## Schedule

Titles, abstracts and available material are below and can be accessed by clicking on the corresponding name in the schedule.### First week

Tuesday 11 | Wednesday 12 | Thursday 13 | Friday 14 | |

9:00 - 10:30 | Opening & registration | Roxin | Roxin | Löcherbach |

10:30 - 10:45 | Coffee break | Coffee break | Coffee break | |

10:45 - 12:15 | Rodriguez-Bunn | Rodriguez-Bunn | Rodriguez-Bunn | Roxin |

12:15 - 13:45 | Lunch break | Lunch break | Lunch break | Lunch break |

13:45 - 14:30 | Garzón | Martínez | Garnier | Vauchelet |

14:30 - 15:15 | Perasso | Gabriel | Abreu | D'Orsogna |

15:15 - 15:30 | Coffee break | Coffee break | Coffee break | Coffee break |

15:30 - 17:00 | Training sessions | Training sessions | Training sessions | Training sessions |

### Second week

Monday 17 | Tuesday 18 | Wednesday 19 | Thursday 20 | Friday 21 | |

9:00 - 10:30 | Löcherbach | Mouhot | Löcherbach | Mouhot | Mouhot |

10:30 - 10:45 | Coffee break | Coffee break | Coffee break | Coffee break | Coffee break |

10:45 - 11:30 | Schmeiser | Schmeiser | Bardos | Schmeiser | Sánchez |

11:30 - 12:15 | Castillo | ||||

12:15 - 13:45 | Lunch break | Lunch break | Lunch break | Lunch break | Lunch break |

13:45 - 14:30 | Coville | Kavian | Free | Figueroa | |

14:30 - 15:15 | Throm | Chevallier | Egaña | ||

15:15 - 15:30 | Coffee break | Coffee break | Coffee break | Coffee break | |

15:30 - 17:00 | Training sessions | Training sessions | Training sessions | Training sessions |

## Courses

Eva Löcherbach

University of Cergy-Pontoise, France
Modelling interacting networks as processes with variable length

Clément Mouhot

University of Cambridge, UK
The many-particle limit and its mathematical analysis

Nancy Rodriguez-Bunn

The University of North Carolina at Chapel Hill, US
Traveling Fronts in Homogeneous and Heterogeneous Environments

Alexander Roxin

Centre de Recerca Matemàtica, Spain
Mean-field Theory for Networks of Spiking Neurons

Christian Schmeiser

University of Vienna, Austria
Mathematical modelling of actin driven cellular morphodynamics and motility

## Talks

**Ricardo Abreu** (Universidad de Holguín, Cuba)
*Clifford Analysis and its applications in Linear
Elasticity Theory*

Solutions of the sandwich equation $\partial x/ f \partial x = 0$, where $\partial x$ stands for the first-order differential operator (called Dirac operator) in the Euclidean space Rm, are known as inframonogenic functions. These functions generalize in a natural way the theory of kernels associated with $\partial x$, the nowadays well-known monogenic functions, and can be viewed also as a refinement of the biharmonic ones. In this talk we study the connections between inframonogenic functions and the solutions of the homogeneous Lamé-Navier system in $\R^3$. Our findings allow to shed some new light on the structure of the solutions of this fundamental system in 3-dimensional elasticity theory.

**Claude Bardos** (Université Paris 7, France)
*From the d’Alembert paradox to the 1984 Kato criteria via the 1941 1/3 Kolmogorov law and the 1949 Onsager conjecture*

Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation. Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition. Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is equivalent to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d’Alembert Paradox.

**Jorge Castillo Medina** (Universidad Autónoma de Guerrero, México)
*Turing-Hopf patterns on dynamic surfaces*

We will deal with the interaction between two bifurcations Turing and Hopf on surfaces that have isotropic growth in order to simulate a situation - phenomenon in real life.

**Julien Chevallier** (Université de Grenoble, France)
*Hawkes processes as a spike train model*

A spike train is the set of firing times of a given neuron. A simple class of stochastic model we can use to describe spike trains is the class of temporal point processes, that are random sets of times. Hawkes processes have been extensively studied with application to neurosciences in mind due to their relevant, yet rather simple, way to mimic synaptic integration. A review on this model is proposed with emphasis on its several mean-field limits, showing how Hawkes processes can be viewed as the individual-based/microscopic stochastic model associated with several macroscopic deterministic models: age structured model, neural fields and monotone cyclic feedback systems. Moreover, the qualitative behaviour of these three models will be discussed

**Jerôme Coville** (INRA Avignon, France)
*Nonlocal propagation in domain with obstacle*

I will present some recent results on the propagation phenomena in domain with obstacles where the dispersal process is governed by a compound Poisson process. I will briefly present the construction of generalised transition wave as well as the description of the resulting stationnary solutions.

**Maria Rita D'Orsogna** (California State University, US)
*Dynamics of neuroendocrine stress response*

The hypothalamic-pituitary-adrenal (HPA) axis is a neuroendocrine system that regulates numerous physiological processes. Disruptions are correlated with stress-related diseases such as PTSD and major depression. We characterize "normal" and "diseased" states of the HPA axis as basins of attraction of a dynamical system describing the main hormones regulated by the HPA axis. Our model includes hormonal self-upregulation, release, synthesis; interaction, delay and feedback mechanisms. External input is associated to psychological trauma, while parameter changes represent physiological damage. We show that transitions between the "normal" and "diseased" states may be induced solely by external input, with all physiological parameters unchanged, emphasizing the severe consequences of psychological trauma. We also find that the timing and duration of the traumatic event is an important determinant of if and how stress disorders will manifest. Finally, we propose mechanisms whereby exposure therapy may act to normalize downstream dysregulation of the HPA axis.

**Giani Egaña** (Universidad de la Habana, Cuba)
*Turing-Hopf instabilities in Keller-Segel model with glycolytic reaction*

The present work classifies in the study of Keller-Segel drift-diffusion PDE system modelling the driving answer of cell populations to chemical signs, so-called chemotaxis, coupled with the glycolysis model, where glycolysis is the metabolic process through which the cell obtains energy from the glucose. We work on the two-dimensional space. We investigate the presence of Turing instabilities at the Hopf bifurcations and spatiotemporal patterns formation in the glycolysis reaction-diffusion system coupled with the Keller-Segel equation. Here we obtain positive results that depend on the values of the involved parameters.

We analyse the presence of generic Hopf bifurcations and with it the appearance of limit cycle solutions in the system of ordinary equations of glycolysis, in terms of the positive parameters. Also, we obtain an asymptotic expansion of the limit cycle. Using the R. Ricard-Mischler theorem, we analyse too the appearance of strong Turing-Hopf instabilities for the glycolysis system with diffusion in terms of the parameters, the emergence of twinkling patterns product of these diffusive instability of the limit cycle and the influence of such instabilities in the generalized parabolic-parabolic chemotaxis system in the domain.

**Susely Figueroa** (Université de Toulouse, France)
*Evolutionary dynamics for phenotypically structured populations in fluctuating environments*

I will present a study about the long time behavior of a Lotka-Volterra parabolic equation by considering a time-periodic reaction term and a non-local competition. Such an equation describes the dynamics of a phenotypically structured population under the effect of mutations and selection in a fluctuating environment. We prove that the solution of this equation converges in long time to the only periodic solution of the problem that I will then describe asymptotically when the effect of the mutations vanishes. Using a theory based on Hamilton-Jacobi equations with constraint, we prove that the solution concentrates on a single Dirac mass, while the size of the population varies periodically in time. When the effect of mutations is small but non zero, these results can be compared to biological experiments, taking the growth rate in several ways. (joint work with Sepideh MIRRAHIMI)

**Pierre Gabriel** (Université de Versailles Saint-Quentin, France)
*Coupling methods for PDEs*

First we will recall the standard Doeblin contraction argument and its generalization using Lyapunov functions, which allow to obtain the exponential convergence to a unique invariant measure for conservative positive semigroups. Then we will present recent extensions of the method to the non-conservative cases, that often appear in population dynamics. Each case will be illustrated by an example of application in neurosciences or cell division.

**Jimmy Garnier** (CNRS - Chambéry, France)
*Genetic diversity in age-structured populations*

In many population, the behavior of individuals might differ according to their age. This structure of the population have profound influence on the dynamics of the population and in particular, on its genetic diversity. I will first investigate the inside dynamics of expanding solutions to integro--difference equations with stage--structure. I will show that the presence of the structure in the population might modify the nature of the expanding solutions at its leading edge. In particular, expanding solutions with age--structure are pushed in a local sense while in absence of age--structure those solutions are pulled. This work is in collaboration with N Marculis, R Lui and M A Lewis. In a second part, I will investigate the dynamics of the genetic diversity in metapopulations. I show that the age--structure might have profound influence on the local diversity of sub--population composing the metapopulation. This work is in collaboration with P Lafontaine.

**Diego Garzón Alvarado** (Universidad Nacional de Colombia, Colombia)

**Otared Kavian** (Université de Versailles Saint-Quentin, France)
*Remarks on the blowing up of positive solutions to some semilinear diffusion equations*

Let $\Omega \subset {\Bbb R}^N$ be a domain with $N \geq 1$ an integer and $p > 1$ a real number. We consider an initial data $u_{0} \in L^1(\Omega)$ such that $u_{0}\not\equiv 0$ and $u_{0} \geq 0$, and the solution of the semilinear diffusion equation
\begin{equation}\label{eq:Diffusion}
\partial_{t} u = Au + |u|^{p-1}u, \qquad u(0,x) = u_{0}(x) \geq 0.
\end{equation}
In the above equation the linear operator $A$ is a diffusion operator of local or non local type, for instance
$$Au = \Delta u, \quad\mbox{or} \quad Au = K*u - u,$$
with a convolution kernel $K \geq 0$.
In this talk we present a few results showing that, depending on the diffusion operator $A$, and on the nature of the domain $\Omega$, it may happen that, no matter how *small* the nonnegtaive initial data $u_{0}$ is, the solution $u(t,\cdot)$ blows in finite time when $1 < p \leq p_{*}$, for an exponent $p_{*}$ which depends on the domain $\Omega$ and the operator $A$.

**Antoine Perasso** (Université de Besançon, France)
*Identifiability and stability analysis for infection-load structured epidemiological models*

The goal of this talk is to present some results connected to a class of PDE epidemiological models whose infected population is structured according to an exponentially increasing infection load with respect to time. We consider in particular the question of parameter identifiability, an inverse problem that consists in assessing if the parameter to output (observation) map of the system is into, and we study the asymptotic behaviour of solutions through a global analysis of the stability and the attractivity of equilibria.

**Faustino Sánchez Garduño** (Universidad Nacional Autónoma de México, México)
*PDE and pattern formation*

The emergence of patterns at different levels of organization (in living and nonliving systems) is ubiquitous in Nature. The explanation of the mechanisms from which those emerge, had been approached by using different tools. In this talk we preesent and overview of those by using a partial differential equations approach. We begin with a debate based on two schools of though: historial account vs scientific explanation. We continue with the description of a few fundamental processes in morphogenesis. Namely: reaction-diffusion and chemotaxis including their mathematical formulation and some examples as well. We conclude the talk by listing some mathematical open problems in this area.

**Sebastian Throm** (Universidad de Granada, Spain)
*Long-time behaviour for Smoluchowski's coagulation equation*

Smoluchowski's coagulation equation is a mean-field model to describe systems of aggregating particles. This phenomenon is widespread in natural and industrial processes and can be observed on many different length scales (e.g. raindrops, algal growth, formation of planets). An important question for Smoluchowski's equation is how solutions behave for large times. The scaling hypothesis conjectures that the model exhibits a universal long-time behaviour which is described by self-similar profiles. In this talk, we will given an overview of the coagulation model with a particular emphasis on this problem and several recent developments.

**Nicolas Vauchelet** (Université Paris 8, France)
*Mathematical modeling of the spread of Wolbachia for dengue control.*

Due to the numerous diseases that they transmit, mosquitoes is considered as the most dangerous animal species for human. In order to control the transmission of viruses some techniques consist in acting on the mosquitoes population. Among them a technique consists in using the bacteria Wolbachia. Indeed, bacteria Wolbachia has gain a lot of attention since scientists discover that infected mosquitoes with this bacteria cease to transmit some disease like dengue, chikungunya and Zika. Moreover, this bacteria is maternally transmitted from mother to offsprings. Then a strategy of control of dengue transmission consists in releasing Wolbachia infected mosquitoes in the aim to replace the wild population of mosquitoes by Wolbachia-infected mosquitoes. Obviously the safe and efficient use of such strategy relies on careful mathematical studies. In this presentation, we will focus on the spatial spread of Wolbachia infected mosquitoes into a host population. More precisely, we will address the following questions: How the spatial repartition of the releases will influence the spread of the bacteria into the population ? Once the spread is initiated, is it possible that environmental characteristics stop the spread ? How to optimize the success of this strategy ? In order to answer to these questions, some models of partial differential equations of parabolic types will be introduced and studied with relevant mathematical tools.