Vincent Calvez “ Kinetic models for wave propagation in Biology”

1- Kinetic description of bacterial population; chemotactic waves.
2- Kinetic reaction-transport fronts; large scale limit of the traveling waves.
3- Reaction-diffusion fronts with heterogeneous motility, large scale limit of the traveling waves; selection of the highest motilities.

I will present recent advances in modeling propagation phenomena in biology using a kinetic framework.
I will describe a couple of kinetic models, respectively for the propagation of bacteria in micro-channels and for the invasion of cane toads in Northern Australia. I will explain the modeling issues, the mathematical analysis, and the biological outcomes. The first model is concerned with self-organization of cells driven by chemotaxis. The second model is concerned with reaction-diffusion traveling waves with mutations, where the selection acts on the diffusion coefficient.



Jose Antonio Carrillo “PDE models in computational neuroscience”

I will introduce different questions in computational neuroscience where PDE models appear naturally. Different interesting questions arise such as: derivation of models from microscopic descriptions, qualitative behavior of solutions, numerical approximations, periodic and/or stationary solutions,… We will discuss the dimensional reduction of Fokker-Planck dynamics for rate models along their stable manifolds. This allows the understanding of super- and sub-critical bifurcations in terms of performance and reaction times.

1. Rate models, Fokker-Planck equations, and their reduction.
2. Integrate & Fire Models: Qualitative Properties & Steady States.
3. Integrate & Fire Models with conductance: Qualitative Properties.


Odo Diekmann “Delay Equations and Physiologically Structured Population Models”

Notes 1, Figures 1, Notes 2, Notes 3, Notes 4, Notes 5, Notes 6, Notes 7

A delay equation is a rule for extending a function of time towards the future,
on the basis of the known past. Renewal Equations prescribe the current value,
while Delay Differential Equations prescribe the derivative of the current value.
With a delay equation one can associate a dynamical system by translation
along the extended function.
To begin with, I will illustrate by way of examples how such equations arise
in the description of the dynamics of structured populations and the epidemiology
of infectious diseases. (Along the way I shall also describe the relationship
with first order PDE with nonlocal boundary conditions). Next I will briefly
sketch the functional analytic theory, which is based on perturbation theory
for adjoint semigroups of bounded linear operators (so-called sun-star calculus).
Finally, I will make a plea for the development of numerical bifurcation tools.


Philip Maini  “Mathematical modelling in development and cancer”

There is a long history of mathematical modelling in developmental biology
and, in recent years, this has been extended to cancer modelling. There are many
levels at which such models can be developed and many different methodologies
used. Our work focuses on developing models consisting of coupled systems of
partial differential equations and hybrid cellular automaton to address issues of
spatial pattern formation and cell migration. Applications are to, in development,
limb development, animal coat markings and somite formations, and in
cancer, to the acid-mediated invasion hypothesis, somatic evolution, and colorectal
cancer formation.


Stéphane Mischler “Mathematical basis for evolution PDEs”

Chapter 1, Chapter 2, Chapter 3


Lesson 1 - Parabolic equation.
Existence of solutions for parabolic equations by the mean of the
variational approach and the existence Theorem of J.-L. Lions.
A remark on the uniqueness of solutions and the semigroup theory.

Lesson 2 - Transport equation.
Existence of solutions by the mean of the characterics method and
renormalization theory of DiPerna-Lions.
A remark on the transport theory and solutions built by duality.
 
Lesson 3 & 4 - Uniqueness and complements.
Uniqueness of solutions thanks to Gronwall's argument and duality argument.
Duhamel formula and existence of solutions for equations with (possibly nonlinear) source term.
Positivity, conservation laws and Lyapunov functionals.
A priori estimates (moment and Lebesgue norm) and compactness method.
Tutorial on the well-posedness problem for (simple) biological partial differential evolution
equations (if possible, in relation with the models studied in the other courses).

Lesson 5 - More about the heat equation.
Smoothing effect thanks to (robust) Nash's argument.
Rescaled (self-similar) variables and Fokker-Planck equation.
Poincaré inequality and long time asymptotic (with rate).

For the second week I thought about two possibilities:

Lesson 6 - Entropy of Boltzmann.
Fisher information, log Sobolev inequality and long time
convergence to the equilibrium (with rate).

Lesson 7 - Spectral analysis for semigroups.
Resolvants and spectral analysis of generators.
Spectral mapping theorem.
Rate of convergence for solutions to the Fokker-Planck equation
in a weighted Lebesgue space thanks to the extension of the
spectral gap approach.

or

Lesson 6 bis - Remarkable solutions problem.
Existence of stationary solutions to the (possibly rescaled)
evolution problem: steady states, self-similar solutions
and first eigenfunctions.
 
Lesson 7 bis. Entropy of Boltzmann.
General relative entropy methods and long time asymptotic to
a suitable remarkable solution.