13:30 -- 14:20  Izumi Takagi (Tohoku University)

Abstract: The diffusion driven instability, DDI for short, by Alan Turing says that when two chemicals with different diffusion rates react, the spatially homogeneous state may be destabilized, and as a result a spatially nontrivial state, a pattern, emerges. In a modern mathematical language this is interpreted as bifurcation from a constant steady-state solution. In this talk we ask whether DDI is necessary or not for pattern formation. An example is given in which nonconstant stable stationary solutions do exist even if DDI does not occur. Therefore, it is much more important to pay attention to the global structure of nonlinearity when we model pattern formation.

14:30 -- 15:20  Norihisa Ikoma(Tohoku University)

Abstract:The eigenvalue problems play an important role in many problems, for instance, in the study of bifurcation and stability problem. This talk is concerned with the eigenvalue problem for the second order operators and we consider both of linear and nonlinear operators under various boundary conditions.
The aim of this talk is to show the existence of eigenvalues and to characterize them (not only the principal eigenvalues but also the higher eigenvalues) by using the existence of sub- and super-solutions and the solvability of the equation with inhomogeneous term. This is joint work with Professor Hitoshi Ishii (Waseda University).

coffee break

15:50 -- 16:40   Keiji Miura (Tohoku University) and Kazuki Nakada (University of Electro-communications)

Abstract: A system of weakly interacting oscillators can generally be reduced to phase oscillators, where a single phase variable for each oscillator suffices for predicting synchronization. Unfortunately, some useful engineering applications of coupled oscillators such as robot walking or analogue digital converters are outside this classical framework (Kuramoto,1984). Here we first review the classical phase reduction theory briefly and then discuss possible extensions to excitable units or strong interactions.

10:00 -- 10:50  Yongtao Zhang (University of Notre Dame)

Abstract: In this talk I shall present two computational studies on developmental patterning.
  The first one is on early dorsal-ventral (DV) patterning of the zebrafish embryo. We developed a computational model in three dimensions of the zebrafish embryo and use it to study molecular interactions in the formation of bone morphogenetic proteins (BMP) morphogen gradients in early DV patterning. Simulation results are presented on the dynamics BMP gradient formation, the cooperative action of two feedback loops from BMP signaling to BMP and its regulator Chordin synthesis, and pattern sensitivity with respect to BMP and Chordin dosage. Computational analysis shows that synergy of the two feedback loops in the zygotic control of BMP and Chordin expression, along with early initiation of localized Chordin expression, is critical for establishment and maintenance of a stable and appropriate BMP gradient in the zebrafish embryo.
  The second problem is on vertebrate limb developmental patterning. Especially we studied how the numbers of skeletal elements along the proximodistal (P-D) and anteroposterior (A-P) axes are determined and how the shape of a growing limb affects skeletal element formation. We have simulated the behavior of the core chondrogenic mechanism of the developing limb in the presence of a fibroblast growth factor (FGF) gradient using a novel computational environment that permits simulation of LALI systems in domains of varying shape and size. The model predicts the normal proximodistal pattern of skeletogenesis.
  A novel numerical method, called Krylov implicit integration factor method, is developed for efficiently solving the reaction-diffusion / advection-reaction-diffusion equations on high dimensional complex domains, which arise from computational modeling of these developmental patterning problems.

coffee break

11:10 -- 12:00   Anna Marciniak-Czochra (Heidelberg University)

Abstract: Recent experimental observations show that although leukemias exhibit clonal heterogeneity, only few clones are detected at diagnosis and at relapses. The pattern of clone selection may be different. To address these questions we propose a mathematical model of dynamics of multiple leukemic clones coupled to dynamics of healthy hematopoiesis. Each cell line goes through several differentiation stages and is characterized by parameters describing proliferation rates and self-renewal potential. Considering a continuum of leukemic clones leads to a structured population model. The model takes a form of a system of integro-differential equations with a nonlinear and nonlocal coupling, which describes regulatory feedback loops in cell proliferation and differentiation process. We show that such coupling leads to mass concentration in points corresponding to maximum of the self-renewal potential. Using a Lyapunov function constructed for a finite dimensional counterpart of the model, we prove that the total mass of the solution converges to a globally stable equilibrium. Mathematical analysis suggests which mechanisms of clonal selection predict clonality observed in the course of disease.

lunch break

13:30 -- 14:20  Sohei Tasaki (Tohoku University)

Abstract: Bacteria form a great variety of colonies. The spatio-temporal population structures have a major impact on many scientific fields, and so it becomes more desirable to predict the development of colonies; nevertheless the prediction still remains difficult because the growth kinetics and its dependence on the environmental conditions have not yet been fully understood. Here we show morphological changes of colonies grown on nutrient agar media due to pH and nutrient-richness (peptone concentration) variations for pre-cultured, motile cells of bacterial species Bacillus subtilis. Then we perform a reproduction analysis consisting of a series of culture and measurement experiments, modelling and simulations on the basis of theoretical considerations. The resulting model enables us to predict the development of colonies of motile bacteria for diverse environmental settings. Furthermore, our findings provide insights into bacterial survival strategies and the role of each bacterial activity.
This is a joint work with Madoka Nakayama (Sendai National College of Technology) and Wataru Shoji (Tohoku University).

14:30 -- 15:20  Shin-Ichiro Ei (Hokkaido University)

Abstract: We consider the motions of front or pulse solutions for Fitzh-Hugh Nagumo equations on one-dimensional heterogeneous media. We derive equations de- scribing the motions of single front or single pulse solution depending on the heterogeneity and also consider the interaction of multi-front solutions. Through the analysis, we show opposite influences for a forward and a back- ward front solution from the heterogeneity in FitzHugh-Nagumo equations.
Some part of this research is a joint work with Chao-Nien Chen, Shyuh-yaur Tzeng and Y. Morita.