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## Mathematical Approaches to Pattern Formation

updated on 25/oct/14 [11]

### 28 October -- 31 October, 2014

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

### Abstracts of Talks

Tuesday, 28 October, 2014

13:20--13:30     opening

13:30--14:30     Anna Marciniak-Czochra (Heidelberg)
Diffusion-driven unbounded growth and dynamical spike patterns in reaction-diffusion-ode models

Abstract: In this talk we explore a mechanism of pattern formation arising in processes described by a system of a single reaction-diffusion equation coupled to ordinary differential equations. Such systems of equations arise, for example, in modeling of interactions between cellular processes and diffusing growth factors. Our theory applies to a wide class of pattern formation models with an autocatalytic non-diffusing component.
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)
Decomposition of Möbius energy and Möbius invariance of decomposed energies

Abstract: The Möbius energy is introduced to determine the most balanced shape of knot in a given knot class. Though several energies were known for this purpose, the Möbius energy is geometrically important because it has invariance under the Möbius transformations. Analytically we must deal with it carefully, because the energy density has singularity on the diagonal part in the integral region. In this talk, we show the decomposition of energy into three parts, which enables us to treat the energy relatively easily. This decomposition does not destroy the geometry of energy, that is, each decomposed energy also has the Möbius invariance. This is a joint work with A. Ishizeki.

15:30--16:00             coffee break

16:00--16:50     Takayoshi Ogawa (Sendai)
Threshold for the global behavior of solution to degenerate Keller-Segel (drift-diffusion) system in between critical exponents

Abstract: We consider the large time behavior of a weak solution for degenerate Keller-Segel (drift-diffusion) system. It is known that there exists two distinct exponents for the problem and the global existence and non-existence are well understood in those cases. Here we present a threshold for the global existence and non-existence of the weak solution for the system with the exponent between those two critical cases. The threshold of the initial data is given by the best possible constant of the modified Hardy-Littlewood-Sobolev inequality. We also discuss the large time asymptotics of the global solution. (joint work with A. Kimijima and K.Nakagawa)

17:30--19:00         welcome party

Wednesday, 29 October, 2014

10:00--10:50     Hideo Ikeda (Toyama)
Stability and bifurcation properties of a standing spot solution in 3-component FitzHugh-Nagumo systems

Abstract: Various localized planar patterns are observed in many reaction-diffusion systems. Especially, two-component systems are well studied so far and several mathematical results are obtained. But, even if such stationary localized planar solutions (stationary spot solutions) exist stably, one do not get stable traveling spot solutions which are bifurcated from stationary spot solutions. This implies that such traveling spot solutions seem to be unstable in two-component systems if they exist. In this talk, we will show the existence of stationary spot solutions and the stability properties of them in three-component FitzHugh-Nagumo systems and consider the possibility of the supercritical drift bifurcation. Note that these results are already obtained by Heijster and Sandstede via the formal analysis (Physica D 275(2014) 19-34). Here, we will give the proof of it by the two methods, one is the construction of the Evans function the other is the SLEP method.

Finite-time blow-up in the parabolic Keller-Segel system

Abstract: We study the Neumann initial-boundary value problem for the fully parabolic Keller-Segel system
$\left\{ \begin{array}{l} u_t=\Delta u - \nabla \cdot (u\nabla v), \qquad x\in\Omega, \ t>0, \\[1mm] v_t=\Delta v-v+u, \qquad x\in\Omega, \ t>0, \end{array} \right. \qquad \qquad (\star)$ 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

Aya Ishizeki (Saitama)
Variational formulae and estimates for decomposed Möbius energies
Abstract: The first and second variational formulae of the Möbius energy were calculated by several mathematicians. Direct calculation produces a lot of terms which are not integrable even in the sense of Cauchy's principal value. By combining several terms appropriately, the integrability recovers, however, it is a quite hard job. Using the decomposition which was given in Nagasawa's talk, we can calculate the variational formulae relatively easily. The explicite forms of the first and second variational formulae of decomposed eneries are given. We also discuss their estimates in several function spaces. This is a joint work with T. Nagasawa.

Shoichi Hasegawa (Sendai)
Liouville theorem for Henon type equation on Riemannian models

Abstract: We discuss the Liouville result for stable solutions of the Henon type equation on Riemannian models. In 2007, Farina showed that Lane-Emden-Fowler equation on Euclidean space has no non-trivial stable solutions if the exponent is smaller than Joseph-Lundgren exponent. On the other hand, Berchio, Ferrero, and Grillo proved that the same equation has stable solutions for any exponent on some manifolds with negative sectional curvature in 2014. This result implies that the structure of stable solutions of the elliptic equation on the manifold is different from that on Euclidean space. In this talk, we study the nonexistence of non-trivial stable solutions of the Henon type equation on some Riemannian models for small exponent. We also mention that the existence of non-trivial stable solutions for large exponent.

Jin Takahashi (Tokyo)
Time-dependent singularities for a semilinear parabolic equation with an absorption term

Abstract: We consider solutions with time-dependent singularities for a semilinear heat equation with a superlinear absorption term. Here, by time-dependent singularity, we mean a singularity with respect to the space variable whose position depends on the time variable. It is shown that if the power of the nonlinearity is in some range, there is no time-dependent singular solution. On the other hand, in other range, two types of time-dependent singular solutions exist. One behaves like the fundamental solution of the Laplace equation near the singular point. The other one behaves like the positive singular steady state. This is a joint work with Eiji Yanagida.

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

Hiroko Yamamoto (Sendai)
Location of concentration points in a spatially heterogeneous semilinear Neumann problem

Abstract: We are interested in a point-condensation phenomenon in solutions of a spatially heterogeneous semilinear Neumann problem. This means that the distribution of a solution concentrates in a very narrow region around a finitely many points. Hence, it is most important to find the candidates for the concentration points. In this talk we consider the singularly perturbed Neumann problem for a semilinear elliptic equation with variable coefficients, and reveal the effects of spatial heterogeneity on the location of concentration points.

Kohta Suzuno (Tokyo)
Modeling bottleneck flow in pedestrian dynamics

Abstract: We clarify the mechanism of the strange "faster-is-slower" effect in a pedestrian simulation via a modeling. Pedestrian dynamics is a discipline that treats collective motion of human and is of great interest for mathematical scientists as it shows various self-organizing phenomena. In this talk we focus on the spontaneous formation of an arch at a bottleneck. It is known that the arch causes the "faster-is-slower" effect, the counter-intuitive relation between the driving force and the flow rate [D. Helbing et al., Nature 407, 487 (2000)]. Through a modeling approach, we show this effect comes from the competition between the driving force and nonlinear friction [K. Suzuno et al., Phys. Rev. E 88, 052813 (2013)].

Kurumi Hiruko (Sendai)
Dynamics of mutable prostate cancer cells under intermittent androgen suppression therapy

Abstract: We are interested in a mathematical model describing dynamics of prostate tumor under intermittent androgen suppression (IAS) therapy. The clinical phenomenon has the following special features:
(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)
The obstacle problem for a fourth order parabolic equation

Abstract: We consider an obstacle problem for the parabolic biharmonic equation. In this talk, we show a recent joint work with Matteo Novaga (Pisa University) on the obstacle problem. We analyze the obstacle problem via an implicit time discretization. The method allows us to prove an existence of long time solution to the obstacle problem and to find its regularity properties. Since the study is motivated by an obstacle problem for curve straightening flow defined on planar curves, we shall also mention the geometric obstacle problem.

Thursday, 30 October, 2014

10:00--10:50     Eiji Yanagida (Tokyo)
Expanding patterns in the Fisher-KPP equation

Abstract: We consider the dynamics of interfaces in the Fisher-KPP equation on $$\Bbb{R}^N$$. Under appropriate rescaling, solutions exhibit interfaces that correspond to transition layers from the trivial steay state to a stable positive steady state. If initial data decay rapidly in space, then the interface moves with a constant speed that is equal to the minimal speed of traveling fronts in one-dimensional space. On the other hand, it is known that for slowly decaying initial data, the interface may expand in a rather irregular way. In this talk, we show that the dynamics of expanding patterns for slowly decaying initial data can be described as a level set of some first order equation of Hamilton-Jacobi type. We also discuss properties of solutions of the Hamilton-Jacobi equation. This is a joint work of Hirokazu Ninomiya of the Meiji University.

11:00--12:00     Yuan Lou (Columbus/Beijing)
Evolution of Dispersal in Advective Environments

Abstract: We consider some mathematical models in one-dimensional advective environments. Individuals are exposed to unidirectional flow, with the possibility of being lost through the boundary. Our analysis suggests that, in contrast to the case of no advection, slow dispersal is generally selected against in advective environments. When the diffusion and advection rates are small and comparable, we determine some criterion for the existence and multiplicity of evolutionarily stable strategies. We also study whether that these strategies are convergent stable.

12:00--13:30             lunch break

13:30--14:20     Mayuko Iwamoto (Tokyo)
Mathematical model for adhesive locomotion in gastropods

Abstract: For many gastropods who are called generally “univalve shell”, adhesive locomotion is driven by a succession of periodic muscular waves (contractions and relaxations) moving along their pedal foot, the force generated by these waves being coupled to the substratum by a thin layer of pedal mucus. Interestingly, gastropod pedal mucus has unusual physical properties [1]; the mucus is a viscoelastic solid at a small deformation and shows a sharp yield point, then at greater strains the mucus is a viscous liquid though the mucus recovers its solidity if allowed to heal for a certain period of time. The recent engineering researches are focused on the role of these properties of mucus as friction control in adhesive locomotion [2].
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

Variational characterizations of the static solutions to the Einstein-Maxwell equation

Abstract: In this talk, we present a set of characterizations of the known static solutions of the Einstein-Maxwell equation, based on a project with Marcus Khuri and Gilbert Weinstein. Those new characterizations are variational in the sense that the static solution appears as the equality/rigidity cases of the so-called Penrose-type inequality. The inequality is among the quantities parameterizing the asymptotic structure of the spacetime as well as the geometric information on the blackhole horizon. We will also discuss the origin of the inequality devised by Penrose, in comparison to the Newtonian gravitational theory.

15:20--16:00             coffee break

16:00--16:50     Izumi Takagi (Sendai)
Control of patterns by way of spatial heterogeneity

Abstract: What Turing tried to say was "even if the environment is perfectly uniform, pattern can emergy as a result of diffusion-driven instability". No biological pattern formation takes place in the perfect spatial homogeneity, rather it happens in a fairly non-uniform environment. Sometimes spatial heterogeneity is very important for a group of cells to develop a form. If the heterogeneity is not strong enough, it takes too long to get started with forming a pattern. We hence face a difficulty of tautological explanation "patterns emergy because there is already a pattern". In this talk we refine the activator-inhibitor system by Gierer and Meinhardt and propose a very simple two-stage model to avoid this "the chicken or the egg" argument. The first stage is to enhance the built-in spatial heterogeneity which is a purely local process, and the second stage is to form a pattern by resorting to a reaction-diffusion mechanism. This talk is based on the joint work with Hiroko Yamamoto.

Friday, 31 October, 2014

10:00--10:50     Sohei Tasaki (Sendai)
Growth prediction of colonies of motile bacteria

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).

11:00--12:00     Peter Bates (East Lansing)
Invariant manifolds of multi interior spike states for the Cahn-Hilliard equation

Abstract: We construct invariant manifolds of interior multi-spike states for the nonlinear Cahn-Hilliard equation and then investigat e the dynamics on it. An equation for the motion of the spikes is derived. It turns out that the dynamics of interior spikes has a global character and each spike interacts with all the others and with the boundary. Moreover, we show that the speed of the interior spikes is super slow, which indicates the long time existence of dynamical multi-spike solutions in both po sitive and negative time. This result is the application of an abstract result concerning the existence of truly invariant m anifolds with boundary when one has only approximately invariant manifolds.

12:00--12:10   closing

### Mini-workshop on Patterns Resulting from Competition between Diffusion and Taxis

Friday, 31 October, 2014
Venue: Kawai Hall, Mathematical Institute, Tohoku University

13:30--14:20     Koichi Osaki (Sanda)
Chemotaxis vs. logistic growth - global existence and pattern formation of solutions

Abstract: Mimura and Tsujikawa[Physica A 230 (1996)] proposed a mathematical model for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis. For this model, Tello and Winkler[Comm.PDE 35 (2007)] obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution, while Kurata et al. [GAKUTO Internat. Ser. Math. Sci. Appl. 29 (2008)] numerically showed several spatial patterns, including stripe or hexagonal patterns, in a rectangle. Motivated by their work, we consider some pattern formation of solutions from local bifurcation viewpoints.

14:40--15:20     Hiroshi Wakui (Sendai)
Boundedness of a weak solution to a degenerate drift-diffusion equation with a critical exponent

Abstract: We consider the initial value problem of a degenerate Keller-Segel (drift-diffusion) system in $$\mathbb{R}^n$$ with $$n \ge 3$$. The equation has a diffusion exponent and there exists a critical exponent that separates the large time behavior of weak solutions. This exponent is called as the mass ($$L^1$$-) critical exponent. In the mass critical case, it is known that there exists a critical mass which classifies a global behavior of weak solutions by the global existence and non-existence. If the mass of initial data is strictly less than the critical mass $$M_*$$, then the weak solution exists globally in time and it decays at $$t \to \infty$$. While if the initial data has larger mass than $$M_*$$, then there exists an initial data such that corresponding weak solution blows up in a finite time. It is also known that if the initial mass coincides the critical mass, the corresponding solutions exists globally in time. In this talk, we discuss on the uniform boundedness for the weak solution with the critical mass. We obtain a partial result for a uniform boundedness of weak solution under some additional conditions and it includes the stationary solution of the system.

15:30--16:10     Yoshifumi Mimura (Sendai)
The critical mass for a degenerate Keller-Segel system from a variational viewpoint

Abstract: We consider a degenerate Keller-Segel system in a bounded domain in $$\mathbb{R}^d$$ and $$d>2$$ with smooth boundary. Our interest in the problem is whether or not a threshold mass of the first variable that separates the existence from non-existence of blow-up solutions exists. We show that a certain critical value appears in the variational construction of solutions called the minimizing movement scheme and present the time-global existence of solutions under the assumption that the mass of the first variable is less than the critical value.

• 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