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

March 21:
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 competition-diffusion system
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.

12:15 Lunch

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 problem in 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

March 22:
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.