arXiv subjects

Hideki Murakawa

Publications and source records attributed to Hideki Murakawa.

8 recordsLinked to original sources

arXiv2026

Relationship between haptotaxis and chemotaxis in cell dynamics

Cell sorting mediated by direct cell--cell contact or cellular protrusions can be described by nonlocal cell--cell adhesion models of haptotaxis type, whereas communication through diffusible chemical signals is described by Keller--Segel type chemotaxis systems. We investigate the mathematical relationship between these two descriptions. We first prove that, in the fast signal diffusion limit, subsequences of weak solutions of a Keller--Segel type parabolic--parabolic chemotaxis system with possibly degenerate nonlinear diffusion and density saturation converge to weak solutions of the corresponding parabolic--elliptic system. Eliminating the elliptic chemical fields rewrites the population equation as a nonlocal adhesion model whose interaction kernel is a finite linear combination of Green functions. We then prove, in arbitrary spatial dimensions, that the periodization of the gradient of a prescribed radially symmetric interaction potential can be approximated by finite linear combinations of gradients of Green functions. Combining these results, we prove that suitable weak solutions of the nonlocal adhesion model and a parabolic--parabolic chemotaxis system can be chosen with an arbitrarily small difference over any fixed finite time interval. Numerical simulations compare the nonlocal, parabolic--elliptic, and parabolic--parabolic models and examine how the number of kernel terms and the relaxation time affect their differences.

math.AP↗

arXiv2026

Bifurcation structure and mesa pattern formation in a one-component nonlocal adhesion model with population pressure and degenerate mobility

We analyze pattern formation from a homogeneous steady state in a one-component nonlocal adhesion model with population pressure and degenerate mobility. First, using linear stability analysis, we derive the instability threshold and a selection rule for the fastest-growing mode, and elucidate the mechanism by which the selected wavenumber shifts toward lower wavenumbers as the mean density increases. We then perform a weakly nonlinear analysis near the critical adhesion strength and derive an explicit expression for the Landau coefficient in the Stuart--Landau equation. This expression shows that the critical bifurcation is classified as supercritical or subcritical according to the mean density, the nonlinear exponent, and the second-harmonic response ratio of the kernel. Furthermore, we show that a large nonlinear exponent promotes a transition to subcriticality and confirm, through numerical bifurcation analysis and time-dependent simulations, a bifurcation structure with a fold point and the formation of mesa patterns. Finally, through the energy limit as \(m\to\infty\), we relate the observed mesa profiles to a capacity-constrained limiting structure.

nlin.PS↗

arXiv2026

Convergence of a discrete-in-time Approximation to a Degenerate Parabolic-Hyperbolic System

In this paper we consider an implicit semi-discrete approximation of a degenerate reaction-cross-diffusion system. Due to the symmetry in the parabolic part, this system is known to preserve segregation of densities -- initially non-overlapping densities belonging to different species remain segregated for all times, which leads to internal layers between different species. We show that time-discrete approximations exist and converge to a weak solution, as the timestep goes to zero.

math.AP↗

arXiv2026

Reaction-Diffusion System Approximation to the Fast Diffusion Equation

This paper proposes a novel reaction-diffusion system approximation tailored for singular diffusion problems, typified by the fast diffusion equation. While such approximation methods have been successfully applied to degenerate parabolic equations, their extension to singular diffusion-where the diffusion coefficient diverges at low densities-has remained unexplored. To address this, we construct an approximating semilinear system characterized by a reaction relaxation parameter and a time-derivative regularizing parameter. We rigorously establish the well-posedness of this system and derive uniform a priori estimates. Using compactness arguments, we prove the convergence of the approximate solutions to the unique weak solution of the target singular diffusion equation under three distinct asymptotic regimes: the simultaneous limit, the limit via a parabolic-elliptic system, and the limit via a uniformly parabolic equation. This approach effectively transfers the diffusion singularity into the reaction terms, yielding a highly tractable system for both theoretical analysis and computation. Finally, we present numerical experiments that validate our theoretical convergence results and demonstrate the practical efficacy of the proposed approximation scheme.

math.AP↗

arXiv2024

Keller-Segel type approximation for nonlocal Fokker-Planck equations in one-dimensional bounded domain

Numerous evolution equations with nonlocal convolution-type interactions have been proposed. In some cases, a convolution was imposed as the velocity in the advection term. Motivated by analyzing these equations, we approximate advective nonlocal interactions as local ones, thereby converting the effect of nonlocality. In this study, we investigate whether the solution to the nonlocal Fokker-Planck equation can be approximated using the Keller-Segel system. By singular limit analysis, we show that this approximation is feasible for the Fokker-Planck equation with any potential and that the convergence rate is specified. Moreover, we provide an explicit formula for determining the coefficient of the Lagrange interpolation polynomial with Chebyshev nodes. Using this formula, the Keller-Segel system parameters for the approximation are explicitly specified by the shape of the potential in the Fokker-Planck equation. Consequently, we demonstrate the relationship between advective nonlocal interactions and a local dynamical system.

math.AP↗

arXiv2022

Convergence of a Fully Discrete and Energy-Dissipating Finite-Volume Scheme for Aggregation-Diffusion Equations

We study an implicit finite-volume scheme for non-linear, non-local aggregation-diffusion equations which exhibit a gradient-flow structure, recently introduced by Bailo, Carrillo, and Hu (2020). Crucially, this scheme keeps the dissipation property of an associated fully discrete energy, and does so unconditionally with respect to the time step. Our main contribution in this work is to show the convergence of the method under suitable assumptions on the diffusion functions and potentials involved.

math.NA↗

arXiv2019

Fast reaction limit of reaction-diffusion systems

Singular limit problems of reaction-diffusion systems have been studied in cases where the effects of the reaction terms are very large compared with those of the other terms. Such problems appear in literature in various fields such as chemistry, ecology, biology, geology and approximation theory. In this paper, we deal with the singular limit of a general reaction-diffusion system including many problems in the literature. We formulate the problem, derive the limit equation and establish a rigorous mathematical theory.

math.AP↗

arXiv2019

A population dynamics model of cell-cell adhesion incorporating population pressure and density saturation

We discuss several continuum cell-cell adhesion models based on the underlying microscopic assumptions. We propose an improvement on these models leading to sharp fronts and intermingling invasion fronts between different cell type populations. The model is based on basic principles of localized repulsion and nonlocal attraction due to adhesion forces at the microscopic level. The new model is able to capture both qualitatively and quantitatively experiments by Katsunuma et al. (2016) [J. Cell Biol. 212(5), pp. 561--575]. We also review some of the applications of these models in other areas of tissue growth in developmental biology. We finally explore the resulting qualitative behavior due to cell-cell repulsion.

q-bio.CB↗