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## Mathematical Approaches to Pattern Formation |

updated on 25/oct/14 [11]

Venue:
Kawai Hall, Mathematical Institute, Tohoku University, Sendai

**Tuesday, 28 October, 2014**

13:20--13:30 opening

13:30--14:30 **Anna Marciniak-Czochra **(Heidelberg)

We show that the lack of diffusion in some model components may lead to singularities which result in instability of all regular stationary patterns. Interestingly, the degeneration of the system yields a continuous spectrum of the linearization operator, which contains positive values. We show that, under some conditions, also all discontinuous stationary solutions are unstable. However, in numerical simulations, solutions having the form of periodic or irregular spikes are observed. We explain this phenomenon using a shadow-type reduction of the reaction-diffusion-ode model. For the resulting system of integro-differential equations, we prove convergence of the model solutions to singular unbounded spike patterns, which location depends on the initial condition. Moreover, we find a class of reaction-diffusion-ode models with diffusion-driven blow-up of spatially heterogenous solutions.

The talk is a based on a joint research with Kanako Suzuki (Ibaraki University) and Grzegorz Karch (University of Wroclaw).

14:40--15:30 **Takeyuki Nagasawa** (Saitama)

15:30--16:00 coffee break

16:00--16:50 **Takayoshi Ogawa** (Sendai)

17:30--19:00 welcome party

**Wednesday, 29 October, 2014**

10:00--10:50 **Hideo Ikeda ** (Toyama)

11:00--12:00 **Michael Winkler** (Paderborn)

\[ \left\{ \begin{array}{l} u_t=\Delta u - \nabla \cdot (u\nabla v), \qquad x\in\Omega, \ t>0, \\[1mm]

v_t=\Delta v-v+u, \qquad x\in\Omega, \ t>0, \end{array} \right. \qquad \qquad (\star) \] in a ball \(\Omega \subset {\mathbb{R}}^n\), where \(n\ge 2\). This system forms the core of numerous models used in mathematical biology to describe the spatio-temporal evolution of cell populations governed by both diffusive migration and chemotactic movement towards increasing gradients of a chemical that they produce themselves.

We demonstrate that in the case \(n\ge 3\), for any prescribed \(m>0\) there exist radially symmetric positive initial data \( (u_0,v_0) \in C^0(\bar\Omega) \times W^{1,\infty}(\Omega) \) with \( \int_\Omega u_0=m \) such that the corresponding solution blows up in finite time. Moreover, by providing an essentially explicit blow-up criterion it is shown that within the space of all radial functions, the set of such blow-up enforcing initial data indeed is large in an appropriate sense; in particular, this set is dense with respect to the topology of \(L^p(\Omega) \times W^{1,2}(\Omega)\) for any \( p \in (1,\frac{2n}{n+2})\). We moreover comment on a corresponding result for \( n=2\) which indicates that finite-time blow-up is a generic phenomenon also in this case, at least within the framework of radial solutions.

One focus of the presentation is on the method through which these results can be obtained. In contrast to previous approaches, it is based on a more elaborate use of the natural energy inequality associated with (\(\star\)), e.g. in the case \( n\ge 3 \) involving an estimate of the form\[ \int_\Omega uv \le C \cdot \bigg( \Big\|\Delta v-v+u\Big\|_{L^2(\Omega)}^{2\theta} + \Big\|\frac{\nabla u}{\sqrt{u}}-\sqrt{u}\nabla v\Big\|_{L^2(\Omega)} +1 \bigg), \]

which is valid with certain \( C>0 \) and \( \theta \in (0,1) \) for a wide class of smooth positive radial functions \( (u,v)=(u(x),v(x)) \).

12:00--13:30 lunch break

13:30--14:30 Short communications, I

14:40--15:40 Short communications, II

(i) There are the competition and mutation interactions between cells in prostate tumor;

(ii) In the IAS therapy, medication is stopped when the size of tumor decreases and attains a lower threshold, and resumed when the size increases and attains an upper threshold.

Due to the feature (ii), the mathematical model is given by a "hybrid system" with second order semilinear parabolic equation. We deal with the hybrid system and prove mathematically that the size of tumor remains in some bounded interval for any time under the IAS therapy.

15:40--16:10 coffee break

16:10--17:00 **Shinya Okabe **(Sendai)

**Thursday, 30 October, 2014**

10:00--10:50 **Eiji Yanagida** (Tokyo)

11:00--12:00 **Yuan Lou ** (Columbus/Beijing)

12:00--13:30 lunch break

13:30--14:20 **Mayuko Iwamoto ** (Tokyo)

To show clearly the role of mucus in adhesive locomotion, we verify using a simple mathematical model that the directional migration can be realized by the interaction between the periodic muscular waves and the specific physical feature of mucus. Our simulations indicate that the hysteresis property of mucus is essential for realization of crawling locomotion. Furthermore, our numerical calculations with the mathematical model show that the hysteresis property of mucus realizes both two locomotion styles, direct wave and retrograde wave which have been understood by different mechanisms until now. As new perspective, our results suggest that the selection decision of locomotion style caused by the differences on the properties of pedal mucus and muscle of foot [3].

References

[1] Denny, M. W., The role of gastropod pedal mucus in locomotion. Nature, 285 (1980) 160-161.

[2] Chan, B., Balmforth, N. J. and Hosoi, A. E., Building a better snail: lubrication and adhesive locomotion. Phys. Fluids 17 (2005) 113101.

[3] Iwamoto M., Ueyama D., Kobayashi R., The advantage of mucus for adhesive locomotion in gastropods, J. Theor. Biol. 353 (2014) 133-141

14:30--15:20 **Sumio Yamada** (Tokyo)

15:20--16:00 coffee break

16:00--16:50 **Izumi Takagi** (Sendai)

**Friday, 31 October, 2014**

10:00--10:50 **Sohei Tasaki** (Sendai)

11:00--12:00 **Peter Bates **(East Lansing)

12:00--12:10 closing

13:30--14:20 **Koichi Osaki** (Sanda)

14:40--15:20 **Hiroshi Wakui** (Sendai)

15:30--16:10 **Yoshifumi Mimura** (Sendai)

- Organizing Committee: Yuan Lou (Ohio State University/Renmin University), Izumi Takagi (Tohoku University; chair) and Eiji Yanagida (Tokyo Institute of Technology)
- This symposium is supported by the Tohoku University Focused Research Project "Interdisciplinary Mathematics toward Smart Innovations", the JSPS Grant-in-Aid for Scientific Research (A) #22244010 "Theory of Differential Equations Applied to Biological Pattern Formation--from Analysis to Synthesis", and the JSPS Grant Chanllenging Exploratory Research #26610027 "Control of Patterns by Multi-component Reaction-Diffusion Systems of Degenerate Type".

created on 19/sep/14