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Networks & Heterogeneous Media

2013 , Volume 8 , Issue 1

Special issue

dedicated to Hiroshi Matano on the occasion of his 60th birthday: Part II

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Henri Berestycki, Danielle Hilhorst, Frank Merle, Masayasu Mimura and Khashayar Pakdaman
2013, 8(1): i-iii doi: 10.3934/nhm.2013.8.1i +[Abstract](41) +[PDF](93.9KB)
Professor Hiroshi Matano was born in Kyoto, Japan, on July 28th, 1952. He studied at Kyoto University, where he prepared his doctoral thesis under the supervision of Professor Masaya Yamaguti. He obtained his first academic position as a research associate at the University of Tokyo. He then moved to Hiroshima University in 1982 and came back to Tokyo in 1988. He is a Professor at the Graduate School of Mathematical Sciences at the University of Tokyo since 1991.

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Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow
Sigurd Angenent
2013, 8(1): 1-8 doi: 10.3934/nhm.2013.8.1 +[Abstract](64) +[PDF](380.5KB)
We provide formal matched asymptotic expansions for ancient convex solutions to MCF. The formal analysis leading to the solutions is analogous to that for the generic MCF neck pinch in [1].
    For any $p, q$ with $p+q=n$, $p\geq1$, $q\geq2$ we find a formal ancient solution which is a small perturbation of an ellipsoid. For $t\to-\infty$ the solution becomes increasingly astigmatic: $q$ of its major axes have length $\approx\sqrt{2(q-1)(-t)}$, while the other $p$ axes have length $\approx \sqrt{-2t\log(-t)}$.
    We conjecture that an analysis similar to that in [2] will lead to a rigorous construction of ancient solutions to MCF with the asymptotics described in this paper.
A nonlinear partial differential equation for the volume preserving mean curvature flow
Dimitra Antonopoulou and Georgia Karali
2013, 8(1): 9-22 doi: 10.3934/nhm.2013.8.9 +[Abstract](66) +[PDF](427.6KB)
We analyze the evolution of multi-dimensional normal graphs over the unit sphere under volume preserving mean curvature flow and derive a non-linear partial differential equation in polar coordinates. Furthermore, we construct finite difference numerical schemes and present numerical results for the evolution of non-convex closed plane curves under this flow, to observe that they become convex very fast.
Reaction-diffusion waves with nonlinear boundary conditions
Narcisa Apreutesei and Vitaly Volpert
2013, 8(1): 23-35 doi: 10.3934/nhm.2013.8.23 +[Abstract](71) +[PDF](345.9KB)
A reaction-diffusion equation with nonlinear boundary condition is considered in a two-dimensional infinite strip. Existence of waves in the bistable case is proved by the Leray-Schauder method.
The degenerate and non-degenerate deep quench obstacle problem: A numerical comparison
L’ubomír Baňas, Amy Novick-Cohen and Robert Nürnberg
2013, 8(1): 37-64 doi: 10.3934/nhm.2013.8.37 +[Abstract](101) +[PDF](2045.6KB)
The deep quench obstacle problem $$ {\rm{\bf{(DQ)}}} \begin{equation}\left\{ \begin{array}{l} \frac{\partial u}{\partial t}=\nabla \cdot M(u) \nabla w, \\ w + \epsilon^2 \triangle u + u \in \partial \Gamma(u), \end{array} \right. \end{equation}$$ for $(x,t) \in \Omega \times (0,T)$, models phase separation at low temperatures. In (DQ), $\epsilon>0,$ $\partial \Gamma(\cdot)$ is the sub-differential of the indicator function $I_{[-1,1]}(\cdot),$ and $u(x,t)$ should satisfy $\nu \cdot \nabla u=0$ on the ``free boundary'' where $u=\pm 1$. We shall assume that $u$ is sufficiently smooth to make these notions well-defined. The problem (DQ) corresponds to the zero temperature ``deep quench'' limit of the Cahn--Hilliard equation. We focus here on a degenerate variant of (DQ) in which $M(u)=1-u^2,$ as well as on a constant mobility non-degenerate variant in which $M(u)=1.$ Although historically more emphasis has been placed on models with non-degenerate mobilities, degenerate mobilities capture some of the underlying physics more accurately. In the present paper, a careful numerical study is undertaken, utilizing a variety of benchmarks as well as new upper bounds for coarsening, in order to clarify evolutionary properties and to explore the differences in the two variant models.
Growth regulation and the insulin signaling pathway
Peter W. Bates, Yu Liang and Alexander W. Shingleton
2013, 8(1): 65-78 doi: 10.3934/nhm.2013.8.65 +[Abstract](65) +[PDF](412.5KB)
The insulin signaling pathway propagates a signal from receptors in the cell membrane to the nucleus via numerous molecules some of which are transported through the cell in a partially stochastic way. These different molecular species interact and eventually regulate the activity of the transcription factor FOXO, which is partly responsible for inhibiting the growth of organs. It is postulated that FOXO partially governs the plasticity of organ growth with respect to insulin signalling, thereby preserving the full function of essential organs at the expense of growth of less crucial ones during starvation conditions. We present a mathematical model of this reacting and directionally-diffusing network of molecules and examine the predictions resulting from simulations.
Traveling fronts guided by the environment for reaction-diffusion equations
Henri Berestycki and Guillemette Chapuisat
2013, 8(1): 79-114 doi: 10.3934/nhm.2013.8.79 +[Abstract](104) +[PDF](648.5KB)
This paper deals with the existence of traveling fronts for the reaction-diffusion equation: $$ \frac{\partial u}{\partial t} - \Delta u =h(u,y) \qquad t\in \mathbb{R}, \; x=(x_1,y)\in \mathbb{R}^N. $$ We first consider the case $h(u,y)=f(u)-\alpha g(y)u$ where $f$ is of KPP or bistable type and $\lim_{|y|\rightarrow +\infty}g(y)=+\infty$. This equation comes from a model in population dynamics in which there is spatial spreading as well as phenotypic mutation of a quantitative phenotypic trait that has a locally preferred value. The goal is to understand spreading and invasions in this heterogeneous context. We prove the existence of threshold value $\alpha_0$ and of a nonzero asymptotic profile (a stationary limiting solution) $V(y)$ if and only if $\alpha<\alpha_0$. When this condition is met, we prove the existence of a traveling front. This allows us to completely identify the behavior of the solution of the parabolic problem in the KPP case.
    We also study here the case where $h(y,u)=f(u)$ for $|y|\leq L_1$ and $h(y,u) \approx - \alpha u$ for $|y|>L_2\geq L_1$. This equation provides a general framework for a model of cortical spreading depressions in the brain. We prove the existence of traveling front if $L_1$ is large enough and the non-existence if $L_2$ is too small.
Homogenized description of multiple Ginzburg-Landau vortices pinned by small holes
Leonid Berlyand and Volodymyr Rybalko
2013, 8(1): 115-130 doi: 10.3934/nhm.2013.8.115 +[Abstract](51) +[PDF](458.9KB)
We consider a homogenization problem for the magnetic Ginzburg-Landau functional in domains with a large number of small holes. We establish a scaling relation between sizes of holes and the magnitude of the external magnetic field when the multiple vortices pinned by holes appear in nested subdomains and their homogenized density is described by a hierarchy of variational problems. This stands in sharp contrast with homogeneous superconductors, where all vortices are known to be simple. The proof is based on the $\Gamma$-convergence approach applied to a coupled continuum/discrete variational problem: continuum in the induced magnetic field and discrete in the unknown finite (quantized) values of multiplicity of vortices pinned by holes.
Modeling contact inhibition of growth: Traveling waves
Michiel Bertsch, Masayasu Mimura and Tohru Wakasa
2013, 8(1): 131-147 doi: 10.3934/nhm.2013.8.131 +[Abstract](64) +[PDF](614.8KB)
We consider a simplified 1-dimensional PDE-model describing the effect of contact inhibition in growth processes of normal and abnormal cells. Varying the value of a significant parameter, numerical tests suggest two different types of contact inhibition between the cell populations: the two populations move with constant velocity and exhibit spatial segregation, or they stop to move and regions of coexistence are formed. In order to understand the different mechanisms, we prove that there exists a segregated traveling wave solution for a unique wave speed, and we present numerical results on the ``stability" of the segregated waves. We conjecture the existence of a non-segregated standing wave for certain parameter values.
Archimedean copula and contagion modeling in epidemiology
Jacques Demongeot, Mohamad Ghassani, Mustapha Rachdi, Idir Ouassou and Carla Taramasco
2013, 8(1): 149-170 doi: 10.3934/nhm.2013.8.149 +[Abstract](94) +[PDF](3153.3KB)
The aim of this paper is first to find interactions between compartments of hosts in the Ross-Macdonald Malaria transmission system. So, to make clearer this association we introduce the concordance measure and then the Kendall's tau and Spearman's rho. Moreover, since the population compartments are dependent, we compute their conditional distribution function using the Archimedean copula. Secondly, we get the vector population partition into several dependent parts conditionally to the fecundity and to the transmission parameters and we show that we can divide the vector population by using $p$-th quantiles and test the independence between the subpopulations of susceptibles and infecteds. Third, we calculate the $p$-th quantiles with the Poisson distribution. Fourth, we introduce the proportional risk model of Cox in the Ross-Macdonald model with the copula approach to find the relationship between survival functions of compartments.
Multiple travelling waves for an $SI$-epidemic model
Arnaud Ducrot, Michel Langlais and Pierre Magal
2013, 8(1): 171-190 doi: 10.3934/nhm.2013.8.171 +[Abstract](92) +[PDF](417.7KB)
In this note we analyze a spatially structured SI epidemic model with vertical transmission, a logistic effect on vital dynamics and a density dependent incidence. The dynamics of the underlying system of ordinary differential equations are first shown to exhibit an infinite number of heteroclinic orbits connecting the trivial equilibrium with an interior equilibrium. Our mathematical study of the corresponding reaction-diffusion system is concerned with travelling wave solutions. Based on a detailed study of the center-unstable manifold around the interior equilibrium, we are able to prove the existence of an infinite number of travelling wave solutions connecting the trivial equilibrium and the interior equilibrium.
Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems
Shin-Ichiro Ei and Toshio Ishimoto
2013, 8(1): 191-209 doi: 10.3934/nhm.2013.8.191 +[Abstract](62) +[PDF](430.8KB)
We consider pulse-like localized solutions for reaction-diffusion systems on a half line and impose various boundary conditions at one end of it. It is shown that the movement of a pulse solution with the homogeneous Neumann boundary condition is completely opposite from that with the Dirichlet boundary condition. As general cases, Robin type boundary conditions are also considered. Introducing one parameter connecting the Neumann and the Dirichlet boundary conditions, we clarify the transition of motions of solutions with respect to boundary conditions.
A spatialized model of visual texture perception using the structure tensor formalism
Grégory Faye and Pascal Chossat
2013, 8(1): 211-260 doi: 10.3934/nhm.2013.8.211 +[Abstract](143) +[PDF](3007.0KB)
The primary visual cortex (V1) can be partitioned into fundamental domains or hypercolumns consisting of one set of orientation columns arranged around a singularity or ``pinwheel'' in the orientation preference map. A recent study on the specific problem of visual textures perception suggested that textures may be represented at the population level in the cortex as a second-order tensor, the structure tensor, within a hypercolumn. In this paper, we present a mathematical analysis of such interacting hypercolumns that takes into account the functional geometry of local and lateral connections. The geometry of the hypercolumn is identified with that of the Poincaré disk $\mathbb{D}$. Using the symmetry properties of the connections, we investigate the spontaneous formation of cortical activity patterns. These states are characterized by tuned responses in the feature space, which are doubly-periodically distributed across the cortex.
Stochastic control of traffic patterns
Yuri B. Gaididei, Carlos Gorria, Rainer Berkemer, Peter L. Christiansen, Atsushi Kawamoto, Mads P. Sørensen and Jens Starke
2013, 8(1): 261-273 doi: 10.3934/nhm.2013.8.261 +[Abstract](74) +[PDF](386.5KB)
A stochastic modulation of the safety distance can reduce traffic jams. It is found that the effect of random modulation on congestive flow formation depends on the spatial correlation of the noise. Jam creation is suppressed for highly correlated noise. The results demonstrate the advantage of heterogeneous performance of the drivers in time as well as individually. This opens the possibility for the construction of technical tools to control traffic jam formation.
A short proof of the logarithmic Bramson correction in Fisher-KPP equations
François Hamel, James Nolen, Jean-Michel Roquejoffre and Lenya Ryzhik
2013, 8(1): 275-289 doi: 10.3934/nhm.2013.8.275 +[Abstract](92) +[PDF](418.7KB)
In this paper, we explain in simple PDE terms a famous result of Bramson about the logarithmic delay of the position of the solutions $u(t,x)$ of Fisher-KPP reaction-diffusion equations in $\mathbb{R}$, with respect to the position of the travelling front with minimal speed. Our proof is based on the comparison of $u$ to the solutions of linearized equations with Dirichlet boundary conditions at the position of the minimal front, with and without the logarithmic delay. Our analysis also yields the large-time convergence of the solutions $u$ along their level sets to the profile of the minimal travelling front.
The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system
Kota Ikeda
2013, 8(1): 291-325 doi: 10.3934/nhm.2013.8.291 +[Abstract](67) +[PDF](976.3KB)
The Gierer-Meinhardt system is a mathematical model describing the process of hydra regeneration. This system has a stationary solution with a stripe pattern on a rectangular domain, but numerical results suggest that such stripe pattern is unstable. In [8], Kolokolnikov et al. proved the existence of a positive eigenvalue, which is called an unstable eigenvalue, for a stationary solution with a stripe pattern by the NLEP method, which implies the instability of the stripe pattern. In addition, the uniqueness of the unstable eigenvalue was shown under some technical assumptions in [8]. In this paper, we prove the existence and uniqueness of an unstable eigenvalue by using the SLEP method without any extra conditions. We also prove the existence of a single-spike solution in one-dimension.
Spread of viral infection of immobilized bacteria
Don A. Jones, Hal L. Smith and Horst R. Thieme
2013, 8(1): 327-342 doi: 10.3934/nhm.2013.8.327 +[Abstract](89) +[PDF](403.9KB)
A reaction diffusion system with a distributed time delay is proposed for virus spread on bacteria immobilized on an agar-coated plate. A distributed delay explicitly accounts for a virus latent period of variable duration. The model allows the number of virus progeny released when an infected cell lyses to depend on the duration of the latent period. A unique spreading speed for virus infection is established and traveling wave solutions are shown to exist.
Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity
John R. King
2013, 8(1): 343-378 doi: 10.3934/nhm.2013.8.343 +[Abstract](94) +[PDF](644.1KB)
We adapt (ray-based) geometrical optics approaches to encompass the formal asymptotic analysis of front propagation in a Fisher-KPP equation with slowly varying spatial inhomogeneities. The wavespeed is shown to be selected by two distinct (and fully constructive) mechanisms, depending on whether the source term is an increasing or decreasing function of the spatial variable. Canonical inner problems, analogous to those arising in the geometrical theory of diffraction, are formulated to give refined expressions for the wavefront location. Additional phenomena, notably the initiation of new fronts and the transitions that occur when the source term is a non-monotonic function of space, are shown to be amenable to the same asymptotic approaches.
Traveling fronts of pyramidal shapes in competition-diffusion systems
Wei-Ming Ni and Masaharu Taniguchi
2013, 8(1): 379-395 doi: 10.3934/nhm.2013.8.379 +[Abstract](82) +[PDF](435.2KB)
It is well known that a competition-diffusion system has a one-dimensional traveling front. This paper studies traveling front solutions of pyramidal shapes in a competition-diffusion system in $\mathbb{R}^N$ with $N\geq 2$. By using a multi-scale method, we construct a suitable pair of a supersolution and a subsolution, and find a pyramidal traveling front solution between them.
Pattern forming instabilities driven by non-diffusive interactions
Ivano Primi, Angela Stevens and Juan J. L. Velázquez
2013, 8(1): 397-432 doi: 10.3934/nhm.2013.8.397 +[Abstract](84) +[PDF](647.9KB)
In analogy to the analysis of minimal conditions for the formation of diffusion driven instabilities in the sense of Turing, in this paper minimal conditions for a class of kinetic equations with mass conservation are discussed, whose solutions show patterns with a characteristic wavelength. The related linearized systems are analyzed, and the minimal number of equations is derived, which is needed for specific patterns to occur.

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