Program of the
Workshop on Free Boundaries and PDEs in Biology
March 21/22, 2006, Wolfgang Pauli Institute Vienna
Organizers: Danielle Hilhorst, Christian Schmeiser, Henrik Shagholian
9:30 Benoit Perthame: Selection, mutation, adaptive dynamics: an asymptotic point of view
Abstract: Living systems are subject to constant evolution.
Their environment can be considered as a nutrient shared by all the
population. This alllows certain individuals, characterized by a 'trait',
to grow faster because they are better adapted to the environment.
This leads to select the 'best adapted trait' in the population (singular
point of the system). On the other hand, the new-born population undergoes
small variance on the trait under the effect of genetic mutations.
In these circumstances, it possible then to describe the dynamical
evolution of the current trait?
We will give a mathematical model of such dynamics and show that an
asymptotic method allows us to describe the evolution of the 'best adapted
trait' and eventually to compute bifurcations which lead to the
cohabitation of two different populations.
In the regular regime, we obtain a canonical equation where the drift is
given by a nonlinear problem.
The asymptotic method leads to evaluate the weight and position of a moving
Dirac mass desribing the population. We will show that a Hamilton-Jacobi
equation with constraints naturally describes this asymptotic.
Some more theoretical questions as uniqueness for the limiting H.-J.
equation will also be addressed.
10:15 Danielle Hilhorst: Fast reaction limit and long time behavior for a
Abstract: We consider a two-component competition-diffusion
system in the case of equal diffusion coefficients
and inhomogeneous Dirichlet boundary conditions. As the reaction coefficient tends to infinity, the
solution converges to that of a Stefan problem with zero latent heat which possesses a Lyapunov functional. However this is not the case for the original competition-diffusion system. Our main result is the following : suppose that all the stationary solutions are nondegenerate. Then if the reaction coefficient is large enough the solutions of the competition-diffusion system converge to a stationary solution.
This is joint work with E.C.M. Crooks and E.N. Dancer.
11:00 Coffee break
11:30 Andrea Malchodi: Concentration phenomena for singularly perturbed elliptic PDEs
Abstract: We consider a class of singularly perturbed nonlinear elliptic (scalar) equations,
which arise in the study of some biological systems, including the Gierer-Meinhardt one, when the
diffusion coefficients are very different from each-other. In particular, when one diffusivity is
very small, solutions are sharply concentrated near some subsets of the domain. We analyze solutions
concentrating at non-trivial sets, proving existence of layers with arbitrary dimension.
14:15 Gianmaria Verzini: Stationary states and asymptotic analysis
for reaction diffusion systems with strong competition
Abstract: see below
15:00 Susanna Terracini: A variational problem for the spatial segregation
of reaction diffusion systems
Abstract: In the first part (G. Verzini) of the talk we consider the segregation
connection with the asymptotic analysis of solutions of competition-diffusion systems.
The asymptotic estimates rely on suitable Liouville--type results based upon variants
and extensions of the monotonicity formula by Alt-Caffarelli-Friedman. Next we present
a class of optimal partition problems associated with the the same segregation problem
in competition-diffusion systems. Beside establishing the existence and uniqueness
of the optimal states, we develop a theory for the extremality conditions and the
regularity of minimizers and the interfaces.
15:45 Coffee break
16:15 Peter Markowich: Kinetic models for chemotaxis
9:30 Masayasu Mimura: Spatial segregation in competition-diffusion systems
Abstract: Understanding of spatial and/or temporal behaviors of ecologically
interacting species is a central problem in population ecology. As for competitive interaction
of ecological species, problems of coexistence or exclusion have been theoretically
investigated by using different types of mathematical models. Especially, variety types
of reaction-diffusion equations (and/or cross-diffusion equations) have been proposed
to study spatial segregation of competing species. Recently, in order to understand the
evolutional behavior of spatially segregating regions of competing species, the singular
limit analysis have been successfully developed. These enable us to derive new equations
to understand spatially segregating regions of competing species. In this lecture, I would
like to focus on qualitative behavior of spatially segregating solutions of
competition–diffusion systems as well as competition-cross-diffusion systems, by using
singular limit analysis.
10:15 Christian Schmeiser: Asymptotic analyis of advection-dominated chemotaxis
models with density control
Abstract: A variant of the classical Keller-Segel model for cell aggregation by
chemotaxis is studied. Its peculiarities are a finite volume effect limiting the cell
density and the assumption of small diffusivity (compared to convection by chemotaxis).
The dynamics involves distinct time scales with fast evolution governed by a nonlinear
hyperbolic conservation law with nonlocal flux. The slow evolution describes the movement
of cell aggregates attracting each other or being attracted by domain boundaries.
A mixture of formal and rigorous asymptotic results as well as numerical computations
will be presented.
This is joint work with M. Burger and Y. Dolak-Struss.
11:00 Coffee break
11:30 Piotr Rybka (joint work with Yoshikazu Giga): Formation of singularities in
the crystalline curvature flow
Abstract: We are interested in the weighted curvature flow with a driving term.
The structure of the driving term is suggested by the physics of the
mono-crystal growth from vapor. We show existence and uniqueness of
solution for initial data which are a perturbation of the Wulff shape,
i.e. the equilibrium configuration. We show what kind of shape emerges
throughout the evolution.