11:00 -- 11:50   Xuefeng Wang (Tulane University)

Abstract: The most important phenomenon about chemotaxis is cell aggregation. The simplest way to model this phenomenon is to use steady states with striking features such as spikes and transition layers. The pioneering work in this direction is due to Ni and Takagi, who developed a powerful variational method that has been successfully applied to other problems. Other researchers have used methods such as Lyapunov--Schmidt method and global bifurcation method to prove the existence of spiky and transition layer steady states. I will survey these results.

13:30 -- 14:20   Akisato Kubo (Fujita Health University)

Abstract: Recently in the field of medicine, there are a large number of mathematical models described directional cell movement. Especially we focus on some models proposed by Mark Chaplain and coworkers and discuss them in a consistent way.

14:30 -- 15:20   Madoka Nakayama (Sendai National College of Technology)

Abstract: Bacteria form various shapes of colonies to perform their collective activity. Today, the bacterial activity has a major impact on many scientific fields, and so it becomes more desirable to predict the development of colonies. Here we report new morphology of colonies of Bacillus subtilis. The colonies are ultra-sensitive to the environmental pH, and illustrate how organisms highly resistant to environmental changes can be highly sensitive not only in the microscopic interior systems realizing the high-resistance but also in the macroscopic self-organization.

15:50 -- 16:40   Sohei Tasaki (Tohoku University)

Abstract: Bacterial colonies expand in different paces and patterns depending upon the environmental conditions. Here we present a mathematical model for the spatio-temporal growth prediction of the pH-sensitive colonies of Bacillus subtilis from assigned environmental conditions. Furthermore, we propose a universal principle of self-organization "input-saturating positive feedback" by means of the prediction model, and explain how the colony growth can be governed by this concept.

10:00 -- 10:50   Xuefeng Wang (Tulane University)

Abstract: The simplest chemotaxis model is Keller-Segel's minimal model. In the case of 1 spatial dimension, It is known that the minimal model has monotone spiky steady states when the chemotactic coefficient is sufficiently large. We prove the local asymptotic stability of these steady states.

11:10 -- 12:00   Tohru Tsujikawa (Miyazaki University)

Abstract: Several mathematical models are proposed for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis. For these model, it is known that there are many mathematical results of nonconstant stationary solutions and several spatio-temporal patterns numerically showed in the bounded domain with 1 and 2 space dimensions. We consider some qualitative behaviors of stationary solutions from global and local (bifurcation) viewpoints and construct local bifurcation branches of stripe and hexagonal stationary solutions in a rectangle. Next, we introduce the asymptotic behavior of stationary solutions as the chemotactic intensity grows to infinity. Moreover, we exhibit several numerical simulation for the stationary and oscillating patterns.

13:30 -- 14:20   Kanako Suzuki (Ibaraki University)

Abstract: One of the central problems in developmental biology is to understand the mechanism of the formation of a spatial pattern of tissue structure, starting from almost homogeneous states. In 1952, A. Turing proposed the notion of “diffusion-driven instability” (Turing instability) in his attempt to model biological pattern formation, which means that the reaction between two chemicals with different diffusion rates may cause the destabilization of the spatially homogeneous state. This phenomenon has inspired a development of a vast number of mathematical models of biological pattern formation. Such models with Turing instability describe a process of a destabilization of stationary spatially homogeneous steady states and evolution of spatially heterogeneous structures towards spatially heterogeneous steady states. I would like to make a brief survey of results on the dynamics and the stability of stationary solutions of reaction-diffusion equations with Turing instability.

14:30 -- 15:20   Hiroki Hoshino (Fujita Health University)

Abstract: This talk is concerned with traveling wave solutions to a hyperbolic system related to a malignant tumor invasion. Properties of smooth and discontinuous waves are discussed.

15:50 -- 16:40   Izumi Takagi (Tohoku University)

Abstract: Real biological phenomena take place usually in spatially heterogeneous environments. We study the Keller-Segel model with a logarithmic sensitivity function under the assumption that the decay rate and the production rate of the chemoattractant depends on the spatial variable. The existence of a stable steady-state is proved.

10:00 -- 10:50   Tohru Tsujikawa (Miyazaki University)

Bifurcation structure of steady states for a bistable equation with nonlocal constraint

Abstract: In this talk, we consider the Neumann problem of a bistable equation with nonlocal constraint which is described as a limiting system of chemotaxis-growth model in some sense. We show the global bifurcation structure of solutions by a level set analysis for the associate integral mapping and singular perturbation method. This structure implies that solutions can form a saddle-node bifurcation curve connecting two boundary layer states in some case. Moreover, we discuss the stability of these boundary layer solutions and conjecture an appearance of periodic solution.

11:10 -- 12:00   Tai-Chia Lin (National Taiwan University)

Abstract: Keller-Segel type systems are important models with many applications in biology, engineering and physics. The Poisson-Nernst-Planck (PNP) system is a well-known model of ion transport, which belongs to Keller-Segel type systems and plays a crucial role in the study of many physical and biological phenomena. Cross-diffusion terms may describe the exclusion of steric effects. In this lecture, I shall introduce cross diffusion terms from the Lennard-Jones potential and show the analytical results as follows: