connected neural network for the dataset thatconsists of pre-computed signature features and the corresponding labels for clas-sification.
Outcome The project will contribute to improving the efficiency and accuracy of thesignature methods for online Chinese hand-writing recognition. Students are expectedto get exposure on the neural network and rough paths theory, and improve their codingskills.
References
[1] LeCun, Y., Bengio, Y., and Hinton, G. (2015) Deep learning, Nature, 521(7553),436–444.
[2] Lyons, T. J. (1998) Differential equations driven by rough signals, Revista MatematicaIberoamericana, 14(2), 215–310.
[3] Levin, D., Lyons, T., and Ni, H. (2013) Learning from the past, predicting the statisticsfor the future, learning an evolving system, arXiv preprint arXiv:1309.0260.
[4] Yin, F., Wang, Q. F., Zhang, X. Y., and Liu, C. L. (2013, August). Icdar 2013Chinese handwriting recognition competition, in Document Analysis and Recognition(ICDAR), 2013 12th International Conference on (pp. 1464–1470). IEEE.
[5] Graham, B. (2013) Sparse arrays of signatures for online character recognition, arXivpreprint arXiv:1308.0371.
[6] Reizenstein, J., and Graham, B. (2014) Signatures in online handwriting recognition,http://www2.warwick.ac.uk/fac/cross_fac/complexity/people/students/dtc/students2013/
reizenstein/handwriting.pdf
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[7] Yang, W., Jin, L., Xie, Z. and Feng, Z. (2015, August) Improved deep convolutionalneural network for online handwritten Chinese character recognition using domain-specificknowledge, in Document Analysis and Recognition (ICDAR), 2015 13th InternationalConference on (pp. 551–555). IEEE.
1.9 Energy Profiling Using Integer Programming
Supervisors: Andrew Thompson and Stephen HabenContact: [email protected] and [email protected]
The increased uptake of low carbon technologies such as electric vehicles and solar panelshas meant that local low voltage (LV) electricity demand and uncertainty is increasing.Modelling such networks may be very important for management and planning of futureLV networks by distribution network operators (DNOs) (the owners of the cables whichsupply the electricity). Such LV networks consist from as few as ten to as many as fiftyhouseholds which require realistic and, if possible, accurate energy profiles. In theory,smart meters will be creating half hourly electricity data for every customer in the UKby 2020. Unfortunately, such data is proprietary and may not be available to DNOsand, even if it is, it could be very expensive to purchase for a large number of customers.However, DNOs do have access to some information about the customers connected totheir cables in the form of their quarterly meter readings which give estimates abouttheir average daily mean usage. In addition, DNOs own the local LV substations andcan therefore install their own monitoring of the half hourly aggregate level (i.e. thetotal demand of all the connected customers). Such monitoring is expensive though so,ideally, will only be used if necessary. Thus the aim of this project is to investigatethe possibility of using quarterly meter readings and substation monitoring to assignunmonitored customers a realistic and, if possible, accurate half hourly electricity profilefrom a limited pool of monitored customers.
The problem can be formulated as a binary-valued integer program (IP) with linearconstraints, and the main focus of this project will be to compare IP-based approaches(for which many off-the-shelf solvers are available) with combinatorial optimization al-gorithms such as genetic algorithms. The algorithms will be implemented (most likelyusing Matlab) and tested on real smart meter customer data, and compared both interms of accuracy and computational efficiency. This project therefore represents aninteresting opportunity to apply optimization theory to a problem of practical inter-est. The project will be extended to consider the more fundamental question of howto formulate the problem. One possible formulation is as a particular kind of IP witha quadratic objective, known as a quadratic assignment problem, for which more spe-cific algorithms have been proposed. At heart, the profiling problem involves a trade-offbetween the two objectives of matching both individual profiles and total demand, andanother possibility is to formulate the problem as a multi-objective optimization. Inaddition, time permitting, the student could also investigate the limits to the modellingwhen substation monitoring is not available.
Likely outcomes of the project are:
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1. An understanding of the advantages/disadvantages of different problem formula-tions and algorithms for the profiling problem.
2. The implementation of an algorithm capable of efficiently solving the profilingproblem on the data provided.
3. Interpretation of the findings in terms of (a) the importance of the constraints atthe substation level and (b) the number of customers required to ensure accuratemodelling.
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2 Biological and Medical Application Projects
2.1 Multi-Species Interaction within Porous Living Tissue
Supervisor: Dr Ricardo Ruiz BaierContact: [email protected]
Reaction-diffusion systems can explain many phenomena taking place in diverse disci-plines such as industrial and environmental processes, biomedical applications, popula-tion dynamics, etc. These models allow to reproduce chaos, spatio-temporal patterns,rhythmic and oscillatory scenarios, and so on. Nevertheless, in most of the applicationsmentioned above, the reactions do not occur in complete isolation. The species are ratherimmersed in a fluid, or they move within (and interact with) a fluid-solid continuum.
The task consists in employing a primal-mixed finite element method [5] to actually solvethe system, extending the existing results to some of the cases listed below. By primal-mixed we mean that, at both continuous and discrete levels, the elasticity equations areset in a mixed form (that is, the associated formulation possesses a saddle-point structureinvolving additional unknowns, as for instance, the strain, or the rotations), whereas theformulation of the reaction-diffusion system is written exclusively in terms of the speciesconcentrations. Such a structure of the governing equations is motivated by the needof recovering strains without postprocessing them from a (typically low-order) discretedisplacement (which usually leads to insufficiently reliable approximations).
Electromechanics of perfused living tissues. We aim at combining a perfusion modelfor of the left ventricle using a poroelasticity-based description at finite strains regime(cf. [7]), with a specialized model for electrical conduction on porous tissues recentlydeveloped in [4]. The expected results are the reproduction of flow impediment andwall thickening phenomena, which are extremely difficult to recover with isolated linearporoelastic models or thermodynamically inconsistent formulations.
Mathematical modelling of biofilm characterization. The way biofilms are considered incontinuum-based models has a huge impact in the possible range of phenomena that canbe represented. In particular, our goal is to describe the intake of nutrients by bacteriaforming biofilms and their detachment due to shear forces. We plan to extend the resultsin e.g. [6] to the case of deformable porous structures.
Pattern formation and limb morphogenesis. Here the focus is on the modelling of organo-genesis based on simple reaction-diffusion systems governing the interaction betweenbone morphogenic proteins and fibroblast growth factors concentration. In [1] the au-thors developed a three-species model of chicken limb where the growth velocity of thetissue is prescribed normal to the limb surface and directly depends on the local concen-tration of fibroblast growth factor. Other models consider the tissue growth as a liquiddisplacement, or it is typically assumed that the position of the cell aggregates can becomputed as a postprocess of some growth factor. However, evidence shows that itsshort-term response corresponds with that of an elastic solid [3], and therefore here weassume that the domain can deform according to its inherent mechanical response, andthat it is affected isotropically by the signaling factor via a forcing term.
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Remarks. The main directions of the project can be discussed and modified accordinglyto the skills or preferences of the student.
References
[1] Badugu, A., Kraemer, C., Germann, P., Manshykau, D., and Iber, D. (2012) Digitpatterning during limb development as a result of the BMP-receptor interaction, Sci Rep.,2, 991.
[2] Burger, R., Ruiz-Baier, R., and Torres, H. (2012) A stabilized finite volume elementmethod for sedimentation-consolidation processes, SIAM J. Sci. Comput., 34, B265–B289.
[3] Dillon R. and Othmer, H. G. (1999) A mathematical model for outgrowth and spatialpatterning of the vertebrate limb bud, J. Theor. Biol., 197, 295–330.
[4] Hurtado, D. E., Castro, S. and Gizzi, A. Computational modeling of non-linear diffu-sion in cardiac electrophysiology: A novel porous-medium approach, Comput. MethodsAppl. Mech. Engrg., to appear.
[5] Ruiz-Baier, R. (2015) Primal-mixed formulations for reaction-diffusion systems ondeforming domains, J. Comput. Phys., 299, 320–338.
[6] Tierra, G., Pavissich, J. P., Nerenberg, R., Xu, Z. and Alber, M. S. (2015) Multicom-ponent model of deformation and detachment of a biofilm under fluid flow, J. R. Soc.Interface, 12, 106–130.
[7] Vuong, A.-T., Yoshihara, L., and Wall, W. A. (2015) A general approach for modelinginteracting flow through porous media under finite deformations, Comput. MethodsAppl. Mech. Engrg., 283, 1240–1259.
2.2 Modelling Syringomyelia
Supervisor: Prof. Ian SobeyContact: [email protected]
Background: Syringomyelia is a progressive pathological condition where fluid filledcavities, called syrinxes, develop and increase in size at the centre of the spinal chord.The origin of these syrinxes and the reason for continued growth is not fully known.In some, but not all cases, the origin may be attributed to an abnormal neurologicalcondition at the base of the brain (a so called Chiari malformation) or to traumaticinjury. Regardless of how the initial cavities are formed, there is no certain explanationfor the continued growth of the syrinxes.
I have proposed one possible explanation for the continued development based aroundtreating the spinal chord as a poroelastic material which in the presence of a syrinx takesthe form of an annulus of poroelastic material with fluid in the centre at rest and thefluid pressure around the outside of the annulus of poroelastic material (in the subduralspace) being oscillatory (the fluid is cerebrospinal fluid [CSF] and there are existing
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observations and models which consider the highly oscillatory flow in the subdural spaceoutside of the spinal chord but within the spinal column).
The postulated mechanism is that during the phase when the annulus pressure is higherthan that in the syrinx, fluid will flow into the porous material and into the syrinx,compressing the poroelastic material and when the pressure in the subdural space fallsbelow the syrinx pressure, there will be an outward flow but because the poroelasticmaterial is compressed, the permability will be lower and the outflow correspondinglylower, resulting in a gradual pumping of fluid into the syrinx.
Project: This project is to develop a model for flow in and around an annulus ofporoelastic material where the permeability of the poroelastic material may be a functionof the fluid pressure and where various outer pressure distributions can be used totest and confirm or discount this, or a similar hypothesis. For example, one can alsohypothesise that a non-sinusoidal outer pressure oscillation could have longer part of aperiod with elevated pressure (and so inflow into the syrinx) and a shorter part of thecycle with decreased pressure (but larger so the pressure varies about an average) andthis, coupled with the non-linear permeability, could also produce a pumping action intothe syrinx.
Outcome: The project will contribute to modelling of syringomyelia by setting theproblem in a poroelastic framework and by confirming or discarding the underlyinghypothesis that the interaction of pressure oscillations in the subdural space with linearor non-linear poroelastic effects will result in syrinx growth.
References
[1] Levine, D. N. (2004) The pathogenesis of syringomyelia associated with lesions at theforamen magnum: a critical review of existing theories and proposal of a new hypothesis,J Neurol Sci., 220(1-2), 3–21.
[2] Carpenter, P. W., et al. (2003) Pressure wave propagation in fluid-filled co-axialelastic tubes. Part 2: Mechanisms for the pathogenesis of syringomyelia, J BiomechEng., 125(6), 857–863.
[3] Bertram, C. D. (2010) Evaluation by fluid/structure interaction spinal-cord simulationof the effects of subarachnoid-space stenosis on an adjacent syrinx, J Biomech Eng., 132,061009-1.
[4] Elliot, N. S. J. (2012) Syrinx fluid transport: Modelling pressure-wave-induced fluxacross the spinal pial membrane, J Biomech Eng., 134, 031006-1.
2.3 High Speed Detachment
Supervisors: Prof. Derek Moulton and Prof. Dominic VellaContact: [email protected] and [email protected]
Seed dispersal is a critically important task in the plant world, and one that is accom-plished in myriad ways. The motivation for this project is the plant Cardamine hirsuta,
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which has a remarkable mechanism of seed dispersal. The fruit consists of two long andthin valves attached to a replum, with seeds between the valves. As the fruit matures,the valves build energy in the form of an elastic bilayer, which when released results inan explosive coiling of the valves outward from the replum and a subsequent catapultingof the seeds. Key to this catapulting mechanism is having an appropriate connection –the right “glue” – between the rapidly coiling valve and the seeds. In this project wewill focus on this aspect, with fluid and solid mechanics modelling as well as from anexperimental point of view.
Figure 1: High-speed video footage of catapulting seed dispersal.
The project will begin by reviewing previously developed work on the coiling dynamicsof an elastic bilayer. These dynamics will then feed to a model of the bond between theseed and the valve, with the aim of determining how the forces of motion trigger thebreaking of the bond and catapulting of the seeds. This will involve aspects of fluid andsolid mechanics, as well as numerical simulation of Lagrangian dynamics. Concurrently,the project will involve development of simple table-top experiments to understand thephenomenon and test the theory. The Mathematical Institute houses a laboratory, calledthe Mathematical Observatory. A key piece of equipment is a high-speed camera, ca-pable of filming at rates of up to 400,000 frames per second. This tool enables a directconnection to be made between theory and experiment. The guiding ideal is to study acomplex system in a simplified form so as to capture its fundamental quality.
Expected outcomes include an understanding of the catapulting trigger as a function ofthe properties of the “glue” and the coiling dynamics, and a comparison of predictionwith table top experiments filmed with the high-speed camera.
References
[1] Callan-Jones, A. C., Brun, P. T., and Audoly, B. (2012). Self-Similar Curling of aNaturally Curved Elastica. Phys Rev Lett, 108(17), 174302.
[2] Majidi, C. (2007). Remarks on formulating an adhesion problem using Euler’s elas-tica, Mechanics Research Communications, 34(1), 85–90.
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2.4 Two-Phase Flow Models for Active Cell Motion
Supervisors: Prof. Jim Oliver and Prof. Andreas MunchContact: [email protected] and [email protected]
Active cell migration is a very common biological phenomenon, and plays an importantrole in biological processes such as including tissue growth, immune surveillance, andfor the survival of unicellular organisms. One approach to understand how this motionis achieved by cells is via continuum, multiphase-model and their thin film approxima-tions. For example, the cell may be considered to behave like a two-phase mixture of a“network” and a “solution” phase, where the former represents the constituents of theactin-myosin network that generates the forces that give rise to the motion, and thelatter is the combination of an aqueous solvent and G-actin complexes that flow throughthe network.
The tasks of this project will include formulation of a model (based on existing litera-ture), investigation of its properties via numerical codes, which candidates will have todevelop or adapt, as well as travelling wave, asymptotic and stability analysis.
2.5 Learning Strategies for an Iterated Game with an Unknown Asym-metry
Supervisors: Dr Cameron Hall and Prof. Mason Porter (with Marian Dawkinsin Zoology as a collaborator)Contact: [email protected] and [email protected]
Various models based on game theory have been used in attempts to describe socialinteractions among animals. One classic example of this is the iterated “prisoner’sdilemma”, which has been used to explain why animals that regularly encounter eachother might be expected to cooperate, while animals that rarely encounter each othermight be expected to fight over resources. However, this model is flawed: true altruismand cooperation are rare, even amongst animals who live together in groups, whereasdominance hierarchies (such as the pecking order in chickens) are much more common.
One of the flaws of the iterated prisoner’s dilemma is that it fails to take into accountindividual asymmetries: if a given individual has a significant “natural advantage” inan iterated game, it has little incentive to cooperate with a weaker player. We havedeveloped a simple model of social interactions that we are using to analyse the effects ofindividual asymmetries. In our model, there are two players who repeatedly encountereach other to compete for a resource, and each player has the option of behaving ina “friendly” manner or an “aggressive” manner to its opponent on each encounter.The distinguishing feature of our model is that the case where both players behaveaggressively is resolved probabilistically: with probability p, Player A is rewarded as thewinner of the conflict, and with probability 1− p, Player B is rewarded as the winner ofthe conflict. Asymmetries between individuals can be accounted for by considering thecase where p 6= 1
2 .
In this project, we are interested in continuing our analysis of the case where neither
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player has perfect prior knowledge of p. What’s a reasonable strategy for playing thisiterated game when you don’t know your chance of winning a fight? How can you bestuse the information from past fights to guide your future behaviour? How many fightsshould you lose before you decide that it’s never a good idea to be aggressive? Whatdefines a good learning strategy? We have a collection of code written in Matlab thatis able to run simulations of the model for a variety of different learning strategies, andwe’ve used the code to obtain partial answers to some of these questions, but there isa lot of potential for new work on investigating model extensions and on exploring howthe “best learning strategy” depends on the model parameters.
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3 Physical Application Projects
3.1 Collective Behaviour: From Robot Experiments to MathematicalModels
Supervisors: Dr Tamsin Lee and Prof. Radek ErbanContact: [email protected] and [email protected]
Background: Swarm robots are used to model agents in an environment where onlyshort-range communication is possible [1]. By assigning a few simple rules to each agent,the robot, a greater group level task can be completed [2]. These rules may be actionsperformed based on short-range signals with the environment and other agents. It isintuitive that algorithms of swarm (collective) robotics have often been motivated bycollective animal behaviour [3], which has been of interest for mathematical researchthroughout the last century [4].
Swarm bots lend themselves well to agent-based modelling. Agent based models areoften used to understand bee behaviour, for example [5] and [6]. A detailed descriptionof reproducing the results of [6] in Netlogo (agent-based modelling software) is availableat [7].
Methods and Techniques: We have a group of mobile e-puck robots [8] which caninteract through a number of different channels (audio, video, bluetooth). In previousdissertations [9-13], MSc students investigated how a group of robots can efficientlyachieve a desired task including target finding algorithms, navigation through unknownenvironments and spreading information. To answer these questions, MSc students haveused a combination of robot experiments, numerical simulations (of both agent-basedmodels and PDEs), optimisation techniques and analysis of PDEs.
Project goals: In this project we will use a combination of experiments with robotsand agent-based modelling in NetLogo (or an alternative). The robots are particularlysuited to modelling collective behaviour of individuals (robots) communicating throughshort-range (proximity sensors) and long-range (auditory and visual cues, bluetooth)means. We would like to understand the advantages of different types of strategies
18
within a group of robots for the successful completion of a pre-defined group task. Wewill be modelling bees pollinating a field of flowers [14]. We would like to compare theefficacy of the stochastic behaviour of many bees with the deterministic behaviour ofa person manually pollinating the flowers with a brush. The optimisation of tasks andthe comparisons will depend on the student interest. A thorough overview of swarmstability and optimisation is [15]. The project will involve analytical and numericalmodelling, Netlogo (or an alternative), microcontroller programming, and efficient sensordata analysis.
References
[1] Reif, J. H., and Wang, H. (1999) Social potential fields: A distributed behavioralcontrol for autonomous robots, Robotics and Autonomous Systems, 27(3), 171–194.
[2] Desai, J. P., Ostrowski, J. P., and Kumar, V. (2001) Modeling and control of forma-tions of nonholonomic mobile robots, Robotics and Automation, IEEE Transactions on,17(6), 905–908.
[3] Garnier, S. (2011). From ants to robots and back: how robotics can contribute to thestudy of collective animal behavior, in Bio-Inspired Self-Organizing Robotic Systems,105–120. Springer Berlin Heidelberg.
[4] Sumpter, D. (2011) Collective Animal Behavior, Princeton University Press.
[5] Faruq, S., McOwan, P. W., and Chittka, L. (2013). The biological significance ofcolor constancy: An agent-based model with bees foraging from flowers under variedillumination, Journal of vision, 13(10), 10.
[6] Hogeweg, P., and Hesper, B. (1983). The ontogeny of the interaction structure inbumble bee colonies: a MIRROR model, Behavioral Ecology and Sociobiology, 12(4)(4),271–283.
[7] http://www.gisagents.org/2015/03/bumble-bee-colonies.html
[8] Mondada, F. et al, (2009) Proc. of 9th Conf. on Autonomous Robot Systems andCompetitions 1, 59–65.
[9] Taylor-King, J. (2013) Hard-Sphere Velocity-Jump Processes: Applications to SwarmRobotics, M.Sc. dissertation.
[10] Taylor-King, J. P., Franz, B., Yates, C. A., and Erban, R. (2015) Mathematicalmodelling of turning delays in swarm robotics, IMA Journal of Applied Mathematics,80(5), 1454–1474.
[11] Kremer Garcıa, D. (2014) Swarm Robotics, M.Sc. dissertation.
[12] Petrat, J. (2015) A Velocity Jump Model for Robot Target-Finding in Environmentswith Obstacles, M.Sc. dissertation.
[13] Malik, A. (2015) Modelling Motion and Propagation of Information in SwarmRobotics, M.Sc. dissertation.
[14] Clarke, D., Whitney, H., Sutton, G., and Robert, D. (2013) Detection and learning
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of floral electric fields by bumblebees, Science, 340(6128), 66–69.
[15] Gazi, V. and Passimo K. (2011) Swarm Stability and Optimization, Springer.
3.2 Algorithmic Classification of Writing Styles via Time-Series Anal-ysis of Punctuation
Supervisors: Prof. Mason Porter and Prof. Sam HowisonContact: [email protected] and [email protected]
Using punctuation only (no words!), how can one algorithmically tell the differencebetween the writing styles of authors such as William Faulkner, Lewis Carrol, and MarkTwain? How can one algorithmically tell the difference, using punctuation only (no wordsor equations!), between the lucid prose of Samuel D. Howison and the cumbersome,seemingly random prattle (full of parentheses, em-dashes, compound adjectives, andOxford commas) of Mason A. Porter?
Recently [1], Adam J. Calhoun produced beautiful visualizations of punctuation in novelsusing both symbol sequences and heat maps, and the differences in different writingstyles are visually striking. (Note that Calhoun has made his python code available tothe public.) Such symbol sequences can be analyzed using time-series analysis, and thesein turns can lead to a classification of different writing styles of novelists, scientists, andothers. In this project, the student will produce and analyze punctuation sequences frommany authors. He/she will compare the authors using basic statistics and time-seriesanalysis. Methods from data clustering and machine learning are likely to be helpfulfor the classification, as well as ideas from Markov chains, regarding the punctuationmarks as states. Ideally, the quantitative classification produced will also be comparedto existing qualitative classifications.
Reference
[1] Calhoun, A. J. (2016) Punctuation in novels. Available at https://medium.com/
@neuroecology/punctuation-in-novels-8f316d542ec4#.mqwxm5cuw
3.3 A Model for Thermoelastic Contact Oscillations
Supervisor: Prof. John OckendonContact: [email protected]
This project concerns the derivation and preliminary analysis of a model for judder indevices such as brakes and clutches. The model is based on the competition betweenwear and thermal expansion which, in the multiple-pin configuration proposed by Barber(1969), leads to a novel, non-local dynamic contact problem. An unpublished papergiving more details is available on request.
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3.4 Homogenized Boundary Conditions for Faraday Cages
Supervisors: Dr David Hewett and Prof. HewittContact: [email protected] and [email protected]
The Faraday cage effect is the phenomenon whereby electric fields and electromagneticwaves can be blocked by a wire mesh. The effect was demonstrated experimentallyby Faraday in 1836, was familiar to Maxwell, and its practical application in isolatingelectrical systems and circuits is well known to modern-day engineers and physicistsalike. An exciting modern application is to the design of “optical metacages”[4].
Two recent publications, [1] and [2], have studied the problem using a continuum approx-imation in which the shielding effect of a large number of discrete wires is replaced bya homogenized boundary condition on an infinitesimally thin interface between the “in-side” and “outside” of the cage. Such boundary conditions can be derived by matchingasymptotic expansions of the field away from the mesh with expansions in a boundarylayer close to the mesh, where a multiple scales approximation can be applied (cf. [1,§5and Appendix C], [2], and the closely related work in [3]). Interestingly, in the elec-tromagnetic case there is the possibility of resonance, where the presence of the cageactually amplifies the incident field, rather than shielding it. Such resonance effectsoccur when the wavenumber is close to a resonant wavenumber of the idealised cage inwhich the wire mesh is replaced by a solid shell.
The aim of this M.Sc. project would be to generalise the analysis of [1,2], which con-sidered only Dirichlet or Neumann boundary conditions, to more realistic impedanceboundary conditions, or perhaps transmission boundary conditions where the field canpenetrate the wires to some degree (here the analysis in [3] may be particularly useful).We would investigate such problems using both numerical simulations and continuummodels, to quantify the shielding effect of the cage, and its dependence on the materialparameters of the wires.
References
[1] Chapman, S. J. , Hewett, D. P., and Trefethen, L. N. (2015) Mathematics of theFaraday cage, SIAM Review, 57(3), 398–417.
[2] Hewett, D. P., and Hewitt, I. J. (2016) Homogenized boundary conditions and reso-nance effects in Faraday cages, arXiv:1601.06944.
[3] Delourme, B., Haddar, H., and Joly, P. (2012) Approximate models for wave propa-gation across thin periodic interfaces, J. Math. Pures Appl., 98, 28–71.
[4] Mirzaei, A., Miroschnichenko, A. E., Shadrivov, I. V., and Kivshar, Y. S. (2015)Optical Metacages, Phys. Rev. Lett., 115, 215501.
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3.5 Fluid System Models for Drip Coffee Makers
Supervisor: Dr Robert Van Gorder (with additional input from industry collabo-rators1)Contact: [email protected]
Background and Problem: Extraction time and temperature are among the mostimportant aspects of coffee brewing. Automatic drip coffee makers brew better coffeethan percolators by avoiding reboiling the coffee and reducing extraction time, thuspreserving the aroma and reducing coffee bitterness [1]. However, for shorter extractiontimes, there may be incomplete flavour extraction. Variability in any part of this processcan positively or negatively influence coffee taste and aroma.
The coffee makers contain no valves and have a water flow meter comprised of a turningpaddle which induces an electrical pull each time a magnet on the paddle passes a givenpoint (see Figure 2). Due to the irregular output flow of vapour and water from theboiler, the flow through the meter isn’t at a constant rate but instead pulses with theboiler action. SharkNinja have the following observations: (i) cold water gives slowerpulsing and higher volume per pulse; (ii) low voltage gives slower pulsing and highervolume per pulse; (iii) lower water head gives slower pulsing and lower volume per pulse.
Approach and Prerequisites: In this project, we will focus on the cause of variabilityin fluid delivery by coffee makers. We will first model the fluid system, taking the geom-etry into account. We will then study the propagation of the pulse solutions. Once wehave a good understanding of the fluid problem, we will consider a coupled fluid-thermalproblem, which we will solve asymptotically and numerically in order to obtain morequantitative results and our aim is to uncover the mechanism for the system instability.Finally, we will apply this knowledge to suggest a method for obtaining optimal fluidvolume delivery. Experience with mathematical modeling and fluid dynamics will beuseful, as will experience with ODEs and PDEs (methods for obtaining solutions as wellas stability).
Results and Deliverables: We seek to develop a model describing the operation ofthe coffee machine, and then to use such a model to identify causes of variability influid delivery. If this goes well, we would aim to develop a controller for optimal fluidvolume delivery. Such results would be of great interest to our industry partner. Inaddition to meeting the requirements for the dissertation, fundamental results (beyondwhat might be of industrial importance) would be prepared for publication in a peerreviewed journal.
Reference
[1] Cai, E. Z. (2002) Fluid delivery system for generating pressure pulses to make bever-ages, U.S. Patent No. 6,405,637.
1Note that this project is in collaboration with SharkNinja. They have given permission to go forwardwith the project, and may provide additional guidance and data as the project progresses.
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Figure 2: Schematic of the coffee maker apparatus.
3.6 Wave Propagation in Granular Crystals
Supervisor: Dr Robert Van GorderContact: [email protected]
Background and Problem: A granular material is a conglomeration of discrete solid,macroscopic particles characterized by a loss of energy whenever the particles inter-act. Propagation of macroscopic wave packets (due to a disturbance or force appliedinternally or at a boundary) within granular media or granular crystals (tightly packedgranular media with sufficient structure) finds applications in areas such as medicalphysics, Bose-Einstein condensates, nonlinear optics, atomic physics, and elastic colli-sions, with specific applications in molecular chains, second-harmonic generation, arraysof repelling magnets, nonlinear resonances, shock mitigation, energy localization, stresswave control, nonlinear acoustic lenses (see [1] and references therein).
Many models of wave propagation through an array of beads on 1D finite or infinitelattices take the form [2]
d2undt2
= V ′(un − un−1)− V ′(un+1 − un) +W (un) .
Here un is the displacement of then nth bead from it’s initial position, V and W arefunctions which govern the particular physics of the problem. Regarding the domain, wehave either n ∈ 0, 1, . . . , N or n ∈ Z. In the former case, boundary conditions wouldbe needed, while in the latter case, solution would be expected to satisfy relevant decaycriteria as n→ ±∞. To study 2D granular crystals, one would consider similar systemson two dimensional finite or infinite lattices. It is not necessary for such domains to berectangular lattices; other lattices are both possible and useful.
There have been a variety of solutions to systems of this type, with solitary waves [3,4],dark breathers [5], plane solitary waves [6], periodic traveling waves and compactons[7] among solutions being found. There have been studies on interactions with linearmedia (such as a wall placed at a boundary) [8], and the scattering of waves at sur-faces or interfaces has also been studied [9]. While many results assume homogeneousmedia, there have been solitary waves observed in heterogeneous yet ordered (periodic)media [10,11]. Furthermore, although one-dimensional or quasi-one-dimensional chainsare most common, some work has been done on fundamentally 2D problems (see, for
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Figure 3: Wave propagation in a hexagonal granular crystal. Taken from Fig. 3 of [12].
example, [12]). One such example of a wave in a hexagonal granular crystal is shown inthe Figure 3.
This project will involve one or more areas within the field of wave propagation ingranular crystals. There are a variety of directions this project can take, and we outlinesome possibilities, below.
• One interesting area to consider would be the propagation of waves in media con-sisting of two different materials, one permitting fast propagation, the other per-mitting slow propagation. With an external potential, we may even be able to slowor even stop a wave, before releasing it. In such cases, it is natural to ask if we canwe preserve the wave structure. Or, will part of the wave always be reflected at aboundary between different media? If we then transition back to the ‘fast’ media,are we able to recover the previous ‘fast’ wave speed?
• Consider the case in which a macroscopic wave packet travels through a granularmedia and then impacts on a wall. Can we create a barrier between the media wecare about and a wall (boundary), so that the wave is dissipated before it impactson the wall? For the 1D problem, it might be possible to get analytical results.
• Consider various 2D lattice domains. It is natural to ask how the shape of thedomain will influence the propagation and reflection of waves. Once this is betterunderstood, then one can consider how such waves would propagate in the presenceof heterogeneous media.
• For 1D models, the location of strikers (used to initiate the wave) can naturallycoincide with a boundary. However, for 2D models, one can place point strikers ormulti-bead strikers along the boundary or in the interior of the domain. How doesthe placement influence the wave propagation?
Approach and Prerequisites: The precise application and physical formulation willbe selected early in consultation with the interested student, and I would encourage the
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student to consider one or more of the potential ideas listed above. Due to the factthat the governing models are fully nonlinear in the spatial differences, the results willprimarily involve numerical simulations. That said, asymptotic results may be soughtin certain physically interesting special cases for which continuum limits (which reducethe spatial difference part of the model into spatial partial derivatives) become relevant.Experience with numerical simulation of PDEs, ODEs, and difference equations will beuseful.
Results and Deliverables: There are a number of directions this project can take,and specific models and geometries will be selected in consultation with the student.Once mathematical results are obtained, a significant part of the project will be linkingthese results back to real scientific applications in areas mentioned above. In additionto meeting the requirements for the dissertation, good results would be prepared forpublication in a peer reviewed journal.
References
[1] Moleron, M., Leonard, A., and Daraio, C. (2014) Solitary waves in a chain of repellingmagnets, Journal of Applied Physics, 115, 184901.
[2] Sen, S., Hong, J., Bang, J., Avalos, E., and Doney, R. (2008) Solitary waves in thegranular chain, Physics Reports, 462, 21–66.
[3] Coste, C., Falcon, E., and Fauve, S. (1997) Solitary waves in a chain of beads underHertz contact, Physical Review E, 56, 6104.
[4] MacKay, R. S. (1999) Solitary waves in a chain of beads under Hertz contact, PhysicsLetters A, 251, 191–192.
[5] Chong, C., Kevrekidis, P. G., Theocharis, G., and Daraio, C. (2013) Dark breathersin granular crystals, Physical Review E, 87, 042202.
[6] Manjunath, M., Awasthi, A. P., and Geubelle, P. H. (2014) Family of plane solitarywaves in dimer granular crystals, Physical Review E, 90, 032209.
[7] James, G. (2012) Periodic travelling waves and compactons in granular chains, Jour-nal of Nonlinear Science, 22, 813–848.
[8] Yang, J., Silvestro, C., Khatri, D., De Nardo, L., and Daraio, C. (2011) Interaction ofhighly nonlinear solitary waves with linear elastic media, Physical Review E, 83, 046606.
[9] Vergara, L. (2005) Scattering of solitary waves from interfaces in granular media,Physical Review Letters, 95, 108002.
[10] Porter, M. A., Daraio, C., Szelengowicz, I., Herbold, E. B., and Kevrekidis, P. G.(2009) Highly nonlinear solitary waves in heterogeneous periodic granular media, PhysicaD, 238, 666–676.
[11] Jayaprakash, K. R., Starosvetsky, Y., and Vakakis, A. F. (2011) New family ofsolitary waves in granular dimer chains with no precompression, Physical Review E, 83,036606.
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[12] Leonard, A., Chong, C., Kevrekidis, P. G., and Daraio, C. (2014) Traveling wavesin 2D hexagonal granular crystal lattices, Granular Matter, 16, 531–542.
3.7 Stability of Initially Slow Viscous Two-Phase Jets Driven by Grav-ity
Supervisor: Dr Robert Van GorderContact: [email protected]
Background and Problem: Two-phase fluids are of extreme importance to industrialapplications. For instance, when extracting iron from raw material through melting,there is a separation of the metal from the leftover material (slag). This mixture isthen poured into a container. While separation occurs when the metal and slag are still,during pouring this separation is disturbed and the two can mix. For a more detaileddiscussion of metal and slag systems, see [1,2]. The motion due to pouring into thecontainer can be modeled as a jet initially at rest which impacts into the container; seeFigure 4. Breakup of this jet is seen to occur in air before impact, and the manner ofbreakup will influence the small-time mixing in the container. To complicate matters,there are a variety of breakup mechanisms possible (see [3] for an excellent review onthe subject), and one must narrow down the dominant effects relevant for a particularapplication of interest. The manner of breakup may have implications for the time ittakes the poured metal-slag mixture to separate once again.
In a recent work, Sauter and Buggisch [4] studied the breakup of an initially slow viscousjets driven by gravity (akin to the pouring problem) for a single fluid. While stabilityanalysis can give one an indication of the onset of instability in a jet flow, determiningthe manner of breakup is much more involved. Indeed, experimental results demonstratethat there can be pinch-off of the jet, and the entire jet can break up before impact withthe fluid surface in the container.
In this project, we aim to consider a two-phase model and generalize the instabilityresults of [4] to determine the manner of breakup for the pouring problem discussedabove. Such results would be applicable to the pouring of a metal-slag mixture. Dueto the fact that the base state is governed by nonlinear PDEs, much of the stabilityanalysis will involve nonlinear eigenvalue problems. Numerical simulations for the fulltime-dependent model can then be used to determine the dynamics of the jet flow duringand following breakup.
Approach and Prerequisites: We will first extend the mathematical model used in[4] to two-phase flows. Then, we would consider a linear stability analysis about the basestate. Since the base state is governed by nonlinear PDEs, this will involve numericallysolving nonlinear eigenvalue problems. Once the stability analysis is performed, theresults are expected to provide insight into parameter regimes for which the jet flowloses stability, numerical simulations for the full time-dependent model can then be usedto verify such results. Furthermore, the numerical simulations of the full model willalso be able to inform us of the dynamics of the jet once breakup begins, allowing us todetermine the manner of breakup one would expect to observe. Knowledge of numericalanalysis and simulation of nonlinear PDEs is therefore essential, while a background in
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Figure 4: (a) A metal/slag pour; photo from http://morriscontrols.com/
industries/shutterstock_97238576/. (b) Evolution of jet flow solutions over indi-cated non-dimensional time; taken from Fig. 5 of [4]. Note that the theoretical resultsshown in (b) do match the observations in (a) in a qualitative sense. Considering atwo-phase jet flow would improve the stability results, quantitatively.
fluid mechanics will prove useful for intuition.
Results and Deliverables: We will obtain the generalization of stability results con-tained in [4] to two-phase flows. This would be of great interest to those studying pouringproblems in the case of fluid mixtures. In addition to meeting the requirements for thedissertation, good results would be prepared for publication in a peer reviewed journal.
References
[1] Shanahan, C. E. A., and Cooke, F. (1957) The study of slag-metal mixing efficiencyby models, Journal of Applied Chemistry, 7, 645–654.
[2] Jakobsson, A., Nasu, M., Mangwiru, J., Mills, K. C., and Seetharaman, S. (1998)Interfacial tension effects on slag-metal reactions, Phil. Trans. R. Soc. Lond. A, 356,995–1002.
[3] Eggers, J., and Villermaux, E. (2008) Physics of liquid jets, Reports on Progress inPhysics, 71, 036601.
[4] Sauter, U. S., and Buggisch, H. W. (2005) Stability of initially slow viscous jets drivenby gravity, Journal of Fluid Mechanics, 533, 237–257.
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3.8 Interaction of Multiple Incoherent Wave-Packets and Optical Tur-bulence
Supervisor: Dr Robert Van GorderContact: [email protected]
Background and Problem: As discussed in the review article [1], turbulence is gen-erally used to describe chaotic behavior of solutions of a system of nonlinear PDEs. Theturbulence associated to nonlinear Schrodinger (NLS) type equations is often referredto as ‘optical turbulence’ because it is useful in describing the propagation of almostmonochromatic light beams in media with a nonlinear refractive index [1]. A good mod-ern review article, focusing on optical wave turbulence, is [2], and Section 5 of that paperwill be of greatest relevance to this project. One way to generate optical turbulence isto consider the interaction of multiple incoherent light waves [2], and this shall be ourfocus.
The interaction of N > 1 incoherent waves in one spatial dimension that propagate withdifferent group velocities can be described by the vector nonlinear Schrodinger equation
i∂Ψ
∂t+
i
µn
∂Ψ
∂x+ βn
∂2Ψ
∂x2+ γn
|Ψn|2 + δ∑m6=n
|Ψm|2Ψn = 0 for 1 ≤ n ≤ N .
Here Ψn ∈ C is the wave function for the nth light wave, µn is the group velocity forthat wave, βn is the dispersion coefficient, γn is a self-focusing parameter, and δ is theratio between the cross and self interaction coefficients. The final term denotes thecross-interaction between the waves, in particular the modulation of Ψn induced by theother wave packets. Such equations are useful for describing vector phenomena in optics,plasma, hydrodynamics, and Bose-Einstein condensates [3].
Note that in order to include damping or amplification, one often considers equationssuch as the complex Ginzburg-Landau equation (CGLE). This would involve consideringsystems like that given above, only with complex coefficients and perhaps quintic orderinteraction terms. (The system of equations written above considers at most cubicinteractions.) Such equations are known to have a wider variety of dynamics whencompared to their NLS counterparts [4]. Relevant to our interests, optical turbulencecan be observed theoretically from CGLEs [5,6]. Sample optical turbulence is shown inFigure 5.
In this project, we shall study the interaction of multiple incoherent wave-packets undera system of coupled CGLEs. As optical turbulence is observed in single CGLEs, wecertainly expect to observe it for such a system. Indeed, for a system of incoherentwaves under CGLE dynamics, we expect a richer variety of dynamics.
Approach and Prerequisites: As mentioned above, solutions to systems of coupledCGLEs will be sought. While there are some analytical results in the literature forsingle CGLEs, we expect that mainly numerical solutions will be attainable for the morecomplicated coupled system. However, under relevant simplifying assumptions, the PDEsystem might be converted into an ODE system, which itself would be more amenable toanalytical treatments. Any simplifying assumptions would need to preserve the salient
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Figure 5: Plot of spatiotemporal turbulence in light intensity obtained (a) experimentallyand (b) numerically from solutions of a system of two CGLEs. Taken from Fig. 2 in [6].
features of the physics. Since this project will involve studying systems of nonlinearPDEs, experience in exact solutions, asymptotics, and numerical simulations for PDEswill be helpful.
Results and Deliverables: While optical turbulence has most often been observed forsystems of NLS equations and single or coherent systems of CGLEs, we shall take thingsa step further and study systems of CGLEs for incoherent waves since not too muchhas been done in this area. This will provide more realistic, albeit more complicated,dynamics (such as those including local damping or amplification), and the results wouldbe of possible interest in the field of nonlinear optics (particularly to those interested inthe interaction of multiple incoherent waves). In addition to meeting the requirementsfor the dissertation, good results would be prepared for publication in a peer reviewedjournal.
References
[1] Dyachenko, S., Newell, A. C., Pushkarev, A., and Zakharov, V. E. (1992) Opti-cal turbulence: weak turbulence, condensates and collapsing filaments in the nonlinearSchrodinger equation, Physica D, 57, 96–160.
[2] Picozzi, A., Garnier, J., Hansson, T., Suret, P., Randoux, S., Millot, G., andChristodoulides, D. N. (2014) Optical wave turbulence: Towards a unified nonequilib-rium thermodynamic formulation of statistical nonlinear optics, Physics Reports, 542,1–132.
[3] Pitois, S., Lagrange, S., Jauslin, H. R., and Picozzi, A. (2006) Velocity locking ofincoherent nonlinear wave packets, Physical Review Letters, 97, 033902.
[4] Akhmediev, N., Soto-Crespo, J. M., and Town, G. (2001) Pulsating solitons, chaoticsolitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach, Physical Review E, 63, 056602.
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[5] Lushnikov, P. M., and Vladimirova, N. (2010) Non-Gaussian statistics of multiplefilamentation, Optics Letters, 35, 1965–1967.
[6] Sugavanam, S., Tarasov, N., Wabnitz, S., and Churkin, D. V. (2015) Ginzburg-Landauturbulence in quasi-CW Raman fiber lasers, Laser & Photonics Reviews, 9(6), L35–L39.
3.9 Nonlinear Waves along Interacting nearly Parallel Vortex Fila-ments
Supervisor: Dr Robert Van GorderContact: [email protected]
Background and Problem: The dynamics of vortex filaments in three dimensionsare of importance to applications in areas of fluid mechanics, condensed matter physics,superfluidity. The chaotic motion of quantized vortex filaments in superfluid helium hasbeen discussed as one mechanism responsible for superfluid turbulence [1]. Indeed, fromstudying quantum turbulence of this nature, it is known that interactions between vortexfilaments (and, in particular, the reconnection process) are one route to such turbulenceseen in experiments [2]. For an example vortex tangle, see Figure 6(a). Often, theseinteractions are simulated numerically, while asymptotic results are most often limitedto the case of single isolated vortex filaments. Regarding mathematical models, one willprefer either a non-local (Biot-Savart, see for instance [3]) or a local (local inductionapproximation, see [4]) formulation, depending on the application.
In order to better understand the dynamics of vortex filament interactions, Klein et al.[5] derive and study a simplified system of equations, based around the local inductionapproximation. For N > 1 interacting vortex filaments which lie roughly parallel to oneanother in R3, these equations can be put into the form
i∂Ψn
∂t+ αnΓn
∂2Ψn
∂s2+
∑k 6=n
ΓkΨn −Ψk
|Ψn −Ψk|2= 0 for 1 ≤ n ≤ N ,
which is a system of N coupled nonlinear PDEs. Here Ψn ∈ C is the wave functionfor the nth filament, αn ∈ R relates to the core structure of the nth filament, Γn ∈R corresponds to the circulation of the nth filament, s ∈ R is a parameterization ofarclength, and t > 0 is time. Each vortex filament curve ξn(s, t) ∈ R3 can be recoveredvia ξn(s, t) = (ReΨn(s, t), ImΨn(s, t), s). Mathematical studies and some particularexact solutions of these equations recently appeared in [6,7]. See Figure 6(b) for asample pair-wise interaction of vortex filaments. Note that as the distance betweenvortex filaments increases, their mutual effect on one another decays like 1/distance.
In this project, we will explore various types of solutions to this model. For example,while solitary wave solutions have been found for the dynamics of single vortex filaments(see, for instance, [4]), these solutions have not been explored in the case of interactingvortex filaments under the Klein et al. model. Interactions between other kinds of non-linear waves may yield chaotic dynamics. Through the study of such chaotic solutions,we might be able to further our knowledge of vortex filament interactions leading toquantum turbulence, as were previously studied experimentally.
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Figure 6: (a) Plot of a vortex tangle at small time before transition to turbulence; takenfrom Fig. 25 of [2]. (b) Plot of pair-wise interacting vortex filaments; taken from Fig. 1of [7].
Approach and Prerequisites: Since this model consists of a system of coupled non-linear Schrodiner equations, there are many approaches we might consider in order toobtain exact or asymptotic solutions under reasonable simplifying assumptions. More-over, numerical simulation of solutions will likely be required for most parameter regimes,particularly when the number of vortex filaments (N) becomes large. A knowledge ofPDEs and perturbation methods would be useful for this project, as would experiencewith numerically solving PDEs and ODEs.
Results and Deliverables: We would obtain an understanding of nonlinear wavespropagating along vortex filaments in the presence of vortex-vortex interactions. Thiswould generalize existing results pertaining to nonlinear waves along single vortex fil-aments. Such results would be of great interest to those studying vortex filament in-teractions and reconnection events in superfluid helium, and may cast light on somefeatures of quantum turbulence (in particular, mechanisms for the onset of such turbu-lence). In addition to meeting the requirements for the dissertation, good results wouldbe prepared for publication in a peer reviewed journal.
References
[1] Nemirovskii, S. K., and Fiszdon, W. (1995) Chaotic quantized vortices and hydrody-namic processes in superfluid helium, Reviews of Modern Physics, 67, 37–84.
[2] Nemirovskii, S. K. (2013) Quantum turbulence: Theoretical and numerical problems,Physics Reports, 524, 85–202.
[3] Van Gorder, R. A. (2015) The BiotSavart description of Kelvin waves on a quantumvortex filament in the presence of mutual friction and a driving fluid, Proceedings of theRoyal Society A, 471(2179), 20150149.
[4] Van Gorder, R. A. (2015) Quantum Hasimoto transformation and nonlinear waves ona superfluid vortex filament under the quantum local induction approximation, PhysicalReview E, 91, 053201.
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[5] Klein, R., Majda, A. J., and Damodaran, K. (1995) Simplified equations for theinteraction of nearly parallel vortex filaments, Journal of Fluid Mechanics, 288, 201–248.
[6] Banica, V. and Miot, E. (2013) Evolution, interaction and collisions of vortex fila-ments, Differential Integral Equations, 26, 355–388.
[7] Banica, V., Faou, E., and Miot, E. (2014) Collisions of vortex filament pairs, Journalof Nonlinear Science, 24, 1263–1284.
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4 Networks
4.1 Continuous-Time Analysis of Temporal Networks
Supervisors: Dr Mariano Beguerisse and Prof. Mason PorterContact: [email protected] and [email protected]
Many systems in the life, social, and engineering sciences consist of several entities(agents) that interact in ways that change with time. The behaviour and propertiesof networked systems and the processes that occur on them have been the subject ofnumerous investigations, but many open questions remain such as: Which componentsform groups of interaction, when do they form them and for how long? and How toincorporate temporal information into
In recent years, the development of techniques to handle temporal [4] and multilayer [5]networks have have been used for investigating temporal systems of interacting agents.For example, there are methods to investigate community structure [1,6,7] or centrality(a proxy for a node’s or edge’s importance) in temporal networks [2,3,8]. These methodsoften rely on multilayer networks in which each layer aggregates all interactions during agiven interval, or on continuous-time formulations that are specific for one type of ques-tion (e.g., computing a specific type of centrality). A significant issue with multilayerrepresentations of temporal networks is the identification of adequate bins (time inter-vals) over which data should be aggregated: intervals that are too coarse will obscureimportant temporal features of the system, overly fine intervals will not contain enoughdata to be informative.
The objective in this project is study a continuous-time formulation for temporal net-works that will allow their investigation without having to aggregate the data into bins.For example, one could represent the edges of an adjacency matrix as continuous-timefunctions that denote the renewal or decay of a connection (e.g., friendships that fadeafter prolonged periods without contact), memory, or the activity of a node. In thisproject, we will consider both theory and applications (e.g., networks in social media orin urban systems).
References
[1] Bazzi, M., Porter, M. A., Williams, S., McDonald, M., Fenn, D. J., and Howison, S.D. (2016) Community detection in temporal multilayer networks, with an application tocorrelation networks, Multiscale Modeling & Simulation, 14(1), 1–41.
[2] Beguerisse-Dıaz, M., McLennan, A. K., Garduno Hernandez, G., Barahona, M., andUlijaszek, S. (2015) The ‘who’ and ‘what’ of #diabetes on twitter, arXiv:1508.05764.
[3] Grindrod, P., and Higham, D. J. (2014) A dynamical systems view of network cen-trality, Proceedings of the Royal Society of London A: Mathematical, Physical and En-gineering Sciences, 470(2165).
[4] Holme, P. (2015) Modern temporal network theory: a colloquium, Eur. Phys. J. B,88(9).
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[5] Kivela, M., Arenas, A., Barthelemy, M., Gleeson, J. P., Moreno, Y. and Porter, M.A. (2014) Multilayer networks, Journal of Complex Networks, 2(3), 203–271.
[6] Mucha, P. J., Richardson, T., Macon, K., Porter, M. A., and Onnela J.-P. (2010)Community structure in time-dependent, multiscale, and multiplex networks, Science,328(5980), 876–878.
[7] Petri G., and Expert, P. (2014) Temporal stability of network partitions, Phys. Rev.E, 90, 022813.
[8] Taylor, D., Myers, S. A., Clauset, A., Porter, M. A., and Mucha, P. J. (2015)Eigenvector-based centrality measures for temporal networks, arXiv:1507.01266.
4.2 Context-Dependent Metabolic Networks: Structure and Dynamics
Supervisors: Dr Mariano Beguerisse and Prof. Eamonn GaffneyContact: [email protected] and [email protected]
Cellular metabolism consists of thousands of enzymatic reactions that extract energyfrom nutrients and assemble macromolecules that are necessary for cell survival. Thesize and complexity of metabolic networks can often obscure the roles of functionallyrelated reactions on the overall distribution of metabolic fluxes. Moreover, different en-vironmental contexts force the cell to reshape its metabolism in order to cope (e.g., aer-obic vs anaerobic conditions, availability of different nutrients). Often studies constructmetabolic reaction networks (e.g., in which reactions are nodes) by creating undirectedconnections between reactions that “share” metabolites [5]. While informative to anextent, such studies neglect key features of metabolism:
• Directionality: Undirected networks are inadequate to describe metabolic fluxesbecause they implicitly assume that all reactions are reversible (i.e., the adjacencymatrices are symmetric) so the nature of the relationship between reactions isobscured; two reactions can be related either consume the same metabolite (com-petition) or produce it, or by consume a metabolite produced by another reaction(supply).
• Physical interpretation of connections: The connections provide a qualitative de-scription of the relationship between two reactions (number of shared metabolites).In reality, the relationship between reactions is one of metabolite flow that can bequantified precisely in physical units (e.g., mmol
gDW·h).
• Biological context: Networks constructed from metabolic models (i.e., lists of re-actions) consider all reactions simultaneously; however, cells can drastically re-balance their metabolism to cope with different environments.
This project will extend a recently formulated metabolic modelling framework [1] thatemploys information from Flux-Balance Analysis (FBA) [4,6] to create context-dependent,directed networks in which the connections between two reactions are precisely defined interms of metabolite flow. In particular the project will explore the following extensionsof the framework:
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• Formulation as a multilayer network: Currently the connections between the reac-tions sum the flow over all metabolites. A more natural way to describe this wouldbe as a multilayer network [3] in which each layer describes the flow pattern of asingle metabolite. Coupling among layers can encode information about reactionsthat involve the metabolite. After an appropriate multilayer reaction network isconstructed, we can, for example, explore methods to discard uninformative layers[2].
• Transitions between states: Different biological contexts produce networks that arequalitatively different (e.g., aerobic vs anaerobic conditions produce networks withdifferent topology and properties). The space of flux constraints that encode eachcontext, is a continuous convex subspace of Rn, which means there are criticaltransitions (akin to bifurcations) between states that have not yet been examinedin detail.
References
[1] Bosque, G., Beguerisse, M., Oyarzun, D., Pico, J., and Barahona, M. Structure andcontext-dependency of metabolic networks. (Unsubmitted draft, available on request.)
[2] De Domenico, M., Nicosia, V., Arenas, A., and Latora, V. (2015) Structural reducibil-ity of multilayer networks, Nat Commun, 6, 6864.
[3] Kivela, M., Arenas, A., Barthelemy, M., Gleeson, J. P., Moreno, Y., and Porter, M.A. (2014) Multilayer networks, Journal of Complex Networks, 2(3), 203–271.
[4] Orth, J. D., Thiele, I., and Palsson, B. (2010) What is flux balance analysis?, NatureBiotechnology, 28(3), 245–248.
[5] Palsson, B. (2006) Systems Biology, Cambridge University Press, 2006.
[6] Rabinowitz, J. D., and Vastag, L. (2012) Teaching the design principles of metabolism,Nat Chem Biol, 8(6), 497–501.
4.3 Optimal Control of Contagions on Networks
Supervisor: Dr Robert Van GorderContact: [email protected]
Background and Problem: Contagions on networks have been studied in a varietyof contexts; see the review [1]. While there have been studies relating to the influence ofnetwork structure on contagions spreading on networks [1], and there have been resultson controlling epidemics at each node [2], optimal control of the spread of contagions onnetworks is not particularly well studied. Particularly complicated is the situation whereaspects of the network change in time (in which case we have dynamical networks —see Sec. VII of the review paper [3] — and references therein — for more information).The focus of this project will be on the optimal control for one or more aspects ofthe dynamics of contagions on networks, with the control being performed in order tominimize the cost or destructiveness of the contagion.
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Consider, for example, an SIS model (susceptible-infected-susceptible: infected individ-uals return to the susceptible class on recovery because the disease confers no immunityagainst reinfection) over a strongly connected graph G = (V,E) with nodes i ∈ V andedges (i, j) ∈ E. While there are perhaps more interesting contagion models, the SISmodel is classical and hence is a reasonable first model to consider. The recovery rateat each note i is given by δi > 0, while the infection rates are assumed to be edge de-pendent: the infection rate at which node i is affected by a specific node j is given byβij > 0 if (i, j) ∈ E while βij = 0 if (i, j) /∈ E. Define pi(t) as the proportion of thepopulation at node i which is infected at a given time t ≥ 0; then, 0 ≤ pi(t) ≤ 1. (Thiscan also be viewed as the probability that an individual at node i will be infected attime t.) The SIS dynamics at node i can then be written [4]
dpidt
= −δipi +
|V |∑j=1
βijpj(1− pi) .
Optimal edge removal is one way to control the spread of a contagion on a network. Un-der one type of SIS dynamics, it was shown in [5] that some network structures amplifiedthe infection, while others mitigated the spread and cost of the infection. This approachwas later applied to the spread of computer viruses since a simple analogy would bea (computer network) routing problem [6] and limiting the spread of diseases throughaltered migration patterns [7]. One can imagine a controller which creates or destroysedges in a network, in order to approximate the theoretically optimal dynamics at eachtime step. However, note that such problems are challenging. Indeed, as discussed in[2], optimal edge removal problems are (in general) NP-hard (see also [8]).
As mentioned in [2], not much has been done in the area of optimal control problemsfor the spread of a contagion on a network. One such optimal control problem for SISdynamics on a network can be written as follows [2]:
Given a linear cost of infection ci and a control di for all i ∈ V , solve the minimizationproblem
min JT =
∫ T
0
|V |∑i=1
cipi(t) + diδi(t)
dt
subject to the SIS dynamics provided above and δi(t) ∈ [δ, δ] for some 0 < δ < δ, for allt ∈ [0, T ].
This problem, in general, is open. In [9], a linear problem is studied (taking the lineariza-tion around the disease-free equilibrium). For such linear dynamics, the optimal controlis a bang-bang controller with at most one switch. Other control problems for simplifieddynamics arising in the context of computer viruses have been considered under SIR(susceptible-infected-recovered) [10,11], and again the optimal controller is a bang-bangcontrol with at most one switch.
Approach and Prerequisites: In this project, we shall look more closely at opti-mal control problems like that given above. Mathematically, the project will involvecontagions on networks [1], temporal dynamics on networks [3], and optimal control on
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networks [2]. Knowledge of dynamical systems is essential, knowledge of networks will beuseful but can be picked up as we go. The ability to perform numerical simulations willalso be essential. The project can be made more theoretical or computational, dependingon the interests of the student.
Results and Deliverables: A goal would be to develop an optimal controller for aspecific type of contagion which is valid for various kinds of network configurations.Such a control would determine how best to modify network properties (either at nodesor along edges) in time so that the propagation or concentration of the contagion ismitigated. In addition to meeting the requirements for the dissertation, good resultswould be prepared for publication in a peer reviewed journal.
References
[1] Newman, M. E. J., (2002) Spread of epidemic disease on networks, Physical ReviewE, 66, 016128.
[2] Nowzari, C., Preciado, V. M., and Pappas, G. J. (2016) Analysis and Control of Epi-demics: A Survey of Spreading Processes on Complex Networks, IEEE Control Systems,36, 26–46.
[3] Porter, M. A., and Gleeson, J. P., (2014) Dynamical systems on networks: A tutorial,arXiv preprint arXiv:1403.7663.
[4] Lajmanovich, A., and Yorke, J. A., (1976) A deterministic model for gonorrhea in anonhomogeneous population, Mathematical Biosciences, 28, 221–236.
[5] Sanders, J., Noble, B., Van Gorder, R. A., and Riggs, C. (2012) Mobility matrixevolution for an SIS epidemic patch model, Physica A: Statistical Mechanics and itsApplications, 391, 6256–6267.
[6] del Rey, A. M. (2015) Mathematical modeling of the propagation of malware: a review,Security and Communication Networks, 8, 2561–2679.
[7] McVinish, R., Pollett, P.K., and Shausan, A. (2016) Limiting the spread of diseasethrough altered migration patterns, Journal of Theoretical Biology, 393, 60–66.
[8] Van Mieghem, P., Stevanovic, D., Kuipers, F., Li, C., Van De Bovenkamp, R., Liu,D., and Wang, H. (2011) Decreasing the spectral radius of a graph by link removals,Physical Review E, 84, 016101.
[9] Khanafer, A., and Basar, T. (2014) An optimal control problem over infected net-works, in Proceedings of the International Conference of Control, Dynamic Systems,and Robotics, Ottawa, Ontario, Canada. Paper 125, pp. 1–6.
[10] Eshghi, S., Khouzani, M., Sarkar, S., and Venkatesh, S. (2016) Optimal Patching inClustered Malware Epidemics, IEEE Transactions on Networking, 24, 283–298.
[11] Bloem, M., Alpcan, T., and Basar, T. (2009) Optimal and robust epidemic responsefor multiple networks, Control Engineering Practice, 17, 525–533.
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4.4 Predator-Prey-Subsidy Population Dynamics on Networks
Supervisor: Dr Robert Van Gorder Contact: [email protected]
Background and Problem: Many empirical studies suggest that the introduction ofallochthonous resources (or resource subsidies) may disrupt otherwise stable food weblinkages [1,2,3]. Holt, in studying a model involving a predator that can move betweena patch containing prey and a second prey-free patch, notes that ‘passive dispersal canstabilize an otherwise unstable predator-prey interaction’ [4]. However, if the movementrate is very small, then the effect may be destabilizing [5]. Nevai and Van Gorder [6]examined a series of predator-prey models in which the predator can also consume anon-living resource subsidy. The motivating example was that of the predation of Arcticfoxes (Alopex lagopus) on lemmings (Cricetidae family). In coastal habitats, arctic foxesare believed to partially subsist in the winter on both the local lemming populationand a resource subsidy, namely seal (Phocidae family) carcasses, which are provided bypolar bear (Ursus maritimus) predation on the sea ice [7]. The following findings wereapparent: (i) At small subsidy input rates, there is a minimum prey carrying capacityneeded to support both predator and prey. (ii) At intermediate subsidy input rates,the predator and prey can always coexist. (iii) At high subsidy input rates, the preycannot persist even at high carrying capacities. (iv) As predator movement increases,the dynamic stability of the predator-prey-subsidy interactions also increases.
A recent MMSC dissertation, which went on to become the paper [8], studied seasonaleffects for this model. It was shown that seasonality can strongly modify populationdynamics for the predator-prey-subsidy system, resulting in stable equilibrium pointsbeing pushed into limit cycles and limit cycles being pushed into chaos. In some cases,a population will be worse off, while in other cases a population will be better off. Fordetails, see [8].
In studying these dynamics, there are three possibilities for the spatial domain. First, onecan assume that the predator, prey, and subsist are all located ‘at a point’ — meaningthat all interactions occur uniformly on some spatial domain. This results in a systemof ODEs for the predator, prey, and subsidy. Secondly, one can assume spatial diffusionover some subset of R2, which will give a system of PDEs for the predator, prey, andsubsidy. Third, one may consider an island model, where movement is between multiplediscrete patches; this results in a network of interconnected patches. This will result incoupled systems of ODEs, one system for the predator, prey, and subsidy at each of thepatches. Papers [6,8] considered both the first and third cases, although at most two orthree patches were considered in the island model simply in order to demonstrate thatdifferences can exist. An exhaustive study of the multiple patch model has not beenconsidered.
In this project, we will be interested in learning how the network structure in a multiplepatch model will influence the dynamics of the predator-prey-subsidy model. Whilehints that the structure will matter have been seen in [6,8], this feature of the multipatch model has not yet been studied in detail.
Approach and Prerequisites: Mathematically, this project will involve the analysisof dynamical systems on networks. A good review article would be [9]. We shall pick a
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variety of network configurations which correspond in some way to what is seen in reallife, and then simulate the predator-prey-subsidy dynamics numerically. For an idea ofthe kinds of dynamics possible in predator-prey-subsidy models in one or a small numberof patches, see [8]. Any biological background can also be found in [6,8], and referencestherein. Knowledge of linear and nonlinear systems of ODEs will be needed, and thestudent will need a good understanding of concepts such as stability and bifurcation forsuch systems. Numerical simulations of nonlinear ODE systems in Matlab or anotherenvironment of the student’s choice will be needed. Analytical or asymptotic resultsmight also be possible in some special or limiting cases.
Results and Deliverables: Any results in this area would inform us of network struc-tures for the multi patch model which may help or hinder the predator or prey popula-tions. This, in turn, may give us insight into non-uniform migration patterns and therole they play in predator-prey-subsidy population dynamics. In addition to meetingthe requirements for the dissertation, good results would be prepared for publication ina peer reviewed journal.
References
[1] Darimont, C. T., Paquet, P. C., and Reimchen, TE. (2008) Spawning salmon disrupttrophic coupling between wolves and ungulate prey in coastal British Columbia, BMCEcol., 8, 1.
[2] Halaj, J., and Wise, D. H. (2002) Impact of a detrital subsidy on trophic cascades ina terrestrial grazing food web, Ecology, 83, 3141.
[3] Henden, J.-A., Ims, R. A., Yoccoz, N. G., Hellstrom, P., and Angerbjorn, A. (2010)Strength of asymmetric competition between predators in food webs ruled by fluctuatingprey: The case of foxes in tundra, Oikos, 119, 27.
[4] Holt, R. D. (1984) Population dynamics in two patch environments — some anoma-lous consequences of an optimal habitat distribution, Theor. Popul. Biol. 28, 181.
[5] Kuang, Y., and Takeuchi, Y. (1994) Predator-prey dynamics in models of prey dis-persal in two-patch environments, Math. Biosci., 120, 77.
[6] Nevai, A. L., and Van Gorder, R. A. (2012) Effect of resource subsidies on predator-prey population dynamics: a mathematical model, J. Biol. Dyn., 6, 891.
[7] Roth, J.D. (2002) Temporal variability in arctic fox diet as reflected in stable-carbonisotopes; the importance of sea ice, Oecologia, 133, 70.
[8] Levy, D., Harrington, H. A., and Van Gorder, R. A. (2016) Role of seasonality onpredator-prey-subsidy population dynamics, Journal of Theoretical Biology, 396, 163–181.
[9] Porter, M. A., and Gleeson, J. P., (2014) Dynamical systems on networks: A tutorial,arXiv preprint arXiv:1403.7663.
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