Yuzo Hosono

Multiple Solutions of Two-Point Boundary Value Problems of Neumann Type with a Small Parameter

This paper studies two-point boundary value problems for two-component systems with a small parameter

Traveling wave solutions for some density dependent diffusion equations

\varepsilon

On the Structure of Multiple Existence of Stable Stationary Solutions in Systems of Reaction-Diffusion Equations

. The boundary conditions are of Neumann type. First it is shown that the reduced problem

Localized cluster solutions of nonlinear degenerate diffusion equations arising in population dynamics

(\varepsilon = 0)

Traveling waves for some biological systems with density dependent diffusion

has multiple solutions. With the aid of this result, the singular perturbation method is used for constructing large amplitude solutions of the original problem

Neumann Layer Phenomena in Nonlinear Diffusion Systems

(\varepsilon > 0)

Propagation speeds of traveling fronts for higher order autocatalytic reaction-diffusion systems

, which possess transition layers. As an application, a model system of prey-predator interaction with diffusion is considered.

The Minimal Propagation Speed of Travelling Waves for Autocatalytic Reaction-Diffusion Equations

In this paper, we show the existence and stability of monotone traveling wave solutions for some nonlinear parabolic equations arising in population dynamics. Using these traveling wave solutions as comparison functions, we investigate the asymptotic behavior of solutions of the pure initial value problem and estimate the support of the solutions.

The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model

Abstract This article is intended to survey the results about pattern formation in a class of reaction-diffusion systems. The focus is on the phenomenon of multiple existence of stable stationary solutions, which has biologically or physically significant consequences. The mathematical structure and stability of stationary solutions is investigated in a certain parameter space. Especially, σ-local stability and instability theorems for D 1 -sheet are given, and stabilization of D 2 -sheet is proved via two approaches: the spectral method and the singular perturbation-theoretic one.

Singular perturbation approach to traveling waves in competing and diffusing species models

The stationary problem of a nonlinear degenerate diffusion equation with nonlocal advection term including the aggregative mechanism is investigated. The spatially clustering phenomena of individuals in population dynamics is modeled. The existence of two kinds of stationary solutions with connected compact support—a large as well as a small cluster solution—is proved by the matched asymptotic expansions method and the geometric singular perturbation method, respectively.