ICMS

Jun 15, 2009 - Jun 20, 2009

ICMS, 14 India Street, Edinburgh, EH3 6EZ

Organisers

Name	Institution
Bogachev, Leonid	University of Leeds
Kondratiev, Yuri	Universität Bielefeld
Law, Richard	University of York

Some group photographs taken outside ICMS at 14 India Street are available here: Photo1, Photo2, Photo3

PDF files of some of the presentations given at the workshop are now available by following the links in the workshop programme.

Scientific Advisory Group (SAG)

Nick Barton (University of Edinburgh)
Research area: Quantitative genetics; Evolution of traits; Speciation
Ben Bolker (University of Florida)
Research area: Theoretical ecology; Spatial population dynamics
Rick Durrett (Cornell University)
Research area: Stochastic spatial models
Alison Etheridge (University of Oxford)
Research area: Infinite-dimensional stochastic processes; Population genetics
Simon Levin (Princeton University)
Research area: Ecosystems and the biosphere; Ecological and evolutionary mechanisms
Terry Lyons (University of Oxford)
Research area: Stochastic analysis and its applications; Stochastic differential equations
Johan A.J. (Hans) Metz (Universiteit Leiden)
Research area: Mathematical biology; Evolutionary dynamics
Michael Röckner (Universität Bielefeld)
Research area: Infinite-dimensional and stochastic analysis

Objectives and Scope
The aim of this workshop is to bring together mathematicians and mathematical biologists who have common interest in dynamical models of random particle configurations. Both communities have developed powerful methods to study the spatial-temporal random dynamics of large particle systems with interaction and adaptation, but the level of mutual understanding of each other’s work is still to be much improved. To bridge this gap and get new insights, it is crucial to share the expertise of the two groups and discuss open problems and possible directions of the future joint research. The workshop will provide a stimulating environment for brain-storming and cross-fertilisation, and may help create critical mass leading to a fruitful interdisciplinary collaboration.

Topics to be discussed include:

• Continuous spatial models in population dynamics

• Measures of spatial structure

• Moment equations for stochastic dynamics

• Closure methods

• Species interaction in space

• Perturbation methods

• Evolutionary processes

• Markov dynamics in individual based models

• Scaling limits and macroscopic equations

• Varying and random environments

Structure of the Workshop
The workshop will start on Monday morning 15 June and finish in the early afternoon on Saturday 20 June. However, presentations and talks will be timetabled from Monday till end of Friday only, whereas the morning of Saturday 20 June will be reserved for optional informal discussions. Participants are advised to take this into account when planning their travel/accommodation arrangements and filling out the online registration form.

To facilitate communication between the two groups and to foster mutual understanding and interaction, the workshop programme will include expository lectures, accessible to a layman, on various aspects of stochastic dynamics modelling and modern ideas in spatial ecology. Further details will be available as arrangements progress.

Arrangements

Participation
Participation is by invitation only. If you are interested in attending the workshop and/or wish to suggest a delegate (eg, a PhD student), please contact one of the Organisers: Leonid Bogachev, Yuri Kondratiev or Richard Law.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and, if so, how to apply for one. If you do require a visa, ICMS can provide a signed invitation letter.

Venue
The workshop will take place at ICMS, 14 India Street, Edinburgh, EH3 6EZ. This house is the birthplace of James Clerk Maxwell and is situated in the historic New Town of Edinburgh, near the city centre.

The ICMS travel pages contain advice on how to travel to Edinburgh. For local information, the finding ICMS page shows the location of ICMS and contains useful maps of the city centre.

The seminar room at 14 India Street has whiteboards, 2 overhead projectors, a data projector and laptop. We ask that you use our ICMS laptop for your presentation which you may bring on a memory stick or CD. Alternatively, a pdf or PowerPoint file may be emailed to Audrey Brown at ICMS by Friday 12 June to be loaded onto the ICMS laptop in advance of the workshop.

Wireless access is available throughout the building. There are also 4 public PCs which may be used at any time for internet access and to check email.

Poster Session
It is anticipated that there will be a poster session during the workshop. Please indicate on your application form if you are interested in presenting a poster, and provide a title and abstract.

Public Lecture
As part of the workshop activities, there will be a public lecture by Professor Charles Godfray FRS, Department of Zoology, University of Oxford, entitled "Malaria, Mosquitoes and Models". (see pdf of abstract) Tickets and seats have been reserved for all participants of this workshop so there is no need for you to arrange this. The lecture which will be held at the Royal Society of Edinburgh (RSE), 22-26 George Street, Edinburgh, EH2 2PQ at 18.00 on Wednesday 17 June 2009. Following the lecture, a wine reception will be held in the Wellcome Rooms of the RSE. All workshop participants are invited to the lecture and the wine reception. RSE report of the Public Lecture.

Accommodation
ICMS has arranged en-suite rooms in hotels nearby for those who requested this. Accommodation is typically about 10 to 20 minutes walk from ICMS. Participants making their own arrangements may claim back the cost, with original receipts, up to a maximum of £70.00 per night bed and breakfast for a maximum of 6 nights. A list of Edinburgh accommodation of various sorts and prices is available here . Sections 1-3 are particularly relevant.

Meals and Refreshments
Morning and afternoon refreshments will be provided on each day of the workshop.

On Monday 15 June, the first day of the workshop, a light buffet lunch will be provided free of charge to participants in the Exhibition Room at 14 India Street. Thereafter, participants are free to explore the many cafés, sandwich shops, restaurants and bars nearby. An informal evening meal has been arranged at Nargile Turkish Restaurant on Tuesday 16 June. Following the public lecture (see above) on Wednesday 17 June, wine and canapés will be served in the Wellcome Rooms of the RSE. The Workshop Dinner will take place on the evening of Thursday 18 June at The Magnum Restaurant. The workshop grant will cover the cost of this catering.

Registration
Registration will take place in the morning on Monday 15 June in the Exhibition Room at 14 India Street.

Financial Arrangements
Unless otherwise specified in your invitation letter, the workshop grant will cover the cost of your bed and breakfast accommodation, tea/coffee Monday to Saturday, lunch on Monday only, the informal evening meal on Tuesday, wine reception on Wednesday, and the Workshop Dinner on Thursday evening.

If we have agreed to pay some of your travel costs, you will be informed by email. Reimbursement will take place after the workshop and will involve payment directly into your bank account. At Registration you will be given an expenses claim form and this should be submitted to ICMS, with original receipts. It would be helpful if you could bring your bank details to the workshop. In addition to the bank account number, participants from the USA and Canada will require their bank’s routing number, those from the UK will be asked for the bank sort code, and those from Europe and the rest of the world, their IBAN and SWIFT/BIC code. We cannot reimburse any item without a signed claim form and original receipts.

Please note that there is no longer a registration fee for this workshop. This fee has been waived for all participants.

Programme

Please note that all talks and presentations during the workshop will take place from Monday 15 June till end of Friday 19 June, with the morning of Saturday 20 June reserved for optional informal discussions.

An allocation of up to 10 minutes has been reserved after each talk for Questions & Answers.

Monday 15 June

08.45 - 09.40	Registration and coffee/tea
09.40 - 09.50	Welcome and Introduction
09.50 - 10.40	Richard Law (University of York) Setting the scene: geometry in ecological dynamic PDF of Presentation
10.50 - 11.20	Coffee/Tea
11.20 - 12.05	Janine Illian (University of St Andrews) Spatial aspects of ecological dynamics - 1 PDF of Presentation
12.15 - 13.00	Janine Illian (University of St Andrews) Spatial aspects of ecological dynamics - 2
13.10 - 14.30	Lunch in the Exhibition Room
14.30 - 15.15	Yuri Kondratiev (Universität Bielefeld) Random evolutions in continuum - 1 PDF of Presentation
15.25 - 16.10	Yuri Kondratiev (Universität Bielefeld) Random evolutions in continuum - 2
16.20 - 16.50	Coffee/Tea
16.50 - 17.30	Martin Grothaus (University of Kaiserslautern) Scaling limits of stochastic processes in continuous particle systems
17.40 - 18.20	Dmitri Finkelshtein (Kiev Institute of Mathematics) Individual based models with competition in spatial ecology

 

Tuesday 16 June

09.00 - 09.45	Benjamin Bolker (University of Florida) Continuous-space moment dynamics of single populations - 1 PDF of Presentation
09.55 - 10.40	Benjamin Bolker (University of Florida) Continuous-space moment dynamics of single populations - 2
10.50 - 11.20	Coffee/Tea
11.20 - 12.00	David Murrell (University College London) Continuous-space models for interacting species: moment closure methods for community dynamics
12.10 - 12.50	Glenn Marion (Biomathematics & Statistics Scotland) Moment-closure approximations in epidemiology and ecology PDF of Presentation
13.00 - 14.30	Lunch break
14.30 - 15.15	Stanislav Molchanov (University of North Carolina at Charlotte) Ideas of random media theory and mathematical models of the population structure dynamics - 1
15.25 - 16.10	Stanislav Molchanov (University of North Carolina at Charlotte) Ideas of random media theory and mathematical models of the population structure dynamics - 2
16.20 - 16.50	Coffee/Tea
16.50 - 17.30	Leonid Bogachev (University of Leeds) Intermittency of higher-order moments in random environments PDF of Presentation
17.40 - 18.30	Dick Dougal (Trustee of the James Clerk Maxwell Foundation) Tour of the birthplace of James Clerk Maxwell and demonstration of experiments
19.00	Informal evening meal at Nargile (Turkish) Restaurant, 73 Hanover Street, Edinburgh

 

Wednesday 17 June

09.00 - 09.40	Terry Lyons (University of Oxford) Selection as a tool in numerical algorithms
09.50 - 10.30	Michael Röckner (Universität Bielefeld) Fokker-Planck equations on Hilbert spaces
10.40 - 11.10	Coffee/Tea
11.10 - 11.50	Alan McKane (University of Manchester) Quasi-cycles in a spatial predator-prey model PDF of Presentation
12.00 - 12.40	Gustav Delius (University of York) Jump-growth model for predator-prey dynamics PDF of Presentation
12.50 - 14.30	Lunch break
14.30 - 15.10	Stephen Cornell (University of Leeds) Exact asymptotic methods for spatially extended stochastic populations PDF of Presentation
15.20 - 16.00	Otso Ovaskainen (University of Helsinki) Evolution of dispersal in heterogeneous environments PDF of Presentation
16.10 - 16.40	Coffee/Tea
16.40 - 17.20	Andrew Nevai (University of Central Florida) A comparison of spatial interactions at different scales PDF of Presentation
18.00 - 19.00	Public Lecture at the Royal Society of Edinburgh (RSE), 22-26 George Street Charles Godfray (University of Oxford) Malaria, Mosquitoes and Models (see pdf of abstract) RSE report of the Public Lecture Chaired by Richard Law (University of York)
19.00 - 20.00	Informal wine reception in The Wellcome Rooms, RSE, 22-26 George Street

 

Thursday 18 June

09.00 - 09.45	Johan Metz (Universiteit Leiden) Effective population sizes and the canonical equation of adaptive dynamics - 1 PDF of Presentation
09.55 - 10.40	Johan Metz (Universiteit Leiden) Effective population sizes and the canonical equation of adaptive dynamics - 2
10.50 - 11.20	Coffee/Tea
11.20 - 12.00	Shuhei Mano (Nagoya City University) Ancestral processes with bias in ectopic gene conversion or migration PDF of Presentation
12.10 - 12.50	Ostap Hryniv (Durham University) Long time behaviour in a model of microtubule growth
13.00 - 14.30	Lunch break
14.30 - 15.10	Denis Mollison (Heriot-Watt University) Pair approximations for spatial structures? PDF of Presentation
15.20 - 16.00	Anja Voss-Böhme (Technische Universität Dresden) Rigorous analysis of stochastic particle-based models in developmental biology - PDF of Presentation
16.10 - 16.40	Coffee/Tea
16.40 - 17.20	Maria João Oliveira (Universidade Aberta/CMAF) Stochastic dynamics of infinite particle systems PDF of Presentation
17.30 - 18.30	Poster Session
19.00	Workshop Dinner at The Magnum Restaurant, 1 Albany Street, Edinburgh (dress informal)

 

Friday 19 June

09.00 - 09.40	Tobias Kuna (University of Reading) Scaling limit for jump, birth and death type dynamics PDF of Presentation
09.50 - 10.30	Eugene Lytvynov (Swansea University) Scaling limits of stochastic dynamics of particle systems in continuum PDF of Presentation
10.40 - 11.10	Coffee/Tea
11.10 - 11.50	Oleksandr Kutovyi (Universität Bielefeld) On ergodic properties of stochastic evolutions in individual based population models
12.00 - 12.40	Torben Fattler (University of Kaiserslautern) An invariance principle for the tagged particle process in continuum
12.50 - 14.30	Lunch break
14.30 - 16.00	Round Table Discussion Future progress in mathematical ecology: frontiers, challenges, and interdisciplinary collaboration
16.00 - 16.30	Coffee/Tea
16.30 - 17.15	Reserve session
17.15 - 18.00	Reserve session

 

Saturday 20 June

09.00 - 10.30	Informal discussions
10.30 - 11.00	Coffee/Tea
11.00 - 12.30	Informal discussion
12.30	Close of Workshop

 

Presentations:

Presentation Details
Adams, Tom
POSTER: Reconstructing Scotland's pine forests: insights from a stochastic model
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Recent conservation efforts for biodiversity in the Caledonian pinewoods have focused on the transformation of existing plantation stands towards a state that is more reminiscent of the native "old-growth" state. The transition from a homogeneous "even-aged" forest to an irregular "uneven-aged" structure is far from simple, however. To tackle the problem, a greater mechanistic understanding of the effects of inhomogeneous interactions upon tree growth, mortality and regeneration, and the consequent population dynamics, is required. This has increased interest in simulation and mathematical models to assist in describing forest stands. This work uses a spatial size-structured model to study the dynamics of a single-species forest stand from plantation conception. The model provides useful insights into data from semi-natural stands in Scotland, and we demonstrate how simulation might be used to inform management aimed at speeding the transformation process.
Atkins, Katie
POSTER: Marek's disease virus: defining virulence and understanding the drivers of evolution
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Marek's disease is an oncogenic poultry disease affecting chickens and is estimated to cost the worldwide industry $1-2 billion annually. The causative agent of Marek's Disease (MD), MDV, provides a well-documented example of virulence evolution occurring over the period of half a century. The reason behind this evolution is unknown, although certain untested hypotheses have been suggested. These include vaccination (with increasingly potent vaccines) and other aspects of industrialisation, including the decreased cohort time of successive generations. Disease control is mainly via vaccination, indeed MDV vaccination was the first such vaccine offering protection against an oncogenic virus. In this study, two sections of work are undertaken. First, estimation of epidemiological parameters is tackled: virulence of MDV is quantified by looking at mortality and virus shedding rates in vaccinated and unvaccinated birds. Both of these are achieved via Maximum Likelihood Estimation and Bayesian MCMC techniques. Second, viral fitness is quantified by defining a lifetime fitness function using the parameters previously estimated and fitness landscapes are calculated to understand the direction and force of selection and the optimum fitness for different cohort times and vaccination strategies. (Joint work with Mark E J Woolhouse & Andrew F Read)
Barraquand, Fred
POSTER: Predator-prey spatial dynamics and predator home range size
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By means of dynamic point processes and moment approximations, we investigate the effect of a behavioural trait, home range size, on the population dynamics and spatial structure of predators and preys.
Berns, Christoph
POSTER: Kawasaki dynamics in continuum
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We propose a construction of a non equilibrium Kawasaki dynamics in continuum. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over Rd. We try to construct such a dynamics by means of a Lyapunov function.
Bogachev, Leonid
Intermittency of higher-order moments in random environments
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The purpose of the talk is to discuss some potential applications of random media theory to the population dynamics modelling, with the emphasis on the so-called "intermittency'' characterized by the emergence of highly erratic spatial-temporal structures. A moment test of intermittency will be explained using a simple model of continuous-time branching random walk with random branching rates, leading to a reaction-diffusion equation with a random potential. The Feynman-Kac representation of the solution allows one to evaluate the Lyapunov exponents responsible for the super-exponential growth of the moments. Higher-order Lyapunov exponents can also be calculated, suggesting that the higher-order intermittency is reduced, in a sense, to that of the mean population density.
Bolker, Benjamin
Continuous-space moment dynamics of single populations
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Basic properties of moment (correlation) equations for populations distributed in continuous space. Starting from a stochastic point process, and assuming homogeneity and isotropy, we can derive dynamical equations for the nth-order spatial moments (1=mean, 2=covariances or correlations, etc.); we then need some form of moment closure in order to make the system analytically or numerically tractable. I will emphasize derivations using different biological and mathematical assumptions, the heuristic properties of the derived equations, and the breadth of biological systems that can be modelled rather than rigorous results.
Cornell, Stephen
Exact asymptotic methods for spatially extended stochastic populations
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Many spatially explicit birth-death processes behave deterministically in the limit where the interactions between individuals have infinite range. When the interactions take place over a large but finite range, the system's properties will be close to the deterministic limit. I shall show how to develop a perturbation expansion about the deterministic limit, which permits the calculation of properties of the system in an asymptotically exact way by using the inverse of an interaction scale as a small parameter. This is a particularly natural approach in many ecological situations, where interactions between individuals often extend beyond nearest neighbours. The method is a convenient way to investigate the effects of space and stochasticity, since it calculates explicitly the corrections to the deterministic and/or spatially uniform limit. I shall illustrate this method by discussing the importance of landscape spatial and temporal structure on the occupancy of metapopulations, and how spatial kernels influence the stability of interacting species. Since the perturbation expansion gives results which are exact in a particular limit, it provides a complementary approach to the moment closure assumption, which - for all its usefullness and intuitive appeal - is an uncontrolled approximation.
Delius, Gustav
Jump-growth model for predator-prey dynamics
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We use a stochastic model to describe the main process determining the size spectrum of organisms in marine ecosystems: larger organisms preying on smaller organisms and growing in size. Instead of the spatial location of the organisms we model their weight, but the techniques are the same. The feeding interaction is non-local in weight-space, determined by a feeding kernel expressing the preference for a certain predator/prey weight ratio. We treat the model both as an individual-based model (stochastic process on configuration space) and as a population model and compare the approaches. The deterministic equation derived from our stochastic model turns out to be a modification of the McKendrick-von Förster equation that has traditionally been used to model size spectra. The steady state is found to be given by a power law weight distribution, in agreement with observation, but we also observe travelling-wave solutions.
Derdziuk, Lukasz
POSTER: Spatially heterogeneous contact models in continuum
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In our work we would like to describe the evolution of some interacting particle systems in continuum, with additional random influence from the environment. Contact model is constructed as a Markov process in the space of configurations (locally finite subsets of Rd) then the evolutions of correlation functions are investigated.
Fattler, Torben
An invariance principle for the tagged particle process in continuum
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We provide an invariance principle for the dynamics of a tagged particle in an infinite particle environment of interacting particles for a large class of interaction potentials. In particular, we can treat interaction potentials having a singularity at the origin, non-trivial negative part and infinite range, as e.g., the Lennard-Jones potential.
Finkelshtein, Dmitri
Individual based models with competition in spatial ecology
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We analyze interacting particle systems with Markov evolutions of birth-and-death type. We have shown that a local competition mechanism leads to a globally regular behavior of the population in course of the stochastic evolution.
Grothaus, Martin
Scaling limits of stochastic processes in continuous particle systems
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Concepts for treating scaling limits of stochastic processes in configuration spaces are discussed. Along an example it is shown that the theory of Dirichlet forms provides many powerful tools for determining the limit and showing convergence.
Hryniv, Ostap
Long time behaviour in a model of microtubule growth
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We study a continuous time stochastic process on strings made of two types of particles, whose dynamics mimics the behaviour of microtubules in a living cell; namely, the strings evolve via a competition between (local) growth/shrinking as well as (global) hydrolysis processes. We give a complete characterization of the phase diagram of the model, and derive several criteria of the transient and recurrent regimes for the underlying stochastic process.
Illian, Janine
Spatial aspects of ecological dynamics - 1 and 2
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Most processes in the natural world take place in a spatial context and are often inherently local, particularly in plant ecology. As a result, spatially explicit data sets are frequently being collected in ecology. Previously this has been perceived as a difficulty since traditional statistical approaches are unsuitable in the presence of spatial autocorrelation. However, recent years have seen an increasing interest in explicitly modelling this spatial dependence. Spatial point processes model the locations of objects such as individual plants, animals, nests in space. They take local dependence structures, the environment around and properties of the objects into account. A suitable model hence characterises the behaviour of every individual object based on a small number of interpretable parameters. A growing number of methods and models have been developed within the statistical literature. However, a large part of this remains unknown to ecologists. This partly due to the fact that fitting these models is not straight forward and often computationally involved. In addition, many approaches have been derived from a rather mathematical perspective and are thus not necessarily appropriate in an ecological context. This talk will provide a general introduction to point process methodology and discusses how it may be suitably applied in ecology. It provides a number of examples of applications, including complex and multivariate point pattern data sets, modelling inhomogeneous data and marked point processes.
Kondratiev, Yuri
Random evolutions in continuum - 1 and 2
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The concept of particle configurations is instrumental in the mathematical modelling of multi-component stochastic systems. The purpose of these lectures is to give a layman overview of the main ideas used in the mathematical treatment of random configuration dynamics. In particular, we shall explain how to construct Markov processes on configuration spaces and review suitable methods of studying their long-term properties (spatiotemporal asymptotics, invariant measures, the spectrum of Markov generators, etc.). Special attention will be given to particular classes of such processes that appear in spatial ecology.
Kuna, Tobias
Scaling limit for jump, birth and death type dynamics
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In order to understand large individual based systems one may derive effective partial differential equations at large scales. I will present diverse scalings for contact type branching processes, free and interacting jump type dynamics and give first results,expected outcomes and discuss possible future investigations.
Kutovyi, Oleksandr
On ergodic properties of stochastic evolutions in individual based population models
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We study the spectral properties of the non-equlibrium Glauber dynamics. We prove ergodicity of this process and show that the speed of convergence to equilibrium is exponential.
Law, Richard
Setting the scene: geometry in ecological dynamics
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This talk sets the scene for the ecological contributions to the meeting. The starting point is that populations and communities are usually structured in space. This is important because ecological interactions tend to take place with neighbours, and the densities of interacting individuals can be quite different from their averages over large areas of space. The effects of such geometry on ecological dynamics can be profound.
Lyons, Terry
Selection as a tool in numerical algorithms
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Monte Carlo is an established methodology for producing numerical approximations to PDE's. It can be used in high dimensional contexts and although it is very slow it is reliable. Following on from work of Kusuoka, Lyons, Victoire et al., it has become apparent that choosing scenarios randomly is inefficient and that by choosing representative samples very carefully there is a potential to achieve (much) more accurate information. However, this approach generates it's own problems in a sequential setting. Recent work of Litterer and Lyons, introducing a selection and re-weighting of the population, allows these methods to work well in the sequential context. The outcome is a new numerical approach, similar in spirit to Monte Carlo, which produces extremely accurate results in some context.
Lytvynov, Eugene
Scaling limits of stochastic dynamics of particle systems in continuum
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We will consider three types of stochastic dynamics of interacting particle systems in continuum. The fist dynamics is diffusive, that is, the trajectory of each separate particle is continuous. An example of such dynamics is given through a system of `big' charged particles suspended in water. Then the `big' particles move due to the energy of pair interaction between the particles, and due to collisions of `small' particles of water with the `big' particles. In the second dynamics, particles randomly hop over the space taking into account position of other particles. In the third dynamics, particles randomly appear (are born) in the space and then die after a random time. We will assume that all three dynamics are in equilibrium, which means that they have the same distribution at each moment of time. We will study two types of scaling limit of the dynamics of hopping particles. In one scaling limit, this dynamics converges to a birth-and-death process, while in the second scaling limit, this dynamics converges to the diffusive dynamics.
Mano, Shuhei
Ancestral processes with bias in ectopic gene conversion or migration
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In a biallelic locus biased gene conversion has exactly the same effect as selection (Nagylaki, 1983). Hear a diffusion model with ectopic biased gene conversion in a multigene family will be presented. It is also a model of a structured population, the island model with biased migration. The fixation probability of an allele is different from that with selection. Then, the ancestral process will be presented. The process is a branching-coalescent process and it is similar to that generates the ancestral selection graph introduced by Krone and Neuhauser (1997).
Marion, Glenn
Moment-closure approximations in epidemiology and ecology
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I will describe the use of moment-closure approximations in a range of recent applications in ecology and epidemiology such as: the SIS model where novel closures characterising both disease persistence and extinction were used for inference; spatial-closures in models of foraging behaviour; and moment-closure for models which seek to characterise the role of social hierarchy in the persistence of diseases in wildlife populations. I will highlight the benefits and problems in applying moment-closures to such models.
McKane, Alan
Quasi-cycles in a spatial predator-prey model
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We use spatial models of simple predator-prey interactions to predict that predator and prey numbers oscillate in time and space. These oscillations are not seen in the deterministic versions of the models, but are due to stochastic fluctuations about the time-independent solutions of the deterministic equations which are amplified due to the existence of a resonance. We calculate the power spectra of the fluctuations analytically and show that they agree well with results obtained from stochastic simulations. This work extends the analysis of these quasi-cycles from that previously developed for well-mixed systems to spatial systems, and shows that the ideas and methods used for non-spatial models naturally generalize to the spatial case.
Metz, Johan
Effective population sizes and the canonical equation of adaptive dynamics - 1 and 2
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Underlying many population dynamical models is a Markov process on a set of ordered n-tuples, n=0, 1, 2, … of doublets (S,X), S the state of an individual, X its position in space. Given their local environmental conditions state transitions of individuals (including death) are Markovian. The environmental conditions are generated as output of an environmental process with (sums of the) outputs of individuals as input. When mating is random, individuals reproduce depending on their individual states, the local availability of mates, and the local environment. Physiologically structured population models supposedly arise as large number limits when the environment is almost constant over areas containing sufficiently many individuals (but proof for all but the simplest cases is still lacking). On such models one can graft evolutionary processes like adaptive evolution as a result of rare repeated substitutions of mutants changing the parameters of the state transition and reproduction processes of individuals or random genetic drift. From this perspective I will heuristically derive some relations between the so-called canonical equation of adaptive dynamics, a differential equation for the change in heritable traits determining those parameters, derived under the additional assumption that mutations have small effect, and the classical genetic diffusion limits.
Molchanov, Stanislav
Ideas of random media theory and mathematical models of the population structure dynamics - 1 and 2
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These lectures will review the main ideas and results of random media theory with a specific focus on the population growth in random environments. We shall present a simple yet powerful method of moment equations which in particular enables one to explain the phenomenon of spatial-temporal intermittency in the population density. As a recent development of these ideas, spatial interaction between diffusive (multi-type) particles (e.g., repulsion between species) may be modelled by a random potential drift. We shall show that this provides a mechanism to balance intermittency and thus avoid the unrealistic overclumping.
Mollison, Denis
Pair approximations for spatial structures?
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This work explores the success of pair approximations in capturing local correlations and the spatial structure of population contact networks, especially in respect of the rate of spread of epidemics. The broad conclusions are that pair approximations can capture the nonlinearity well, significantly improving on simpler deterministic models. But they are unable to capture the slower than exponential spread of spatial epidemics: spatial processes seem to need explicit spatial modelling; possible general reasons for this are discussed.
Murrell, David
Continuous-space models for interacting species: moment closure methods for community dynamics
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Most species are dispersal limited, with offspring dispersing near to their parent(s). Furthermore, ecological interactions tend to be between neighbours that are nearby in space, and in combination with local dispersal this tends to mean that interactions are more frequently between individuals of the same species than would be suggested by their global densities. Incorporating these features into models for interacting species is explored via the use of stochastic models and approximations that use first and second order spatial moments as the state variables to summarise the spatial structure that emerges. It will be shown that spatial structure can have both positive and negative effects on ecological communities
Nevai, Andrew
A comparison of spatial interactions at different scales
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For a stochastic plant population model involving seed dispersal, competition between neighbors, and landscape-induced mortality, we investigate connections between a spatial point process description and a spatial logistic equation with diffusion. The spatial moment equations of the spatial point process consist of an ODE for the mean population density and a pair of coupled PDEs, one for the cross-correlation between population size and landscape mortality and one for the auto-correlation of population size. The spatial logistic equation with diffusion, which amounts to short-range dispersal coupled with local competition and mortality, has corresponding ODE and PDE equations describing the fluctuations of population density and landscape mortality around their mean values. This work is in collaboration with B Bolker.
North, Ace
POSTER: Local adaptation in a changing environment
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Empirical studies of spatially structured populations often reveal patterns of genetic and ecological divergence which are strongly correlated with geographical structure. Divergence may result from adaptive responses to spatially structured variation in habitat, so that resident genotypes in each deme should have on average a higher relative fitness in their local habitat than genotypes originating from other habitats. This pattern, and the process leading to it are known as local adaptation. However, local adaptation may be hindered by gene flow, confounded by genetic drift, opposed by natural selection due to temporal environmental variability, and constrained by lack of genetic variation or by the genetic architecture of underlying traits. This poster outlines a spatial and stochastic modelling framework we have constructed to investigate how dispersal, reproductive mechanism (sexual or asexual), landscape structure and habitat loss influence the balance between local adaptation and (generalist or plastic) global adaptation. Analysis is not complete, yet preliminary results show how populations are inclined to locally adapt in sparse and temporally stable landscapes.
Oliveira, Maria João
Stochastic dynamics of infinite particle systems
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General birth-and-death as well as hopping stochastic dynamics of infinite particle systems in the continuum are considered. The corresponding evolution equations for correlation functions and generating functionals are derived. All considerations are illustrated in a number of concrete examples of Markov evolutions appearing in applications. An extension to multicomponent systems is also considered.
Ovaskainen, Otso
Evolution of dispersal in heterogeneous environments
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The classical work of Hamilton and May showed that the avoidance of kin-competition selects for dispersal even in a uniform environment. In heterogeneous and unpredictable environments, the evolution of dispersal is governed by a complex interplay between landscape structure and the properties of a species, making it difficult to obtain general insights. In particular, earlier work on dispersal evolution has shown that habitat loss can select for increased or decreased dispersal. Here, we unify these apparent inconsistencies into a general framework which shows how dispersal evolution depends on the nature of population dynamics and the manner in which habitat is lost. Our results are based on a mathematical method (perturbation expansion around the mean-field) that makes it possible to derive - in spatial and stochastic models - both the evolutionarily stable strategy that is resistant to invasions, and the evolutionarily stable frequency distribution that evolves in a polymorphic population.
Röckner, Michael
Fokker-Planck equations on Hilbert spaces
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We consider a stochastic differential equation in Hilbert space with timedependent coefficients for which no general existence and uniqueness results are known. We prove, under suitable assumptions, existence and uniqueness of a measure valued solution, for the corresponding Fokker-Planck equation. In particular, we verify the Chapman-Kolmogorov equations and get an evolution system of transition probabilities for the stochastic dynamics informally given by the stochastic differential equation.
Voss-Böhme, Anja
Rigorous analysis of stochastic particle-based models in developmental biology
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The emergence of spatial structures within originally homogeneous cell populations is a major object of study in developmental biology. To understand these processes, it is necessary to develop mathematical models where the assumed concepts of cell level kinetics are formulated explicitly. The analysis of these models should be concentrated on predicting the emergent dynamical behaviour at the tissue scale. Existing models are mostly borrowed from statistical physics. They are often constructed by exploiting phenomenological analogies between cell systems and closed thermodynamical systems. However, it is not clear to what extend physical concepts such as energy, pressure, temperature, surface tension can be applied to cell populations. One reason for this is that cells have a very complex inner structure which responds to intercellular as well as extracellular signals. In addition, these models are equilibrium models in the sense that they describe the interdependence structure once the temporal evolution has stabilized. Therefore they do not contribute enough to an understanding of the dynamical features. It is argued that the model class of interacting particle systems is well-suited for the study of pattern formation in developmental biology. For two exemplary problems, cell sorting and collective migration, cell based lattice models are developed which describe the major details of the respective intercellular interaction. If suitably simplified, these models are analytically tractable. Several results concerning the long-time behavior and the emergence of structure are presented and interpreted in biological terms. Challenging mathematical problems that require further theoretical developments are identified.

Participants

Name	Institution
Adams, Tom	University of Edinburgh
Atkins, Katie	University of Edinburgh
Barraquand, Fred	CNRS Chizé / Université de Paris VI
Berns, Christoph	Universität Bielefeld
Bogachev, Leonid	University of Leeds
Bolker, Benjamin	University of Florida
Cornell, Stephen	University of Leeds
Delius, Gustav	University of York
Derdziuk, Lukasz	Universität Bielefeld
Fattler, Torben	University of Kaiserslautern
Finkelshtein, Dmitri	Institute of Mathematics, Kiev
Foss, Serguei	Heriot-Watt University
Godfray, Charles	University of Oxford
Grothaus, Martin	University of Kaiserslautern
Hryniv, Ostap	Durham University
Illian, Janine	University of St Andrews
Kondratiev, Yuri	Universität Bielefeld
Konstantopoulos, Takis	Heriot-Watt University
Kuna, Tobias	University of Reading
Kutovyi, Oleksandr	Universität Bielefeld
Law, Richard	University of York
Lyons, Terry	University of Oxford
Lytvynov, Eugene	Swansea University
Mano, Shuhei	Nagoya City University
Marion, Glenn	Biomathematics & Statistics Scotland
McKane, Alan	University of Manchester
Metz, Johan	Universiteit Leiden
Molchanov, Stanislav	University of North Carolina at Charlotte
Mollison, Denis	Heriot-Watt University
Murrell, David	University College London
Nevai, Andrew	University of Central Florida
North, Ace	University of Helsinki
Oliveira, Maria João	Universidade Aberta/CMAF
Ovaskainen, Otso	University of Helsinki
Röckner, Michael	Universität Bielefeld
Voss-Böhme, Anja	Technische Universität Dresden