# Macroscopic dynamics and bifurcations of active particle systems

Lead Research Organisation:
Imperial College London

Department Name: Dept of Mathematics

### Abstract

The living world presents many examples of large assemblies of coordinated agents such as insect swarms, bird flocks or crowds and, at a more microscopic scale, swarming bacterial colonies or collectively migrating cells. These agents resemble particles composing inert matter but a striking difference is that they produce their own motion. They are generically referred to as active particles.

Like herds and flocks, most active particle systems exhibit self-organized collective motion. The mechanisms by which self-organization emerges are still poorly understood. Current research on this question is intense. In this work, we view the emergence of self-organization as a bifurcation from a non-coordinated state of the system to a collectively coordinated one. Bifurcations are intimately related to what physicists call phase transitions, i. e. abrupt changes of the state of a system when its environmental parameters are changed. Everyday examples are changes of state of matter such as water changing from liquid to vapor when its temperature crosses the boiling temperature. In nature, animals groups may change from a random motion state (when they are foraging for food for instance) to a coordinated motion state (when they want to escape the attack of a predator) in a similar way.

Our goal is to study mathematical models for active particle systems. We aim to develop macroscopic descriptions of these systems when the number of particles is large and to analyse their bifurcation from disordered to collective motion. Indeed, when the number of agents is large, it is neither possible nor efficient to follow each agent individually. Macroscopic models describe the evolution of statistical averages such as the mean density or velocity of the particles and are computationally much more efficient. Their rigorous derivation involves complex mathematical tools of kinetic theory but they give rise to an efficient way of analysing bifurcations.

Like for matter, there are many different types of bifurcations in active particle systems. In this proposal, we will focus on two specific but important examples. The first one is symmetry-breaking bifurcations when a system state changes its underlying symmetry. The second one is bifurcation due to jamming, which occurs when finite sized particles reach the density where they are all in contact with each other as in dense crowds for instance. To test the general character of our findings, we will also investigate other kinds of bifurcations, by looking at systems of rigid bodies interacting through attitude coordination, having collective sperm-cell dynamics as an application in mind.

The nature of mathematical models varies according to which state of the system they are adapted to. When several states are present simultaneously, they are separated by abrupt transition interfaces. To numerically approximate such situations, numerical methods that are uniformly accurate across the transition interface will be developed. They will allow us to validate the models by comparing them with real data in two selected applications, namely collective sperm-cell dynamics and pedestrian dynamics. In these two examples, we will showcase the usefulness of the models by using them to anticipate the outcome of various strategies of action aiming to change the collective behaviour of the system.

Like herds and flocks, most active particle systems exhibit self-organized collective motion. The mechanisms by which self-organization emerges are still poorly understood. Current research on this question is intense. In this work, we view the emergence of self-organization as a bifurcation from a non-coordinated state of the system to a collectively coordinated one. Bifurcations are intimately related to what physicists call phase transitions, i. e. abrupt changes of the state of a system when its environmental parameters are changed. Everyday examples are changes of state of matter such as water changing from liquid to vapor when its temperature crosses the boiling temperature. In nature, animals groups may change from a random motion state (when they are foraging for food for instance) to a coordinated motion state (when they want to escape the attack of a predator) in a similar way.

Our goal is to study mathematical models for active particle systems. We aim to develop macroscopic descriptions of these systems when the number of particles is large and to analyse their bifurcation from disordered to collective motion. Indeed, when the number of agents is large, it is neither possible nor efficient to follow each agent individually. Macroscopic models describe the evolution of statistical averages such as the mean density or velocity of the particles and are computationally much more efficient. Their rigorous derivation involves complex mathematical tools of kinetic theory but they give rise to an efficient way of analysing bifurcations.

Like for matter, there are many different types of bifurcations in active particle systems. In this proposal, we will focus on two specific but important examples. The first one is symmetry-breaking bifurcations when a system state changes its underlying symmetry. The second one is bifurcation due to jamming, which occurs when finite sized particles reach the density where they are all in contact with each other as in dense crowds for instance. To test the general character of our findings, we will also investigate other kinds of bifurcations, by looking at systems of rigid bodies interacting through attitude coordination, having collective sperm-cell dynamics as an application in mind.

The nature of mathematical models varies according to which state of the system they are adapted to. When several states are present simultaneously, they are separated by abrupt transition interfaces. To numerically approximate such situations, numerical methods that are uniformly accurate across the transition interface will be developed. They will allow us to validate the models by comparing them with real data in two selected applications, namely collective sperm-cell dynamics and pedestrian dynamics. In these two examples, we will showcase the usefulness of the models by using them to anticipate the outcome of various strategies of action aiming to change the collective behaviour of the system.

### Planned Impact

Impact on society: Active particle systems are ubiquitous in biology and social sciences. Many animals live in large coordinated societies. Understanding collective behaviour is important for managing biological resources. Tissues and cells are composed of many individual entities which constantly coordinate to fulfill the needs of the organism. It is of key importance for medicine to understand this coordination as its disruption results in diseases such as cancer. In economics or social dynamics, similar collective phenomena occur and better understanding them would help in preventing financial or social crises. One focus of this project concerns collective sperm-cell dynamics which is a tracer of semen fertility. Collective dynamics triggers turbulent motion of the semen and is easily observed under the microscope. Automatized objective measurements of this turbulence are needed in artificial insemination centers to improve the quality of the selected semen samples. A patent application has already been filed (of which the PI holds 40 \% of the invention shares). Future developments in the area of human fertility are envisioned. A second focus is about pedestrian dynamics. In densely populated areas, pedestrian safety represents a major challenge. Efficiency issues are equally important in places like walkways or underground corridors. To forecast future occupation of walkways in real-time so as to guide people towards less crowded routes, the use of efficient models, such as macroscopic models is necessary. The models developed in the present project will greatly contribute to the advancement of our knowledge in these two selected applications.

Exploitation: Collective dynamics can be used as a probe of the proper functioning of a system, like in the above example, where collective sperm-cell dynamics is used as a tracer of semen fertility. To the largest possible extent, other patents will be sought in the application areas targeted by this proposal. In particular, the PI is currently envisioning a system for crowd forecasting. As often as possible, the PI will look for ways to transfer the basic knowledge acquired in the project into practical innovations. The existence of a dedicated structure at Imperial will greatly facilitate this process. In addition, each faculty has a Knowledge Transfer Fellow and Intellectual Property is protected through Imperial Innovations Ltd.

Human resources: Young researchers hired on this project will be offered a stimulating environment, with a unique opportunity to work at the leading edge of research in applied mathematics and modeling. They will be encouraged to develop an autonomous thinking and research methodology and to follow their own threads of research. They will receive full support to attend the largest international conferences in the field and to present their own work. They will also be encouraged to visit other teams with the objective of building their own network of collaborations. On general aspects of staff development and support, they will find an extensive programme offered by Imperial College. By the combination of all these factors, young researchers will acquire a greater scientific maturity which will considerably improve their future hiring prospects. The PI has a long track-record of post-doc advising. Most of his former post-docs have found positions in academia, research related institutions or industry. People with a strong experience in mathematical and numerical modeling are highly demanded in UK and in Europe. So, it is expected that the young researchers hired on the project will subsequently be in a pole position in the market of highly skilled jobs, be they in academia or in industry.

Exploitation: Collective dynamics can be used as a probe of the proper functioning of a system, like in the above example, where collective sperm-cell dynamics is used as a tracer of semen fertility. To the largest possible extent, other patents will be sought in the application areas targeted by this proposal. In particular, the PI is currently envisioning a system for crowd forecasting. As often as possible, the PI will look for ways to transfer the basic knowledge acquired in the project into practical innovations. The existence of a dedicated structure at Imperial will greatly facilitate this process. In addition, each faculty has a Knowledge Transfer Fellow and Intellectual Property is protected through Imperial Innovations Ltd.

Human resources: Young researchers hired on this project will be offered a stimulating environment, with a unique opportunity to work at the leading edge of research in applied mathematics and modeling. They will be encouraged to develop an autonomous thinking and research methodology and to follow their own threads of research. They will receive full support to attend the largest international conferences in the field and to present their own work. They will also be encouraged to visit other teams with the objective of building their own network of collaborations. On general aspects of staff development and support, they will find an extensive programme offered by Imperial College. By the combination of all these factors, young researchers will acquire a greater scientific maturity which will considerably improve their future hiring prospects. The PI has a long track-record of post-doc advising. Most of his former post-docs have found positions in academia, research related institutions or industry. People with a strong experience in mathematical and numerical modeling are highly demanded in UK and in Europe. So, it is expected that the young researchers hired on the project will subsequently be in a pole position in the market of highly skilled jobs, be they in academia or in industry.

### Organisations

- Imperial College London, United Kingdom (Collaboration, Lead Research Organisation)
- North Carolina State University, United States (Collaboration)
- Case Western Reserve University (Collaboration)
- Granada University (Collaboration)
- University of Strasbourg, France (Collaboration)
- Stuttgart University, Germany (Collaboration)
- French Institute for Research in Computer Science and Automation (Collaboration)
- RWTH Aachen University, Germany (Collaboration)
- University of Ferrara (Collaboration)
- French National Institute of Agricultural Research (Collaboration)
- Arizona State University (Collaboration)
- Carnegie Mellon University, United States (Collaboration)
- University of Vienna (Collaboration)
- Aix-Marseille University, France (Collaboration)
- University of Toulouse III Paul Sabatier, France (Collaboration)
- University of Orleans (Collaboration)
- University of Cambridge (Collaboration)
- Duke University, United States (Collaboration)
- University of Aberdeen, United Kingdom (Collaboration)
- University of Paris South 11, France (Collaboration)
- Paris Dauphine University (Collaboration)
- National Center for Scientific Research (Centre National de la Recherche Scientifique CNRS) (Collaboration)
- University of Toulouse (Collaboration)
- University of Heidelberg, Germany (Collaboration)

### Publications

Aceves-Sanchez P
(2020)

*Large-Scale Dynamics of Self-propelled Particles Moving Through Obstacles: Model Derivation and Pattern Formation.*in Bulletin of mathematical biology
Aceves-Sánchez P
(2019)

*Hydrodynamic limits for kinetic flocking models of Cucker-Smale type.*in Mathematical biosciences and engineering : MBE
Barbaro A
(2016)

*Phase Transitions in a Kinetic Flocking Model of Cucker--Smale Type*in Multiscale Modeling & Simulation
Barré J
(2018)

*Particle Interactions Mediated by Dynamical Networks: Assessment of Macroscopic Descriptions.*in Journal of nonlinear science
Barré J
(2017)

*Kinetic Theory of Particle Interactions Mediated by Dynamical Networks*in Multiscale Modeling & Simulation
Blanchet A
(2016)

*Topological Interactions in a Boltzmann-Type Framework*in Journal of Statistical Physics
Blanchet A
(2017)

*Kinetic Models for Topological Nearest-Neighbor Interactions.*in Journal of statistical physics
Chertock A
(2017)

*An asymptotic-preserving method for a relaxation of the Navier-Stokes-Korteweg equations*in Journal of Computational Physics
Chertock A
(2019)

*Incompressible limit of a continuum model of tissue growth with segregation for two cell populations.*in Mathematical biosciences and engineering : MBE
Creppy A
(2016)

*Symmetry-breaking phase transitions in highly concentrated semen.*in Journal of the Royal Society, InterfaceDescription | Important findings are 1. Emergence of symmetry breaking phase transitions in concentrated semen: Semen confined in an annular channel exhibit spontaneous collective rotation. We have shown that this is linked to a transition from disordered sperm cell motion at low concentration (when the sperm is dilute) to collective motion at high concentration (when the sperm is undiluted). Indeed, self-propulsion of the sperm-cells combines with the effect of confinment in the annular chamber to force all sperm-cell to move in the same direction, leading to the spontaneous collective rotation of the sperm-cells around the inner radius of the annulus. We expect that the measurement of the rotation speed will provide an easy way to select the most active sperm. In a related publication we have shown that this activity level is correlated to a higher fertility. Hence, we can use this apparatus as a fertility diagnostic of semen samples in animal artificial reproduction industry. 2. We have found new models for systems of self-propelled particles interacting through full body attitude coordination i.e. agents that do not only try to align their velocity to their neighbors but align the full attitude of their bodies modelled as rotation matrices. These models feature new mathematical objects such as spatio-temporally dependent rotation matrix or unitary quaternion fields, the properties of which are almost unexplored so far. |

Exploitation Route | Findings 1 (symmetry-breaking phase transitions in concentrated semen) could be used by the animal artificial insemination industry to improve the fertility diagnostics of semen samples. Also, these findings have laid down the foundations of a new award 'sperm-mucus interaction' targetting human fertility and still active. Findings 2 (self-propelled particles interacting through full body attitude coordination) These models could have applications for instance, in the understanding of bird flocks of schools of fish and could be usefully reproduced for the control of fleets of drones or submarines. It could also be useful to understand collective sperm dynamics as sperm cells have complex shapes (they look like tennis rackets) which could have significant influence on their dynamics. |

Sectors | Aerospace, Defence and Marine,Agriculture, Food and Drink,Digital/Communication/Information Technologies (including Software),Education,Environment,Healthcare,Leisure Activities, including Sports, Recreation and Tourism,Manufacturing, including Industrial Biotechology,Transport |

URL | https://sites.google.com/site/degond/Home/scientific-interests |

Description | Our work on mass motility of seminal plasma has raised the interest of the french society IMV-technologies which look for ways to improve the selection of fertile semen samples in animal artificial insemination. One of the modalities is described in the collaboration ''Symmetry-breaking phase-transitions in highly concentrated semen''. Our new numerical algorithm for packing has triggered new collaborations in biology and medicine, for the modelling of epithelial development and the understanding of the epithelial-mesenchymal transition in cancer. I have also undertaken several outreach activities (such as inaugural lectures, featured articles in Imperial College News, public lecture to high school students), etc. |

First Year Of Impact | 2015 |

Sector | Education,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology |

Impact Types | Cultural,Societal,Economic |

Description | Crick PhD Studentship |

Amount | £160,000 (GBP) |

Organisation | Francis Crick Institute |

Sector | Academic/University |

Country | United Kingdom |

Start | 10/2015 |

End | 09/2019 |

Description | Mathematical and In- silico modelisation of normal and malignant HSC in their niche and of their interactions with stromal cells. |

Amount | £160,000 (GBP) |

Organisation | Francis Crick Institute |

Sector | Academic/University |

Country | United Kingdom |

Start | 10/2019 |

End | 09/2023 |

Description | Royal Society - CNRS exchange program |

Amount | £12,000 (GBP) |

Funding ID | IE160750 |

Organisation | The Royal Society |

Sector | Charity/Non Profit |

Country | United Kingdom |

Start | 03/2017 |

End | 02/2019 |

Description | Standard grant |

Amount | £850,713 (GBP) |

Funding ID | EP/P013651/1 |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Public |

Country | United Kingdom |

Start | 04/2017 |

End | 03/2021 |

Description | center |

Amount | £2,570,063 (GBP) |

Funding ID | EP/N014529/1 |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Public |

Country | United Kingdom |

Start | 01/2016 |

End | 12/2019 |

Title | New computational methods for packings |

Description | We have provided a new computational method to minimize the volume occupied by an ensemble of extended objects under non-orvelapping constraints. This involves the minimization of a confinment energy under these constraints. Standard minimization methods are provable convergent in convex cases. Unfortunately, the non-overlapping constraint is not convex and this leads to standard algorithms to be unstable. We have provide a novel minimization algorithm that is stable. This can be used for instance in the computation of the configurations of cells in solid tumors. |

Type Of Material | Improvements to research infrastructure |

Year Produced | 2016 |

Provided To Others? | Yes |

Impact | This method has been published in a high impact factor journal: P. Degond, M. A. Ferreira, S. Motsch, Damped Arrow-Hurwicz algorithm for sphere packing. Journal of Computational Physics, 332 (2017), pp. 47-65 (open access). It is now used in different projects related to cell packings in biology (development of epithelial tissues, Epithelial-Mesenchymal transition in cancer, etc). |

URL | http://www.sciencedirect.com/science/article/pii/S0021999116306398 |

Title | Macroscopic models for self-propelled particles interacting through body attitude coordination |

Description | We have developed a model of self-propelled particles interacting through body alignment. We have developed both the individual-based model and the associated macroscopic model. Both rely on the use of unitary quaternions to encode the rotation describing the body attitude. |

Type Of Material | Computer model/algorithm |

Year Produced | 2016 |

Provided To Others? | Yes |

Impact | A paper concerning this model is about to appear: P. Degond, A. Frouvelle, S. Merino-Aceituno, A new flocking model through body attitude coordination. To appear in Mathematical Models and Methods in Applied Sciences. Manuscript on arXiv. A second one has been submitted: P. Degond, A. Frouvelle, S. Merino-Aceituno, A. Trescases, Quaternions in collective dynamics. Submitted. Manuscript on arXiv. |

Title | Model of tissue self-organization |

Description | This model enables us to study the self-organization of tissue as the result of the interaction between collagen fibers and cells. The model computes the minimal of a mechanical interaction energy between the fibers and the cells subject to the cell-cell non-overalapping constraint. |

Type Of Material | Computer model/algorithm |

Year Produced | 2016 |

Provided To Others? | Yes |

Impact | We have further elaborated this model in many directions: - modelling of epithelial tissue growth (with the Crick Institute) - modelling epithelial-mesenchymal transition in cancer (with University of Toulouse) - modelling vascular growth All these models are under current development and result from the earlier development of this model |

Title | Sperm massal motility |

Description | This data base contains results of experimental measurements related to our paper A. Creppy, F. Plouraboué, O. Praud, X. Druart, S. Cazin, H. Yu, P. Degond, Symmetry-breaking phase-transitions in highly concentrated semen, Journal of the Royal Society Interface, 13 (2016), 20160575 (open access). |

Type Of Material | Database/Collection of data |

Year Produced | 2016 |

Provided To Others? | Yes |

Impact | This research data base increased our visiblity and enabled us to put up a new EPSRC project (successful) entitled ''Sperm-mucus interactions across scales''. |

URL | http://datadryad.org/resource/doi:10.5061/dryad.7309b |

Title | macroscopic model for self-propelled particles interacting through alignment |

Description | self-propelled particles interacting through alignment are commonplace in biology. An example is provided by myxobacteria colonies. We have designed a new macroscopic model for such systems. It describes the mean direction of motion of the particles and the mean density of particles flowing in each direction. |

Type Of Material | Computer model/algorithm |

Year Produced | 2015 |

Provided To Others? | Yes |

Impact | Macroscopic models are cost-effective simulation methods. They make simulations of colonies consisting of a large nuimber of indviduals (several bilions) feasible while individual based models are too CPU intensive. A publication has appeared: P. Degond, A. Manhart, H. Yu, A continuum model for nematic alignment of self-propelled particles, Discrete and Continuum Dynamical Systems Series B, 22 (2017), pp. 1295-1327 (open access). Another one has been submitted: P. Degond, A. Manhart, H. Yu, An age-structured continuum model for myxobacteria, submitted. Manuscript on arXiv |

URL | http://aimsciences.org/journals/pdfs.jsp?paperID=13658&mode=full |

Description | A hybrid agent-based/continuum model of blood network formation |

Organisation | Imperial College London |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | We propose a new model for blood network formation. |

Collaborator Contribution | A new model of blood network formation is introduced. The model is an hybrid agent-based/continuum model: capillaries are modelled as segments modifying locally blood and oxygen flows through conductivity and diffusivity tensors. Simulations clearly show that many properties of networks, such as branching angle, tortuosity or branches merging, could be emergent in response of the environment, and do not have to be specified. Using skeletization techniques, we perform a statistical analysis of the networks we obtain by numerical simulation. Due to the few assumptions needed, this model is a general model of network formation and may be applied other natural networks. Its main advantage is its dynamical adaptability and the emergent properties that it features. |

Impact | We are currently developing this project and exploring new features that could be incorporated into our model to explain the emergence of blood networks in a particular setting. This project is a multi-disciplinary one and involves techniques such as image analysis, numerical solution of Partial Differential Equations and experimental biology. |

Start Year | 2016 |

Description | A hybrid agent-based/continuum model of blood network formation |

Organisation | University of Vienna |

Country | Austria |

Sector | Academic/University |

PI Contribution | We propose a new model for blood network formation. |

Collaborator Contribution | A new model of blood network formation is introduced. The model is an hybrid agent-based/continuum model: capillaries are modelled as segments modifying locally blood and oxygen flows through conductivity and diffusivity tensors. Simulations clearly show that many properties of networks, such as branching angle, tortuosity or branches merging, could be emergent in response of the environment, and do not have to be specified. Using skeletization techniques, we perform a statistical analysis of the networks we obtain by numerical simulation. Due to the few assumptions needed, this model is a general model of network formation and may be applied other natural networks. Its main advantage is its dynamical adaptability and the emergent properties that it features. |

Impact | We are currently developing this project and exploring new features that could be incorporated into our model to explain the emergence of blood networks in a particular setting. This project is a multi-disciplinary one and involves techniques such as image analysis, numerical solution of Partial Differential Equations and experimental biology. |

Start Year | 2016 |

Description | Asymptotic-Preserving Method for a Relaxation of the Navier-Stokes-Korteweg Equations |

Organisation | North Carolina State University |

Country | United States |

Sector | Academic/University |

PI Contribution | The Navier-Stokes-Korteweg (NSK) equations are a classical diffuse-interface model for compressible two-phase flow. As direct numerical simulations based on the NSK system are quite expensive and in some cases even impossible, we consider a relaxation of the NSK system, for which robust numerical methods can be designed. However, time steps for explicit numerical schemes depend on the relaxation parameter and therefore numerical simulations in the relaxation limit are very inefficient. To overcome this restriction, we propose an implicitexplicit asymptotic-preserving finite volume method. We prove that the new scheme provides a consistent discretization of the NSK system in the relaxation limit and demonstrate that it is capable of accurately and efficiently computing numerical solutions of problems with realistic density ratios and small interfacial widths. |

Collaborator Contribution | All partners have contributed equally at all stages of this research |

Impact | A research paper has appeared: A. Chertock, P. Degond, J. Neusser, An Asymptotic-Preserving method for a relaxation of the Navier-Stokes-Korteweg equations, Journal of Computational Physics, 335 (2017), pp. 387-403. (open access). |

Start Year | 2015 |

Description | Asymptotic-Preserving Method for a Relaxation of the Navier-Stokes-Korteweg Equations |

Organisation | University of Stuttgart |

Country | Germany |

Sector | Academic/University |

PI Contribution | The Navier-Stokes-Korteweg (NSK) equations are a classical diffuse-interface model for compressible two-phase flow. As direct numerical simulations based on the NSK system are quite expensive and in some cases even impossible, we consider a relaxation of the NSK system, for which robust numerical methods can be designed. However, time steps for explicit numerical schemes depend on the relaxation parameter and therefore numerical simulations in the relaxation limit are very inefficient. To overcome this restriction, we propose an implicitexplicit asymptotic-preserving finite volume method. We prove that the new scheme provides a consistent discretization of the NSK system in the relaxation limit and demonstrate that it is capable of accurately and efficiently computing numerical solutions of problems with realistic density ratios and small interfacial widths. |

Collaborator Contribution | All partners have contributed equally at all stages of this research |

Impact | A research paper has appeared: A. Chertock, P. Degond, J. Neusser, An Asymptotic-Preserving method for a relaxation of the Navier-Stokes-Korteweg equations, Journal of Computational Physics, 335 (2017), pp. 387-403. (open access). |

Start Year | 2015 |

Description | Asymtotic analysis of a fractional generalized Langevin equation |

Organisation | Imperial College London |

Department | Department of Mathematics |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | We have done the asymptotic analysis of a fractional Fokker-Planck equation with external forcing using entropy production techniques. |

Collaborator Contribution | My colleague has introduced in a paper which just appear a variational approach for rigorous asymptotic analysis of equations with a variational evolution structure. We are aiming at tackling the ssymtotic analysis of a fractional generalized Langevin equation. |

Impact | So far we are trying to apply the techniques mentioned above to tackle the coarse-graining of a fractional generalized Langevin equation. We expect to have a paper in the near future. |

Start Year | 2017 |

Description | Coagulation-fragmentation model for animal group-size statistics |

Organisation | Carnegie Mellon University |

Department | Department of Mathematical Sciences |

Country | United States |

Sector | Academic/University |

PI Contribution | We study coagulation-fragmentation equations inspired by a simple model proposed in fisheries science to explain data for the size distribution of schools of pelagic fish. Although the equations lack detailed balance and admit no H-theorem, we are able to develop a rather complete description of equilibrium profiles and large-time behavior, based on recent developments in complex function theory for Bernstein and Pick functions. In the large-population continuum limit, a scaling-invariant regime is reached in which all equilibria are determined by a single scaling profile. This universal profile exhibits power-law behavior. |

Collaborator Contribution | All partners have contributed equally at all stages of this research |

Impact | A research paper has appeared: P. Degond, J-G. Liu, R. L. Pego, Coagulation-fragmentation model for animal group-size statistics, Journal of Nonlinear Science, 27 (2017), pp 379-424 (open access). Another one is in press: P. Degond, M. Engel, Numerical approximation of a coagulation-Fragmentation Model for Animal Group Size Statistics. Networks and Heterogeneous Media, to appear. Manuscript on arXiv. Other papers are on the way. |

Start Year | 2013 |

Description | Coagulation-fragmentation model for animal group-size statistics |

Organisation | Duke University |

Department | Department of Mathematics |

Country | United States |

Sector | Academic/University |

PI Contribution | We study coagulation-fragmentation equations inspired by a simple model proposed in fisheries science to explain data for the size distribution of schools of pelagic fish. Although the equations lack detailed balance and admit no H-theorem, we are able to develop a rather complete description of equilibrium profiles and large-time behavior, based on recent developments in complex function theory for Bernstein and Pick functions. In the large-population continuum limit, a scaling-invariant regime is reached in which all equilibria are determined by a single scaling profile. This universal profile exhibits power-law behavior. |

Collaborator Contribution | All partners have contributed equally at all stages of this research |

Impact | A research paper has appeared: P. Degond, J-G. Liu, R. L. Pego, Coagulation-fragmentation model for animal group-size statistics, Journal of Nonlinear Science, 27 (2017), pp 379-424 (open access). Another one is in press: P. Degond, M. Engel, Numerical approximation of a coagulation-Fragmentation Model for Animal Group Size Statistics. Networks and Heterogeneous Media, to appear. Manuscript on arXiv. Other papers are on the way. |

Start Year | 2013 |

Description | Coarse-graining of kinetic equation of Cucker-Smale type |

Organisation | Aix-Marseille University |

Country | France |

Sector | Academic/University |

PI Contribution | The PI (Prof. P. Degond) introduced a new method in 2008 to do the coarse-graining of kinetic equations without moment conservation. We have used this method to obtain the macroscopic limit of a kinetic equation of Cucker-Smale type. |

Collaborator Contribution | Our colleagues introduced another way of doing the coarse-graining of kinetic equation without momentum reservation and we have made a connection between their method an the one mentioned in the previous section. |

Impact | Currently we are writing a paper and we expect to have it in about 3 months. As a follow up we are expecting to extend our method to other type of equations. |

Start Year | 2017 |

Description | Coarse-graining of kinetic equation of Cucker-Smale type |

Organisation | Imperial College London |

Department | Department of Mathematics |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | The PI (Prof. P. Degond) introduced a new method in 2008 to do the coarse-graining of kinetic equations without moment conservation. We have used this method to obtain the macroscopic limit of a kinetic equation of Cucker-Smale type. |

Collaborator Contribution | Our colleagues introduced another way of doing the coarse-graining of kinetic equation without momentum reservation and we have made a connection between their method an the one mentioned in the previous section. |

Impact | Currently we are writing a paper and we expect to have it in about 3 months. As a follow up we are expecting to extend our method to other type of equations. |

Start Year | 2017 |

Description | Crowd dynamics |

Organisation | Arizona State University |

Country | United States |

Sector | Academic/University |

PI Contribution | We investigate the development of a "crowd forecasting system", whereby real-time observations of crowds are coupled to fast and reliable models to produce rapid predictions of the crowd movement and eventually help crowd managers choose between tailored optimization strategies. Here, we propose a Bidirectional Macroscopic (BM) model as the core of such a system. Its key input is the fundamental diagram for bi-directional ?ows, i.e. the relation between the pedestrian ?uxes and densities. We design and run a laboratory experiments and show that the model is able to accurately capture the experimental data in a typical crowd forecasting situation. |

Collaborator Contribution | All partners contributed equally to this research |

Impact | A research paper has been submitted |

Start Year | 2009 |

Description | Crowd dynamics |

Organisation | French Institute for Research in Computer Science and Automation |

Country | France |

Sector | Private |

PI Contribution | We investigate the development of a "crowd forecasting system", whereby real-time observations of crowds are coupled to fast and reliable models to produce rapid predictions of the crowd movement and eventually help crowd managers choose between tailored optimization strategies. Here, we propose a Bidirectional Macroscopic (BM) model as the core of such a system. Its key input is the fundamental diagram for bi-directional ?ows, i.e. the relation between the pedestrian ?uxes and densities. We design and run a laboratory experiments and show that the model is able to accurately capture the experimental data in a typical crowd forecasting situation. |

Collaborator Contribution | All partners contributed equally to this research |

Impact | A research paper has been submitted |

Start Year | 2009 |

Description | Crowd dynamics |

Organisation | Paul Sabatier University (University of Toulouse III) |

Country | France |

Sector | Academic/University |

PI Contribution | We investigate the development of a "crowd forecasting system", whereby real-time observations of crowds are coupled to fast and reliable models to produce rapid predictions of the crowd movement and eventually help crowd managers choose between tailored optimization strategies. Here, we propose a Bidirectional Macroscopic (BM) model as the core of such a system. Its key input is the fundamental diagram for bi-directional ?ows, i.e. the relation between the pedestrian ?uxes and densities. We design and run a laboratory experiments and show that the model is able to accurately capture the experimental data in a typical crowd forecasting situation. |

Collaborator Contribution | All partners contributed equally to this research |

Impact | A research paper has been submitted |

Start Year | 2009 |

Description | Crowd dynamics |

Organisation | University Paris Sud |

Country | France |

Sector | Academic/University |

PI Contribution | |

Collaborator Contribution | All partners contributed equally to this research |

Impact | A research paper has been submitted |

Start Year | 2009 |

Description | Flocking through body attitude coordination |

Organisation | Paris Dauphine University |

Country | France |

Sector | Academic/University |

PI Contribution | We study new models for multi-agent dynamics where each agent is described by its position and body attitude: agents travel at a constant speed in a given direction and their body can rotate around it adopting different configurations. Agents try to coordinate their body attitudes with those of their neighbours. We introduce the Individual Based Model for this dynamics and derive its corresponding kinetic and macroscopic equations. This is a new model where collective motion is reached through body attitude coordination, which has never been considered. |

Collaborator Contribution | All partners contributed equally to this research |

Impact | A paper is going to appear : P. Degond, A. Frouvelle, S. Merino-Aceituno, A new flocking model through body attitude coordination. To appear in Mathematical Models and Methods in Applied Sciences. Manuscript on arXiv. Another paper has been submitted: P. Degond, A. Frouvelle, S. Merino-Aceituno, A. Trescases, Quaternions in collective dynamics. Submitted. Manuscript on arXiv. |

Start Year | 2015 |

Description | Flocking through body attitude coordination |

Organisation | University of Cambridge |

Department | Department of Zoology |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | We study new models for multi-agent dynamics where each agent is described by its position and body attitude: agents travel at a constant speed in a given direction and their body can rotate around it adopting different configurations. Agents try to coordinate their body attitudes with those of their neighbours. We introduce the Individual Based Model for this dynamics and derive its corresponding kinetic and macroscopic equations. This is a new model where collective motion is reached through body attitude coordination, which has never been considered. |

Collaborator Contribution | All partners contributed equally to this research |

Impact | A paper is going to appear : P. Degond, A. Frouvelle, S. Merino-Aceituno, A new flocking model through body attitude coordination. To appear in Mathematical Models and Methods in Applied Sciences. Manuscript on arXiv. Another paper has been submitted: P. Degond, A. Frouvelle, S. Merino-Aceituno, A. Trescases, Quaternions in collective dynamics. Submitted. Manuscript on arXiv. |

Start Year | 2015 |

Description | Kinetic theory of particle interactions mediated by dynamical networks. |

Organisation | University of Orleans |

Country | France |

Sector | Academic/University |

PI Contribution | We develop a multiscale analysis of a system of particles interacting through a dynamical network of links. Starting from a microscopic model, via the mean field limit, we derive coupled kinetic equations for the particle and link densities, Assuming that the process of remodelling the network is very fast, we simplify the description to a macroscopic model taking the form of single aggregation-diffusion equation for the density of particles. We analyze qualitatively this equation, addressing the stability of a homogeneous distribution of particles for a general potential. For the Hookean potential we obtain a precise condition for the phase transition, and, using the central manifold reduction, we characterize the type of bifurcation at the instability onset. |

Collaborator Contribution | All partners contributed equally to this research |

Impact | Two papers have been sumbitted: J. Barré, P. Degond, E. Zatorska, Kinetic theory of particle interactions mediated by dynamical networks. Submitted. Manuscript on arXiv and J. Barré, J. A. Carrillo de la Plata, P. Degond, D. Peurichard, E. Zatorska, Particle interactions mediated by dynamical networks: assessment of macroscopic descriptions, Submitted. Manuscript on arXiv Further studies are ongoing. |

Start Year | 2015 |

Description | Kinetic theory of particle interactions mediated by dynamical networks. |

Organisation | University of Vienna |

Department | Faculty of Mathematics |

Country | Austria |

Sector | Academic/University |

PI Contribution | We develop a multiscale analysis of a system of particles interacting through a dynamical network of links. Starting from a microscopic model, via the mean field limit, we derive coupled kinetic equations for the particle and link densities, Assuming that the process of remodelling the network is very fast, we simplify the description to a macroscopic model taking the form of single aggregation-diffusion equation for the density of particles. We analyze qualitatively this equation, addressing the stability of a homogeneous distribution of particles for a general potential. For the Hookean potential we obtain a precise condition for the phase transition, and, using the central manifold reduction, we characterize the type of bifurcation at the instability onset. |

Collaborator Contribution | All partners contributed equally to this research |

Impact | Two papers have been sumbitted: J. Barré, P. Degond, E. Zatorska, Kinetic theory of particle interactions mediated by dynamical networks. Submitted. Manuscript on arXiv and J. Barré, J. A. Carrillo de la Plata, P. Degond, D. Peurichard, E. Zatorska, Particle interactions mediated by dynamical networks: assessment of macroscopic descriptions, Submitted. Manuscript on arXiv Further studies are ongoing. |

Start Year | 2015 |

Description | Models for self-propelled disks interacting through alignment and volume exclusion |

Organisation | University of Strasbourg |

Department | Department of Mathematics and Computer Science |

Country | France |

Sector | Academic/University |

PI Contribution | We study individual-based models describing disk-like self-propelled particles. The disk directions of motion follow alignment rules and volume exclusion interactions with their neighbors. We formally derive a macroscopic model based on self-organized hydrodynamic (SOH) models describing the transport of mass and evolution of mean direction of motion of the disks. Numerical comparisons between the individual-based and macroscopic models are carried out. These models could be applicable, for instance, to describe sperm-cell collective dynamics. |

Collaborator Contribution | All partners contributed equally to this research |

Impact | A paper has appeared: P. Degond, L. Navoret, A multi-layer model for self-propelled disks interacting through alignment and volume exclusion, Mathematical Models and Methods in Applied Sciences, 25 (2015), 2439-2475 (open access). A second one is being written. |

Start Year | 2015 |

Description | Numerial algorithms for packing. |

Organisation | Arizona State University |

Country | United States |

Sector | Academic/University |

PI Contribution | We consider algorithms that, from an arbitrarily sampling of N spheres (possibly overlapping), find a close packed configuration without overlapping. These problems can be formulated as minimization problems with non-convex constraints. For such packing problems, we observe that the classical algorithms do not converge. We derive a novel algorithm and compare this algorithm with classical algorithms. We provide an analysis of the convergence of these algorithms. We investigate the behaviour of our algorithm when the number of spheres is large in two and three spatial dimensions |

Collaborator Contribution | Each partner contributed equally to this research |

Impact | A research paper has appeared: P. Degond, M. A. Ferreira, S. Motsch, Damped Arrow-Hurwicz algorithm for sphere packing. Journal of Computational Physics, 332 (2017), pp. 47-65 (open access). A second paper is being written and future works are in preparation. |

Start Year | 2015 |

Description | Numerical and mathematical model of tumour growth |

Organisation | Paul Sabatier University (University of Toulouse III) |

Country | France |

Sector | Academic/University |

PI Contribution | This collaboration investigates cells proliferation dynamics in small tumor cell aggregates using an individual based model (IBM). The simulation model is designed to study the morphology of the cell population and of the cell lineages as well as the impact of the orientation of the division plane on this morphology. Our IBM model is based on the hypothesis that cells are incompressible objects that grow in size and divide once a threshold size is reached, and that newly born cell adhere to the existing cell cluster. We performed comparisons between the simulation model and experimental data by using several statistical indicators. The results suggest that the emergence of particular morphologies can be explained by simple mechanical interactions. |

Collaborator Contribution | Paul Sabatier did the experiments. Ferrara and us contributed to the model. |

Impact | A paper has been submitted: M. Leroy-Lerêtre, G.Dimarco, M. Cazales, Marie-Laure Boizeau, B. Ducommun, V. Lobjois, P. Degond, Are tumor cell lineages solely shaped by mechanical forces ?, submitted. Manuscript on arXiv Another one is on the way. |

Start Year | 2015 |

Description | Numerical and mathematical model of tumour growth |

Organisation | University of Ferrara |

Country | Italy |

Sector | Academic/University |

PI Contribution | This collaboration investigates cells proliferation dynamics in small tumor cell aggregates using an individual based model (IBM). The simulation model is designed to study the morphology of the cell population and of the cell lineages as well as the impact of the orientation of the division plane on this morphology. Our IBM model is based on the hypothesis that cells are incompressible objects that grow in size and divide once a threshold size is reached, and that newly born cell adhere to the existing cell cluster. We performed comparisons between the simulation model and experimental data by using several statistical indicators. The results suggest that the emergence of particular morphologies can be explained by simple mechanical interactions. |

Collaborator Contribution | Paul Sabatier did the experiments. Ferrara and us contributed to the model. |

Impact | A paper has been submitted: M. Leroy-Lerêtre, G.Dimarco, M. Cazales, Marie-Laure Boizeau, B. Ducommun, V. Lobjois, P. Degond, Are tumor cell lineages solely shaped by mechanical forces ?, submitted. Manuscript on arXiv Another one is on the way. |

Start Year | 2015 |

Description | Phase Transitions in a kinetic flocking model of Cucker-Smale type |

Organisation | Case Western Reserve University |

Department | Department of Mathematics, Applied Mathematics and Statistics |

Country | United States |

Sector | Academic/University |

PI Contribution | We consider a collective behavior model in which individuals try to imitate each others' velocity and have a preferred speed. We show that a phase change phenomenon takes place as diffusion decreases, bringing the system from a "disordered" to an "ordered" state. This effect is related to recently noticed phenomena for the diffusive Vicsek model. We also carry out numerical simulations of the system and give further details on the phase transition. |

Collaborator Contribution | All partners have contributed equally at all stages of this research |

Impact | An article has appeared; A. B. T. Barbaro, J. A. Cañizo, J. A. Carrillo, P. Degond, Phase Transitions in a kinetic flocking model of Cucker-Smale type. Multiscale Modelling and Simulation 14 (2016), pp. 1063-1088 (open access). Other works are on the way. |

Start Year | 2011 |

Description | Phase Transitions in a kinetic flocking model of Cucker-Smale type |

Organisation | University of Granada |

Department | Department of Mathematical Analysis |

Country | Spain |

Sector | Academic/University |

PI Contribution | We consider a collective behavior model in which individuals try to imitate each others' velocity and have a preferred speed. We show that a phase change phenomenon takes place as diffusion decreases, bringing the system from a "disordered" to an "ordered" state. This effect is related to recently noticed phenomena for the diffusive Vicsek model. We also carry out numerical simulations of the system and give further details on the phase transition. |

Collaborator Contribution | All partners have contributed equally at all stages of this research |

Impact | An article has appeared; A. B. T. Barbaro, J. A. Cañizo, J. A. Carrillo, P. Degond, Phase Transitions in a kinetic flocking model of Cucker-Smale type. Multiscale Modelling and Simulation 14 (2016), pp. 1063-1088 (open access). Other works are on the way. |

Start Year | 2011 |

Description | Self-Organized Hydrodynamics with nonconstant velocity |

Organisation | RWTH Aachen University |

Department | Department of Mathematics |

Country | Germany |

Sector | Academic/University |

PI Contribution | Motivated by recent experimental and computational results that show a motility-induced clustering transition in self-propelled particle systems, we study an individual model and its corresponding Self-Organized Hydrodynamic model for collective behaviour that incorporates a density-dependent velocity, as well as inter-particle alignment. The modal analysis of the hydrodynamic model elucidates the relationship between the stability of the equilibria and the changing velocity, and the formation of clusters. We find, in agreement with earlier results for non-aligning particles, that the key criterion for stability is. a nondecreasing mass flux with respect to the density. Numerical simulation for both the individual and hydrodynamic models with a velocity function inspired by experiment demonstrates the validity of the theoretical results. |

Collaborator Contribution | All partners contributed equally to this research. |

Impact | A paper has appeared: P. Degond, S. Henkes, H. Yu, Self-Organized Hydrodynamics with nonconstant velocity, Kinetic and Related Models, 10 (2017), pp. 197-213 (open access). Others are on the way. |

Start Year | 2015 |

Description | Self-Organized Hydrodynamics with nonconstant velocity |

Organisation | University of Aberdeen |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Motivated by recent experimental and computational results that show a motility-induced clustering transition in self-propelled particle systems, we study an individual model and its corresponding Self-Organized Hydrodynamic model for collective behaviour that incorporates a density-dependent velocity, as well as inter-particle alignment. The modal analysis of the hydrodynamic model elucidates the relationship between the stability of the equilibria and the changing velocity, and the formation of clusters. We find, in agreement with earlier results for non-aligning particles, that the key criterion for stability is. a nondecreasing mass flux with respect to the density. Numerical simulation for both the individual and hydrodynamic models with a velocity function inspired by experiment demonstrates the validity of the theoretical results. |

Collaborator Contribution | All partners contributed equally to this research. |

Impact | A paper has appeared: P. Degond, S. Henkes, H. Yu, Self-Organized Hydrodynamics with nonconstant velocity, Kinetic and Related Models, 10 (2017), pp. 197-213 (open access). Others are on the way. |

Start Year | 2015 |

Description | Simple mechanical cues could explain adipose tissue morphology |

Organisation | Paul Sabatier University (University of Toulouse III) |

Department | Biology and Life Sciences |

Country | France |

Sector | Academic/University |

PI Contribution | We have designed and implemented an Individual Based Model describing the interactions between the adipocytes and the collagen fibers |

Collaborator Contribution | Our partners have performed experiments (Stromalab) or assisted us in the development of the numerical model (Institute of Mathematics). |

Impact | A research paper has appeared: P. Degond, F. Delebecque, D. Peurichard, Continuum model for linked fibers with alignment interactions, Mathematical Models and Methods in Applied Sciences, 26 (2016) pp. 269-318 (open access). Another one is under review |

Start Year | 2012 |

Description | Simple mechanical cues could explain adipose tissue morphology |

Organisation | Paul Sabatier University (University of Toulouse III) |

Department | Institute of Mathematics |

Country | France |

Sector | Academic/University |

PI Contribution | We have designed and implemented an Individual Based Model describing the interactions between the adipocytes and the collagen fibers |

Collaborator Contribution | Our partners have performed experiments (Stromalab) or assisted us in the development of the numerical model (Institute of Mathematics). |

Impact | A research paper has appeared: P. Degond, F. Delebecque, D. Peurichard, Continuum model for linked fibers with alignment interactions, Mathematical Models and Methods in Applied Sciences, 26 (2016) pp. 269-318 (open access). Another one is under review |

Start Year | 2012 |

Description | Social dynamics |

Organisation | Duke University |

Department | Department of Mathematics |

Country | United States |

Sector | Academic/University |

PI Contribution | We investigate opinion formation models inspired by a work by Borghesi, Bouchaud and Jensen. We derive a Fokker-Planck-type equation and study conditions under which it has non-trivial equilibria. We derive the macroscopic limit for the evolution of the mean opinion. Finally, we compare different types of interactions leading to different consensus speeds. |

Collaborator Contribution | All partners contributed equally to this research |

Impact | A paper has appeared: P. Degond, J-G. Liu, S. Merino-Aceituno, T. Tardiveau, Continuum dynamics of the intention field under weakly cohesive social interactions. Mathematical Models and Methods in Applied Sciences 27 (2017), pp. 159-182 (open access). Other works are in preparation. |

Start Year | 2016 |

Description | Symmetry-breaking phase-transitions in highly concentrated semen |

Organisation | French National Institute of Agricultural Research |

Country | France |

Sector | Academic/University |

PI Contribution | The project is about producing an automated assessment process of semen fertility, based on the massal motility as a fertility indicator. Depositing fresh semen sample in an annular shaped microfluidic chip leads to a spontaneous vortex state of the fluid at sufficiently large sperm concentration. The rotation occurs unpredictably clockwise or counterclockwise and is robust and stable. The rotation speed gives an objective measurement of mass motility. Our team proposed the design of the experiment, conceived the model and did the numerical simulations. |

Collaborator Contribution | They realized the experiments and the measurements. |

Impact | A paper has appeared: A. Creppy, F. Plouraboué, O. Praud, X. Druart, S. Cazin, H. Yu, P. Degond, Symmetry-breaking phase-transitions in highly concentrated semen, Journal of the Royal Society Interface, 13 (2016), 20160575. A European patent has been filed. This collaboration is multidisciplinary: it involves mathematicians (our team), physicists (the team from IMFT) and biologists (the team from INRA). |

Start Year | 2011 |

Description | Symmetry-breaking phase-transitions in highly concentrated semen |

Organisation | National Center for Scientific Research (Centre National de la Recherche Scientifique CNRS) |

Department | UMR 5502 Institute of Fluid Mechanics of Toulouse |

Country | France |

Sector | Public |

PI Contribution | The project is about producing an automated assessment process of semen fertility, based on the massal motility as a fertility indicator. Depositing fresh semen sample in an annular shaped microfluidic chip leads to a spontaneous vortex state of the fluid at sufficiently large sperm concentration. The rotation occurs unpredictably clockwise or counterclockwise and is robust and stable. The rotation speed gives an objective measurement of mass motility. Our team proposed the design of the experiment, conceived the model and did the numerical simulations. |

Collaborator Contribution | They realized the experiments and the measurements. |

Impact | A paper has appeared: A. Creppy, F. Plouraboué, O. Praud, X. Druart, S. Cazin, H. Yu, P. Degond, Symmetry-breaking phase-transitions in highly concentrated semen, Journal of the Royal Society Interface, 13 (2016), 20160575. A European patent has been filed. This collaboration is multidisciplinary: it involves mathematicians (our team), physicists (the team from IMFT) and biologists (the team from INRA). |

Start Year | 2011 |

Description | Three Dimensional Vlasov-Poisson System for Large Magnetic Field: |

Organisation | Paul Sabatier University (University of Toulouse III) |

Department | Research Centre on Animal Cognition |

Country | France |

Sector | Academic/University |

PI Contribution | We establish the asymptotic limit of the three dimensional Vlasov-Poisson equation with strong external magnetic field. The guiding center approximation is investigated in the three dimensional case with a non-constant magnetic field. In the long time asymptotic limit, the motion can be split in two parts: one stationary flow along the lines of the magnetic field and the guiding center motion in the orthogonal plane of the magnetic field where classical drift velocities and invariants (magnetic moment) are recovered. |

Collaborator Contribution | All partners have contributed equally to this research |

Impact | A paper has appeared: P. Degond, F. Filbet, On the asymptotic limit of the three dimensional Vlasov-Poisson system for large magnetic field : formal derivation, Journal of Statistical Physics, 165 (2016), pp. 765-784 (open access). Another one is on the way. |

Start Year | 2015 |

Description | Topological interactions in a Boltzmann-type framework |

Organisation | University of Toulouse |

Country | France |

Sector | Academic/University |

PI Contribution | We consider a model for particles interacting through topological interactions. This means that the interaction probability of a particle with another one is a function of the proximity rank of the latter with respect to the former, and not of the metric distance between them. Topological interactions have been shown to rule many natural phenomena, such as the interactions between birds in a flock. We study the limit of a system size going to infinity and show that the limit model is ruled by an equation of the Boltzmann type. This is the first time that a Boltzmann type equation is obtained for topological interactions |

Collaborator Contribution | All partners have contributed equally at all stages of this partnership. |

Impact | A publication has appeared: A. Blanchet, P. Degond, Topological interactions in a Boltzmann-type framework, Journal of Statistical Physics, 163 (2016), pp. 41-60 (open access). Another publication will soon be finalized |

Start Year | 2015 |

Description | Transport of congestion in two-phase compressible/incompressible flows |

Organisation | Heidelberg University |

Department | Mathematical Institute |

Country | Germany |

Sector | Academic/University |

PI Contribution | We study a two-phase model of crowd motion model which encompasses the flow in the uncongested regime (compressible) and the congested one (incompressible) with the free boundary separating the two phases. The congested density is prescribed initially and transported along with the flow. We prove that this system can be approximated by the fully compressible Navier-Stokes system with a singular pressure, supplemented with transport equation for the congestion density. We also investigate the application of this approximation for the purposes of numerical simulations. |

Collaborator Contribution | All partners contributed equally to this research |

Impact | A paper has been submitted: P. Degond, P. Minakowski, E. Zatorska, Transport of congestion in two-phase compressible/incompressible flows. Submitted. Manuscript on arXiv. A second one is in the completion phase |

Start Year | 2015 |

Description | Transport of congestion in two-phase compressible/incompressible flows |

Organisation | University of Strasbourg |

Department | Department of Mathematics and Computer Science |

Country | France |

Sector | Academic/University |

PI Contribution | We study a two-phase model of crowd motion model which encompasses the flow in the uncongested regime (compressible) and the congested one (incompressible) with the free boundary separating the two phases. The congested density is prescribed initially and transported along with the flow. We prove that this system can be approximated by the fully compressible Navier-Stokes system with a singular pressure, supplemented with transport equation for the congestion density. We also investigate the application of this approximation for the purposes of numerical simulations. |

Collaborator Contribution | All partners contributed equally to this research |

Impact | A paper has been submitted: P. Degond, P. Minakowski, E. Zatorska, Transport of congestion in two-phase compressible/incompressible flows. Submitted. Manuscript on arXiv. A second one is in the completion phase |

Start Year | 2015 |

Description | continuum model for nematic alignment of self-propelled particles |

Organisation | University of Vienna |

Country | Austria |

Sector | Academic/University |

PI Contribution | A continuum model for a population of self-propelled particles interacting through nematic alignment is derived from an individual-based model. The methodology consists of introducing a hydrodynamic scaling of the corresponding mean-field kinetic equation. The resulting perturbation problem is solved thanks to the concept of generalized collision invariants. It yields a hyperbolic but non-conservative system of equations for the nematic mean direction of the flow and the densities of particles flowing parallel or anti-parallel to this mean direction. Diffusive terms are introduced under a weakly non-local interaction assumption and the diffusion coefficient is proven to be positive. An application to the modeling of myxobacteria is outlined. |

Collaborator Contribution | All partners have contributed equally at all stages of this research. |

Impact | A paper has appeared: P. Degond, A. Manhart, H. Yu, A continuum model for nematic alignment of self-propelled particles, Discrete and Continuum Dynamical Systems Series B, 22 (2017), pp. 1295-1327 (open access). Another one has been submitted: P. Degond, A. Manhart, H. Yu, An age-structured continuum model for myxobacteria, submitted. Manuscript on arXiv. |

Start Year | 2015 |

Description | (In preparation) Oberwolfach snapshot (article for broad audience) |

Form Of Engagement Activity | A press release, press conference or response to a media enquiry/interview |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Other audiences |

Results and Impact | Article in the "Oberwolfach snapshot" explaining for a broad audience the applications of kinetic theory to biology. This article is in preparation and it started at a workshop in Oberwolfach in December 2017. |

Year(s) Of Engagement Activity | 2018 |

Description | Article in 'Imperial College News': Spinning semen provides a measurement of fertility by Hayley Dunning 14 October 2016 |

Form Of Engagement Activity | A press release, press conference or response to a media enquiry/interview |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Media (as a channel to the public) |

Results and Impact | Posting a media articfle about a significant result of my research |

Year(s) Of Engagement Activity | 2016 |

URL | http://www3.imperial.ac.uk/newsandeventspggrp/imperialcollege/newssummary/news_14-10-2016-14-36-56 |

Description | Inaugural lecture at IMperial |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Public/other audiences |

Results and Impact | I gave an inaugural lecture at imperial college. This is a public lecture with broad audience. High school students were present. |

Year(s) Of Engagement Activity | 2016 |

URL | https://www.youtube.com/watch?v=F9YEDteHzWo |

Description | Public lecture at ICMS |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Regional |

Primary Audience | Public/other audiences |

Results and Impact | Public lecture at the occasion of the workshop 'Collective dynamics and self-organization in biological sciences', International Cente for Mathematical Sciences (ICMS), Edinburgh, May 2018. |

Year(s) Of Engagement Activity | 2018 |

Description | Public lecture at UFF, Brasil |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Undergraduate students |

Results and Impact | Public lecture for mathematics undergraduate students at Instituto de Matema´tica e Estati´stica (IME) da Universidade Federal Fluminense, Campus do Gragoata´, Nitero´i, Brasil, August 2018. |

Year(s) Of Engagement Activity | 2018 |

Description | Public lecture in High School 'North London Collegiate School (NLCS)', in Edgware |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Schools |

Results and Impact | I gave a short presentation (45 minutes including questions) to high school students about my research. The talk was based on my inaugural lecture but shortened. After the talk, I had lunch with the members of the math club in the school and could answer their questions. I gave them advice for literature search in relation to their projects. |

Year(s) Of Engagement Activity | 2017 |