Kousuke Kuto

Positive solutions for Lotka–Volterra competition systems with large cross-diffusion

This paper discusses the stationary problem for the Lotka–Volterra competition systems with cross-diffusion under homogeneous Dirichlet boundary conditions. Although some sufficient conditions for the existence of positive solutions are known, the information for their structure is far from complete. In order to get better understanding of the competition system with cross-diffusion, we study the effects of large cross-diffusion on the structure of positive solutions and focus on the limiting behaviour of positive solutions by letting one of the cross-diffusion coefficients to infinity. Especially, it will be shown that positive solutions of the competition system converge to a positive solution of a suitable limiting system. We will also derive some satisfactory results on positive solutions for this limiting system. These results give us important information on the structure of positive solutions for the competition system when one of the cross-diffusion coefficients is sufficiently large.

Numerical Analysis of Optical Waveguides Based on Periodic Fourier Transform

Periodic Fourier transform is formally introduced to anal- yses of the electromagnetic wave propagation in optical waveguides. The transform make the field components periodic and they are then expanded in Fourier series without introducing an approximation of artificial periodic boundary. The proposed formulation is applied to two-dimensional slab waveguide structures, and the numerical results evaluate the validity and show some properties of convergence.

Limiting Structure of Shrinking Solutions to the Stationary Shigesada--Kawasaki--Teramoto Model with Large Cross-Diffusion

This paper is concerned with the limiting behavior of coexistence steady states of the Lotka--Volterra competition model as a cross-diffusion term tends to infinity. Under the Neumann boundary condition, Lou and Ni [J. Differential Equations, 154 (1999), pp. 157--190] derived a couple of limiting systems, which characterize the limiting behavior of coexistence steady states. One of two limiting systems characterizing the segregation of the competing species has been studied by Lou, Ni, and Yotsutani [Discrete Contin. Dyn. Syst., 10 (2004), pp. 435--458], and their work revealed the detailed bifurcation structure for the one-dimensional (1D) case. This paper focuses on the other limiting system characterizing the shrinkage of the species which is not endowed with the cross-diffusion effect. The bifurcation structure of positive solutions to the limiting system is stated. In particular, for the 1D case, we obtain a global connected set of solutions that bifurcates from a point on the line of constant soluti...

Stationary patterns for an adsorbate-induced phase transition model: II. Shadow system

This paper is concerned with stationary solutions of a reaction–diffusion-advection system arising in surface chemistry. Hildebrand et al (2003 New J. Phys. 5 61) have constructed stationary stripe (or spot) solutions of the system in the singular perturbation case and shown a numerical result that the set of stripe (or spot) solutions forms a saddle-node bifurcation curve with respect to a diffusion coefficient. In this paper, we introduce a shadow system in the limiting case that another diffusion and an advection coefficient tend to infinity. Furthermore we obtain the bifurcation structure of stationary solutions of the shadow systems in the one-dimensional case. This structure involves saddle-node bifurcation curves which support the above numerical result in Hildebrand et al (2003 New J. Phys. 5 61, figure 9). Our proof is based on the combination of the bifurcation, the singular perturbation and a level set analysis.

Coexistence states for a prey-predator model with cross-diffusion

This paper discusses a prey-predator system with cross-diffusion. We can prove that the set of coexistence steady-states of this system contains an S or

Multiple coexistence states for a prey-predator system with cross-diffusion

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Stability of steady-state solutions to a prey–predator system with cross-diffusion☆

-shaped branch with respect to a bifurcation parameter in a large cross-diffusion case. We give also some criteria on the stability of these positive steady-states. Furthermore, we find the Hopf bifurcation point on the steady-state solution branch in a certain case.