Updated on 2024/05/20

写真a

 
KUTO, Kousuke
 
Affiliation
Faculty of Science and Engineering, School of Fundamental Science and Engineering
Job title
Professor
Degree
修士(理学) ( 早稲田大学 )
博士(理学) ( 早稲田大学 )

Research Experience

  • 2019.04
    -
    Now

    Waseda University   Faculty of Science and Engineering

  • 2016.04
    -
    2019.03

    The University of Electro-Communications   Faculty of Informatics and Engineering   Professor

  • 2010.10
    -
    2016.03

    The University of Electro-Communications   Faculty of Informatics and Engineering   Associate Professor

  • 2007.04
    -
    2010.09

    Fukuoka Institute of Technology   Faculty of Engineering   Associate Professor

  • 2005.04
    -
    2007.03

    Fukuoka Institute of Technology   Faculty of Engneering   Lecturer

  • 2004.04
    -
    2005.03

    Waseda University   Faculty of Science and Engineering   Assistant Professor

  • 2000.04
    -
    2003.03

    Japan Society for the Promotion of Science   Research Fellowship for Young Scientists

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Education Background

  • 2000.04
    -
    2003.07

    Waseda University   Graduate School, Division of Science and Engineering   Department of Mathematical Sciences  

Professional Memberships

  •  
     
     

    The Mathematical Society of Japan

  •  
     
     

    THE JAPAN SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS

Research Areas

  • Basic analysis / Mathematical analysis

Research Interests

  • degree

  • singular limit

  • spiky solution

  • transition layer

  • bifurcation

  • reaction diffusion equations

  • nonlinear diffusion

  • 写像度

  • 特異摂動

  • スパイク解

  • 遷移層

  • 分岐

  • 反応拡散方程式

  • 非線形拡散

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Papers

  • Coexistence-segregation dichotomy in the full cross-diffusion limit of the stationary SKT model

    Jumpei Inoue, Kousuke Kuto, Homare Sato

    Journal of Differential Equations   373   48 - 107  2023.11  [Refereed]

    DOI

    Scopus

    1
    Citation
    (Scopus)
  • Global structure of steady-states to the full cross-diffusion limit in the Shigesada-Kawasaki-Teramoto model

    Kousuke Kuto

    Journal of Differential Equations   333   103 - 143  2022.10  [Refereed]

    DOI

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    3
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    (Scopus)
  • Bifurcation structure of coexistence states for a prey–predator model with large population flux by attractive transition

    Kousuke Kuto, Kazuhiro Oeda

    Proceedings of the Royal Society of Edinburgh: Section A Mathematics   152 ( 4 ) 965 - 988  2022.08  [Refereed]

     View Summary

    This paper is concerned with a prey–predator model with population flux by attractive transition. Our previous paper (Oeda and Kuto, 2018, Nonlinear Anal. RWA, 44, 589–615) obtained a bifurcation branch (connected set) of coexistence steady states which connects two semitrivial solutions. In Oeda and Kuto (2018, Nonlinear Anal. RWA, 44, 589–615), we also showed that any positive steady-state approaches a positive solution of either of two limiting systems, and moreover, one of the limiting systems is an equal diffusive competition model. This paper obtains the bifurcation structure of positive solutions to the other limiting system. Moreover, this paper implies that the global bifurcation branch of coexistence states consists of two parts, one of which is a simple curve running in a tubular domain near the set of positive solutions to the equal diffusive competition model, the other of which is a connected set characterized by positive solutions to the other limiting system.

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  • Full cross-diffusion limit in the stationary Shigesada-Kawasaki-Teramoto model

    Kousuke Kuto

    Annales de l'Institut Henri Poincaré C, Analyse non linéaire   38 ( 6 ) 1943 - 1959  2021.12  [Refereed]

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    6
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  • On the unboundedness of the ratio of species and resources for the diffusive logistic equation

    Jumpei Inoue, Kousuke Kuto

    Discrete & Continuous Dynamical Systems - B   26 ( 5 ) 2441 - 2450  2021.05  [Refereed]

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    14
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  • Impact of Regional Difference in Recovery Rate on the Total Population of Infected for a Diffusive SIS Model

    Jumpei Inoue, Kousuke Kuto

    Mathematics   9 ( 8 ) 888 - 888  2021.04  [Refereed]

     View Summary

    This paper is concerned with an SIS epidemic reaction-diffusion model. The purpose of this paper is to derive some effects of the spatial heterogeneity of the recovery rate on the total population of infected and the reproduction number. The proof is based on an application of our previous result on the unboundedness of the ratio of the species to the resource for a diffusive logistic equation. Our pure mathematical result can be epidemically interpreted as that a regional difference in the recovery rate can make the infected population grow in the case when the reproduction number is slightly larger than one.

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  • Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation

    Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani

    Discrete & Continuous Dynamical Systems - A   40 ( 8 ) 4907 - 4925  2020  [Refereed]

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    2
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  • Positive steady states for a prey-predator model with population flux by attractive transition

    Kazuhiro Oeda, Kousuke Kuto

    Nonlinear Analysis: Real World Applications   44   589 - 615  2018.07  [Refereed]

  • Global solution branches for a nonlocal Allen-Cahn equation

    Kousuke Kuto, Tatsuki Mori, Tohru Tsujikawa, Shoji Yotsutani

    Journal of Differential Equations   264 ( 9 ) 5928 - 5949  2018.02  [Refereed]

  • Bifurcation structure of stationary solutions for a chemotaxis system with bistable growth

    Hirofumi Izuhara, Kousuke Kuto, Tohru Tsujikawa

    Japan Journal of Industrial and Applied Mathematics   Online First Articles   1 - 35  2018.01  [Refereed]

  • Concentration profile of endemic equilibrium of a reaction-diffusion-advection SIS epidemic model

    Kousuke Kuto, Hiroshi Matsuzawa, Rui Peng

    Calculus of Variations and Partial Differential Equations   56 ( Article No. 112 ) 28 - pages  2017.07  [Refereed]

  • Secondary bifurcation for a nonlocal Allen-Cahn equation

    Kousuke Kuto, Tatsuki Mori, Tohru Tsujikawa, Shoji Yotsutani

    Journal of Differential Equations   263 ( 5 ) 2687 - 2714  2017.04  [Refereed]

  • EXACT MULTIPLICITY OF STATIONARY LIMITING PROBLEMS OF A CELL POLARIZATION MODEL

    Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani

    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS   36 ( 10 ) 5627 - 5655  2016.10  [Refereed]

     View Summary

    We show existence, nonexistence, and exact multiplicity for stationary limiting problems of a cell polarization model proposed by Y. Mori, A. Jilkine and L. Edelstein-Keshet. It is a nonlinear boundary value problem with total mass constraint. We obtain exact multiplicity results by investigating a global bifurcation sheet which we constructed by using complete elliptic integrals in a previous paper.

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    7
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  • Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization

    Tatsuki Mori, Kousuke Kuto, Masaharu Nagayama, Tohru Tsujikawa, Shoji Yotsutani

    Discrete and Continuous Dynamical Systems   Special Issue   861 - 877  2015.11  [Refereed]  [Invited]

    DOI

  • STATIONARY SOLUTIONS FOR SOME SHADOW SYSTEM OF THE KELLER-SEGEL MODEL WITH LOGISTIC GROWTH

    Tohru Tsujikawa, Kousuke Kuto, Yasuhito Miyamoto, Hirofumi Izuhara

    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S   8 ( 5 ) 1023 - 1034  2015.10  [Refereed]

     View Summary

    From a viewpoint of the pattern formation, the Keller-Segel system with the growth term is studied. This model exhibited various static and dynamic patterns caused by the combination of three effects, chemotaxis, diffusion and growth. In a special case when chemotaxis effect is very strong, some numerical experiment in [1],[22] showed static and chaotic patterns. In this paper we consider the logistic source for the growth and a shadow system in the limiting case that a diffusion coefficient and chemotactic intensity grow to infinity. We obtain the global structure of stationary solutions of the shadow system in the one-dimensional case. Our proof is based on the bifurcation, singular perturbation and a level set analysis. Moreover, we show some numerical results on the global bifurcation branch of solutions by using AUTO package.

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    3
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  • Limiting structure of steady-states to the Lotka-Volterra competition model with large diffusion and advection

    Kousuke Kuto, Tohru Tsujikawa

    JOURNAL OF DIFFERENTIAL EQUATIONS   258 ( 5 ) 1801 - 1858  2015.03  [Refereed]

     View Summary

    This paper is concerned with the Neumann problem of a stationary Lotka-Volterra competition model with diffusion and advection. First we obtain some sufficient conditions of the existence of nonconstant solutions by the Leray-Schauder degree theory. Next we derive a limiting system as diffusion and advection of one of the species tend to infinity. The limiting system can be reduced to a semilinear elliptic equation with nonlocal constraint. In the simplified 1D case, the global bifurcation structure of nonconstant solutions of the limiting system can be classified depending on the coefficients. For example, this structure involves a global bifurcation curve which connects two different singularly perturbed states (boundary layer solutions and internal layer solutions). Our proof employs a levelset analysis for the associate integral mapping. (C) 2014 Elsevier Inc. All rights reserved.

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    26
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  • LIMITING STRUCTURE OF SHRINKING SOLUTIONS TO THE STATIONARY SHIGESADA-KAWASAKI-TERAMOTO MODEL WITH LARGE CROSS-DIFFUSION

    Kousuke Kuto

    SIAM JOURNAL ON MATHEMATICAL ANALYSIS   47 ( 5 ) 3993 - 4024  2015  [Refereed]

     View Summary

    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 solutions and blows up where the birth rate of the species is equal to the least positive eigenvalue of -Delta subject to the homogeneous Neumann boundary condition.

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    22
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  • Bifurcation structure of steady-states for bistable equations with nonlocal constraint

    Kousuke Kuto, Tohru Tsujikawa

    Discrete and Continuous Dynamical Systems   Suppl. Vol.   455 - 464  2013.11  [Refereed]

  • Bifurcation structure of stationary layers for generalized Allen-Cahn equations with nonlocal constraint

    Kousuke Kuto

    数理解析研究所講究録   1838   102 - 115  2013.06

    CiNii

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

    Kousuke Kuto, Tohru Tsujikawa

    Nonlinearity   26 ( 5 ) 1313 - 1343  2013.05  [Refereed]

     View Summary

    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. © 2013 IOP Publishing Ltd &amp
    London Mathematical Society.

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    6
    Citation
    (Scopus)
  • ON LIMIT SYSTEMS FOR SOME POPULATION MODELS WITH CROSS-DIFFUSION

    Kousuke Kuto, Yoshio Yamada

    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B   17 ( 8 ) 2745 - 2769  2012.11  [Refereed]

     View Summary

    This paper deals with the following reaction-diffusion system
    (SP) {Delta[(1 + alpha nu)u] + u(a - u - cv) = 0
    Delta[(1 + beta u)v] + v(b - du - v) = 0
    in a bounded domain of R-N with homogeneous Neumann boundary conditions or Dirichlet boundary conditions. Our main purpose is to understand the structure of positive solutions of (SP) and know the effects of cross-diffusion coefficients alpha and beta. For this purpose, our strategy is to study limiting behavior of positive solutions when alpha or beta goes to infinity and derive the corresponding limit systems. We will obtain a priori estimates of u and v independently of beta (resp. alpha) with small alpha >= 0 (resp. beta >=> 0) in case 1 <= N <= 3 under Neumann boundary conditions, while we will obtain a priori estimates of u and v independently of alpha and beta in case 1 <= N <= 5 under Dirichlet boundary conditions. These a priori estimates allow us to investigate limiting behavior of positive solutions. When alpha = 0 and beta -> infinity, we can derive two limit systems for Neumann conditions and one limit system for Dirichlet conditions. We will also give some results on the structure of positive solutions for such limit systems.

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    18
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  • Spatial pattern formation in a chemotaxis-diffusion-growth model

    Kousuke Kuto, Koichi Osaki, Tatsunari Sakurai, Tohru Tsujikawa

    PHYSICA D-NONLINEAR PHENOMENA   241 ( 19 ) 1629 - 1639  2012.10  [Refereed]

     View Summary

    Minima and one of the authors (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 (2007) [22] obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution, while Kurata et al. (2008) numerically showed several spatio-temporal patterns in a rectangle. Motivated by their work, we consider some qualitative behaviors of stationary solutions from global and local (bifurcation) viewpoints in the present paper. First we study the asymptotic behavior of stationary solutions as the chemotactic intensity grows to infinity. Next we construct local bifurcation branches of stripe and hexagonal stationary solutions in the special case when the habitat domain is a rectangle. For this case, the directions of the branches near the bifurcation points are also obtained. Finally, we exhibit several numerical results for the stationary and oscillating patterns. (C) 2012 Elsevier B.V. All rights reserved.

    DOI

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    68
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  • Universal bound for stationary patterns of an adsorbate-induced phase transition model

    Kousuke Kuto, Tohru Tsujikawa

    Journal of Math-for-Industry   3 ( C-9 ) 69 - 72  2011.12  [Refereed]

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  • STATIONARY PATTERNS FOR AN ADSORBATE-INDUCED PHASE TRANSITION MODEL I: EXISTENCE

    Kousuke Kuto, Tohru Tsujikawa

    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B   14 ( 3 ) 1105 - 1117  2010.10  [Refereed]

     View Summary

    We are concerned with a reaction-diffusion-advection system proposed by Hildebrand [4]. This system is a phase transition model arising in surface chemistry. For this model, several stationary patterns have been shown by the numerical simulations (e. g., [15]). In the present paper, we obtain sufficient conditions for the existence (or nonexistence) of nonconstant stationary solutions. Our proof is based on the Leray-Schauder degree theory. Some a priori estimates for solutions play an important role in the proof.

    DOI

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    5
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  • Positive solutions for Lotka-Volterra competition systems with large cross-diffusion

    Kousuke Kuto, Yoshio Yamada

    APPLICABLE ANALYSIS   89 ( 7 ) 1037 - 1066  2010  [Refereed]

     View Summary

    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.

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    38
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  • Coexistence problem for a prey-predator model with density-dependent diffusion

    Kousuke Kuto, Yoshio Yamada

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   71 ( 12 ) E2223 - E2232  2009.12  [Refereed]

     View Summary

    We study a prey-predator model with nonlinear diffusions. In a case when the spatial dimension is less than 5, a universal bound for coexistence steady-states is found. By using the bound and the bifurcation theory, we obtain the bounded continuum of coexistence steady-states. (C) 2009 Elsevier Ltd. All rights reserved.

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    21
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  • LIMITING CHARACTERIZATION OF STATIONARY SOLUTIONS FOR A PREY-PREDATOR MODEL WITH NONLINEAR DIFFUSION OF FRACTIONAL TYPE

    Kousuke Kuto, Yoshio Yamada

    DIFFERENTIAL AND INTEGRAL EQUATIONS   22 ( 7-8 ) 725 - 752  2009.07  [Refereed]

     View Summary

    We consider the following quasilinear elliptic system:
    {Delta u + u(a - u - cv) = 0 in Omega, Delta[(1 + gamma/1+beta u)v] + v(b + du - v) = 0 in Omega, u = v = 0 on partial derivative Omega,
    where Omega is a bounded domain in R(N). This system is a stationary problem of a prey-predator model with non-linear diffusion Delta(v/1 + beta u) and u (respectively v) denotes the population density of the prey (respectively the predator). Kuto [15] has studied this system for large beta under the restriction b > (1 + gamma)lambda(1), where lambda(1) is the least eigenvalue of -Delta with homogeneous Dirichlet boundary condition. The present paper studies two shadow systems and gives the complete limiting characterization of positive solutions as beta -> infinity without any restriction on b.

  • STABILITY AND HOPF BIFURCATION OF COEXISTENCE STEADY-STATES TO AN SKT MODEL IN SPATIALLY HETEROGENEOUS ENVIRONMENT

    Kousuke Kuto

    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS   24 ( 2 ) 489 - 509  2009.06  [Refereed]

     View Summary

    This paper is concerned with the following Lotka-Volterra cross-diffusion system
    ut = Delta[(1 + k rho(x)v)u] + u(a - u - c(x)v) in Omega x (0, infinity),
    tau v(t) = Delta v + v(b + d(x)u - v) in Omega x (0, infinity)
    in a bounded domain Omega subset of R(N) with Neumann boundary conditions partial derivative(nu)v = 0 on partial derivative Omega. In the previous paper [18], the author has proved that the set of positive stationary solutions forms a fishhook shaped branch Gamma under a segregation of rho(x) and d(x). In the present paper, we give some criteria on the stability of solutions on Gamma. We prove that the stability of solutions changes only at every turning point of Gamma if tau is large enough. In a different case that c(x) > 0 is large enough, we find a parameter range such that multiple Hopf bifurcation points appear on Gamma.

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    14
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  • Bifurcation branch of stationary solutions for a Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment

    Kousuke Kuto

    NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS   10 ( 2 ) 943 - 965  2009.04  [Refereed]

     View Summary

    This paper is concerned with the following Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment
    (SP) {Delta[(1+k rho(x)v)u] + u(a - u - c(x)v) = 0 in Omega,
    Delta v + v(b + d(x)u - v) = 0 in Omega,
    partial derivative(v)u = partial derivative(v)u = 0 on partial derivative Omega.
    Here Omega is a bounded domain in R(N)(N <= 3), a and k are positive constants, b is a real constant, c(x) > 0 and d(x) >= 0 are continuous functions and rho(x) > 0 is a smooth function with partial derivative(v)rho = 0 on partial derivative Omega. From a viewpoint of the mathematical ecology, unknown functions u and v, respectively, represent stationary population densities of prey and predator which interact and migrate in Omega. Hence, the set Gamma(rho) of positive solutions (with bifurcation parameter b) forms a bounded line in a spatially homogeneous case that rho, c and d are constant. This paper proves that if a and |b| are small and k is large, a spatial segregation of rho(x) and d(x) causes Gamma(rho) to form a subset of-shaped curve with respect to b. A crucial aspect of the proof involves the solving of a suitable limiting systent as a, |b| -> 0 and k -> infinity by using the bifurcation theory and the Lyapunov-Schmidt reduction. (C) 2007 Elsevier Ltd. All rights reserved.

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    36
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  • Shadow system for adsorbate-induced phase transition model

    Kousuke Kuto, Tohru Tsujikawa

    RIMS Kôkyûroku Bessatsu   B15   1 - 14  2009  [Refereed]

  • Bifurcation structure of steady-states for an adsorbate-induced phase transition model

    Kousuke Kuto

    数理解析研究所講究録   1640   129 - 143  2009

    CiNii

  • Bifurcation phenomena of pattern solution to Mimura-Tsujikawa model in one dimension

    Naoko Kurata, Kousuke Kuto, Koichi Osaki, Tohru Tsujikawa, Tatsunari Sakurai

    GAKUTO International Series. Mathematical Sciences and Applications   29   265 - 278  2008  [Refereed]

    CiNii

  • A strongly coupled diffusion effect on the stationary solution set of a prey-predator model

    Kousuke Kuto

    Advances in Differential Equations   12 ( 2 ) 145 - 172  2007  [Refereed]

  • Pattern formation for adsorbate-induced phase transition model

    Kousuke Kuto, Tohru Tsujikawa

    RIMS Kôkyûroku Bessatsu   B3   43 - 58  2007  [Refereed]

  • A Lotka-Volterra cross-diffusion model in spatially heterogeneous environments

    Kousuke Kuto

    数理解析研究所講究録   1542   41 - 57  2007

    CiNii

  • Positive steady states for a prey-predator model with some nonlinear diffusion terms

    Tomohito Kadota, Kousuke Kuto

    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   323 ( 2 ) 1387 - 1401  2006.11  [Refereed]

     View Summary

    This paper discusses a prey-predator system with strongly coupled nonlinear diffusion terms. We give a sufficient condition for the existence of positive steady state solutions. Our proof is based on the bifurcation theory. Some a priori estimates for steady state solutions will play an important role in the proof. (c) 2005 Elsevier Inc. All rights reserved.

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    52
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  • Numerical analysis of optical waveguides based on periodic Fourier transform

    K. Watanabe, K. Kuto

    PROGRESS IN ELECTROMAGNETICS RESEARCH-PIER   64   1 - 21  2006  [Refereed]

     View Summary

    Periodic Fourier transform is formally introduced to analyses 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.

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    13
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  • Bifurcation structure of the stationary solution set to a strongly coupled diffusion system

    Kousuke Kuto

    数理解析研究所講究録   1475   73 - 90  2006

    CiNii

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

    Kousuke Kuto, Yoshio Yamada

    Discrete and Continuous Dynamical Systems   Suppl. Vol.   536 - 545  2005  [Refereed]

  • Positive solutions to some cross-diffusion systems in population dynamics

    Kousuke Kuto

    数理解析研究所講究録   1416   64 - 84  2005

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

    K Kuto, Y Yamada

    JOURNAL OF DIFFERENTIAL EQUATIONS   197 ( 2 ) 315 - 348  2004.03  [Refereed]

     View Summary

    We study the multiple existence of positive solutions for the following strongly coupled elliptic system:
    Delta[(1 + alphaupsilon)u] + u(a - u - cupsilon) = 0 in Omega,
    Delta[(1 + betau)upsilon] + upsilon(b + du - upsilon) = 0 in Omega,
    u = v = 0 on Omega,
    where alpha, beta, a, b, c, d are positive constants and Omega is a bounded domain in R-N. This is the steadystate problem associated with a prey-predator model with cross-diffusion effects and u (resp. upsilon) denotes the population density of preys (resp. predators). In particular, the presence of beta represents the tendency of predators to move away from a large group of preys. Assuming that a is small and that beta is large, we show that the system admits a branch of positive solutions, which is S or D shaped with respect to a bifurcation parameter. So that the system has two or three positive solutions for suitable range of parameters. Our method of analysis uses the idea developed by Du-Lou (J. Differential Equations 144 (1998) 390) and is based on the bifurcation theory and the Lyapunov-Schmidt procedure. (C) 2003 Elsevier Inc. All rights reserved.

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

    K Kuto

    JOURNAL OF DIFFERENTIAL EQUATIONS   197 ( 2 ) 293 - 314  2004.03  [Refereed]

     View Summary

    This paper is concerned with a cross-diffusion system arising in a prey-predator population model. The main purpose is to discuss the stability analysis for coexistence steady-state solutions obtained by Kuto and Yamada (J. Differential Equations, to appear). We will give some criteria on the stability of these coexistence steady states. Furthermore, we show that the Hopf bifurcation phenomenon occurs on the steady-state solution branch under some conditions. (C) 2003 Elsevier Inc. All rights reserved.

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    86
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  • Multiple existence and stability of steady-states for a prey-predator system with cross-diffusion

    Kousuke Kuto, Yoshio Yamada

    Banach Center Publications   66   536 - 545  2004  [Refereed]

  • Large-time behavior of solutions of diffusion equations with concave-convex reaction term

    Kousuke Kuto

    Advances in Mathematical Sciences and Applications   12 ( 1 ) 307 - 325  2002  [Refereed]

  • Stabilization of solutions of the diffusion equation with a non-Lipschitz reaction term

    K Kuto

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   47 ( 2 ) 789 - 800  2001.08  [Refereed]

     View Summary

    In this paper we are concerned with the reaction-diffusion equation u(t) = Deltau + f(u) in a ball of RN with Dirichlet boundary condition. We assume that f satisfies the concave-convex condition. A typical example is f (u) = \u \ (q-1)u + \u \ (p-1)u (0 < q < 1 < p < (N+2)/(N-2)). First we obtain the complete structure of positive solutions to the stationary problem; Delta phi + f (phi) = 0. Next we state the relations between this structure and time-depending behaviors of nonnegative solutions (global existence or blow up) to the non-stationary problem.

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  • Diffusion problems with concave-convex nonliearities

    Kousuke Kuto

    数理解析研究所講究録   1237   83 - 96  2001

    CiNii

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Research Projects

  • 交差拡散を伴う数理生物学モデルの近平衡系に対する解析基盤の構築

    日本学術振興会  科学研究費助成事業 基盤研究(C)

    Project Year :

    2022.04
    -
    2025.03
     

    久藤 衡介

  • ロトカ・ボルテラ系における交差拡散極限が導く定常解の多層構造の解明

    日本学術振興会  科学研究費助成事業 基盤研究(C)

    Project Year :

    2019.04
    -
    2023.03
     

    久藤 衡介

     View Summary

    有界領域において共通の資源を取り合う関係にある2種類の競争種にとって,合理的に資源を摂取するには,競争相手の種が多い場所ほど空間的拡散を促進させた方が好戦略に思われる.この戦略を「交差拡散」とよばれる拡散の相互作用を表す非線形項として,従来のロトカ・ボルテラ系に加味したモデルが,重定,川崎,寺本(1979)によって提唱され,現在では「SKTモデル」とよばれている.SKTモデルにおいては,定常問題の解構造の解明が重要な問題として残されており,本研究課題においては,交差拡散係数を増大させたときの,定常解の大域分岐構造の導出を主目標としてきた.
    前年度までの研究によって,両種の交差拡散係数を同程度に大きくしたとき,ノイマン境界条件の下では,競争種どうしの空間的棲み分けは不完全排他の形で起こることが分かった,すなわち,片方の種の生存地域において,競争相手である他種のが,比較的少ないながらも生存している状況が定常解で再現された.さらに,非定数な定常解の集合は,定数解からのピッチフォーク分岐枝で構成されていることが分かった.
    当該年度においては,ディレクレ境界条件の下で,両種の交差拡散係数を同程度の大きくすると,定常解の大域分岐枝は「競争種どうしの完全排他的な棲み分けを特徴づける部分」と「両種がともに領域の中心付近に小さいピークをとる小丘共存を特徴づける部分」が繋がる形で出現することが分かった.非線形楕円型方程式の解構造の観点では,小丘共存の部分は零解からの1次分岐枝で構成されていて,その枝から対称性を壊す2次分岐もしくは不完全分岐が起こり,完全排他の部分が2次分岐枝で構成されることを明らかにした.
    また,attractive transition 型の非線形拡散項を伴うロトカ・ボルテラ共生系の定常解の研究に従事し,非定数定常解が出現するメカニズムを分岐理論の立場で明らかにしている.

  • 非線形拡散を伴うSKTモデルに現れる定常パターンの大域分岐構造

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2015.04
    -
    2019.03
     

    Kuto Kousuke, YOTSUTANI shoji

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    In this research, we studied a reaction-diffusion system that describes the spatiotemporal population dynamics of two competitive species. Individually, for the Lotka-Volterra system with a non-linear diffusion term called the cross diffusion term (the Shigesada-Kawasaki-Teramoto model, 1979), the asymptotic behavior of the steady-state solutions was analyzed as a coefficient of the cross diffusion term tends to infinity. We clarified the global bifurcation structure of the solution for the approximation problem called "the second limit system," for which there had been not many results. This result revealed that the steady-state solution set of the Shigesada-Kawasaki-Teramoto model forms a hook-like curve called a saddle-node bifurcation curve.

  • Nonlinear analysis for a parabolic-parabolic chemotaxis-growth system of equations

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2014.04
    -
    2019.03
     

    Osaki Koichi, Nakaguchi Etsushi, Tsujikawa Tohru, Kuto Kousuke, Akagi Goro

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    We studied a parabolic-parabolic chemotaxis system with logistic growth. We showed the global-in-time existence of solutions to the chemotaxis system which has subquadratic degradation and nonlinear secretion. In addition, we showed the bifurcation of nontrivial solutions from the uniform state of the system, which indicates pattern formations, for instance, hexagonal and regular nesting patterns.

  • Elucidation of phenomena in the higher dimensional domain applying the reduced system and construction of the mathematical method

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2014.04
    -
    2018.03
     

    Tsujikawa Tohru, KUTO Kousuke, EI Shin-ichiro, SAKURAI Tatsunari

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    The study of Reaction-Diffusion Equation is important to elucidate the pattern formation. This research is to determine the global structure of nonconstant stationary solutions of Lotka-Volterra competition model, which describes the population dynamics of some biology. Under Neumann boundary condition, we show the sufficient condition of the existence of nonconstant solutions for coefficient parameters by Leray-Shauder degree theory. On the other hand, we know that the solution structure is complex by numerical computations. In order to show the global solution structure, we introduce a limiting system by using some reduction to the model equation. It is a scalar equation with an integral constraint. Since the solution structure of this scalar equation is well known by the bifurcation theory, we obtain the global solution structure due to solve the integral constraint by using Levelset analysis.

  • Study on reaction-diffusion equations and related free boundary problems

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2012.04
    -
    2015.03
     

    YAMADA Yoshio, OTANI Mitsuharu, TANAKA Kazunaga, HIROSE Munemitsu, NAKASHIMA Kimie, TAKEUCHI Shingo, KUTO Kousuke, WAKASA Tohru, OEDA Kazuhiro, KANEKO Yuki

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    This research is concerned with a free boundary problem for reaction-diffusion equations in mathematical ecology. This problem models the invasion or migration of a certain biological species. Our main interest is to study the evolution of the population density and habitat of the species. The population density is described by a reaction-diffusion equation and the boundary (or a part of the boundary) of the habitat is controlled by a free boundary condition of Stefan type. We could obtain theoretical understanding on asymptotic behaviors of solutions for free boundary problems of various types: whether the species vanishes eventually or the species persists with spreading free boundary. Moreover, we got precise results on the spreading speed of the free boundary.

  • Mathematical analysis for the Lotka-Volterra system with nonlinear diffusion

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)

    Project Year :

    2012.04
    -
    2015.03
     

    KOUSUKE Kuto

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    This research studied the global structure of stationary solutions to the Lotka-Volterra system with nonlinear diffusion. Among other things, this research focused on the limiting system as the nonlinear diffusion term tends to infinity, which characterizes the limiting behavior of stationary solutions, and derived the curve of the set of non-constant solutions to the limiting system (the global bifurcation curve) in a functional space. As an example of results, this research proved that an element of unknown functions blows up as a bifurcation parameter approaches the second eigenvalue of the Laplace operator.

  • Analysis of Reaction-Diffusion Systems and Related Nonlinear Problems

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2009
    -
    2011
     

    YAMADA Yoshio, OTANI Mitsuharu, TANAKA Kazunaga, HIROSE Munemitsu, NAKASHIMA Kimie, TAKEUCHI Shingo, KUTO Kousuke, WAKASA Tohru, OHYA Hirokazu, OEDA Kazuhiro, SATO Norihiro

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    This research project is concerned with the mathematical formulation of non-uniformity of species in mathematical ecology such as the segregation of two competing species and the spreading of invasive species. These phenomena are described by reaction-diffusion equations with population densities as unknown functions. We have obtained satisfactory results on the structure of positive steady-states for two-species models with nonlinear diffusion and the mechanism of spreading and vanishing for free boundary problems in invasion models.

  • Study on spatial pattern solutions for the Lotka-Volterra system with advection

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)

    Project Year :

    2009
    -
    2011
     

    KUTO Kousuke

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    We derive some mathematical information on the global bifurcation structure of stationary solutions to the Lotka-Volterra system with nonlinear diffusion. First we study the Lotka-Volterra competition system with cross-diffusion under the Dirichlet boundary conditions to obtain the global bifurcation branch of a subset of stationary solutions to the limiting system as a cross-diffusion term tends to infinity. We also study a reaction-diffusion-advection system related to the Lotka-Volterra system and obtain the global bifurcation structure of stationary solutions to the limiting system as the diffusion and advection terms tend to infinity.

  • Research on the structure of solutions for nonlinear systems of reaction-diffusion equations

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2006
    -
    2008
     

    YAMADA Yoshio, OTANI Mitsuharu, TANAKA Kazunaga, NAKASHIMA Kimie, TAKEUCHI Shingo, KUTO Kousuke, OHYA Hirokazu, SATO Norihiro, WAKASA Tohru

  • Study on the structure of solutions for the Lotka?Volterra system with cross?diffusion

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)

    Project Year :

    2006
    -
    2008
     

    KUTO Kousuke

  • Research of System of Nonlinear Diffusion Equations and Related Elliptic Differential Equations

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2003
    -
    2005
     

    YAMADA Yoshio, OTANI Mitsuharu, TANAKA Kazunaga, NAKASHIMA Kimie, TAKEUCHI Shingo, KUTO Kousuke

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    In this project, we have studied the structure of solutions for the following two types of equations : (a) reaction diffusion systems with nonlinear diffusion in mathematical biology and (b) semilinear diffusion equations describing phase transition phenomena
    The first problem in mathematical biology is given by a system of differential equations with quasilinear diffusion of the form
    u_t=Δ[φ(u,v)u]+au(1-u-v), v_t=Δ[ψ(u,v)v]+bv(1+du-v),
    under homogeneous Dirichlet boundary conditions. Here u and v denote population densities of prey and predator species, respectively. It is well known that the corresponding stationary problem has a positive steady-state under a suitable condition. Our main interest is to derive useful information on profile and stability of each positive steady-state. In case φ(u,v)=1 and 4,φ(u,v=1+β u, we have shown that the stationary problem has at least three positive solutions if β is sufficiently large and some other conditions are imposed. Moreover, stability or instability of each positive solution is also investigated.
    The second problem is given by u_t=ε^2u_<xx>+u(1-u)(u-a(x)) with homogeneous Neumann boundary condition, where 0<a(x)<1. When ε is sufficiently small, it is known that this problem admits various kinds of steady-state solutions. In particular, we are interested in steady state with transition layers and spikes. Here transition layer for a solution means a part of u(x) where u(x) drastically changes from 0 to 1 or 1 to 0 in a very short interval. Such oscillating solutions have been studied by Ai-Chen-Hastings and our group, independently. It has been proved that any transition layer appears only in a neighborhood of x such that a(x)=1/2 and that any spike appears only in a neighborhood of x such that a(x) takes its local maximum or minimum. We have also established more information on profiles of multi-transition layers and multi-spikes, their location and the relationship between profile and stability of steady-state solution with transition layers.

  • 滑らかでない反応項を伴う非線型放物型方程式及び楕円型方程式の解構造の研究

    日本学術振興会  科学研究費助成事業 特別研究員奨励費

    Project Year :

    2000
    -
    2002
     

    久藤 衡介

     View Summary

    平成14年度においては、非線形放物型偏微分方程式、および関連する非線形楕円型偏微分方程式に対する解構造の研究に従事した。とりわけ、非線形項を滑らかな関数に限らない点が当該研究の特徴である。滑らかさを欠く非線形項を伴う微分方程式は、被食生物-捕食生物系などの多くの数理現象のモデル方程式に現れうる。一方で、数学的には「滑らかな非線形項を伴う微分方程式」と比べて解析が著しく困難になるケースが多く、未解決問題を多く残しており、その解明のため平成12年度より研究を推進してきた。その環として本年度は、被食生物(プレイ)と捕食生物(プレデター)の個体数密度のダイナミクスを記述する反応拡散方程式の解構造の解明に従事した。本年度の具体的成果として、次の1および2が挙げられる。
    1 正値定常解(関連する非線型楕円型方程式の正値解)が複数個存在することを数学的に証明した。
    2 ホップ分岐現象による時間周期解の存在を証明した。
    3 正値定常解の漸近安定性の判定に成功した。
    1と2は、相互拡散のケースでのみ起こりうる数理現象を示したことに意義がある。この結果は、解のもつ時空的なダイナミクス(被食生物と捕食生物の個体数密度のダイナミクス)が、相互拡散効果の有無によって本質的に異なることを示唆する。3については、局所的な安定性のみを判定しており、解のもつ挙動と共存定常解の大域的関係の解明については、今後の課題となる。

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Misc

 

Syllabus

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Teaching Experience

  • 微積分3, 応用数学6

    Waseda University  

  • Calculus II

    The University of Electro-Communications  

  • Calculus Ⅰ

    The University of Electro-Communications  

  • 微分積分学第一

    電気通信大学  

  • Linear Algebra II

    The University of Electro-Communications  

  • 線形代数学第二

    電気通信大学  

  • 現代数学入門A

    電気通信大学  

  • Mathematics Exercise I

    The University of Electro-Communications  

  • Calculus Ⅱ

    The University of Electro-Communications  

  • 微分積分学第二

    電気通信大学  

  • Exercise in Mathematics I

    The University of Electro-Communications  

  • 数理科学特殊講義VIII(集中講義)

    関西学院大学  

  • 数理科学特殊講義VIII(集中講義)

    関西学院大学  

  • 解析学1A

    早稲田大学  

  • Exercise in Mathematics Ⅱ

    The University of Electro-Communications  

  • Analysis

    The University of Electro-Communications  

  • Exercise in Mathematics Ⅰ

    The University of Electro-Communications  

  • 数学演習第一

    電気通信大学  

  • Linear Algebra I

    The University of Electro-Communications  

  • 線形代数学第一

    電気通信大学  

  • Advanced Topics in Analysis

    The University of Electro-Communications  

  • 解析学特論

    電気通信大学  

  • Exercise in Mathematics II

    The University of Electro-Communications  

  • 数学演習第二

    電気通信大学  

  • Analysis

    The University of Electro-Communications  

  • 解析学

    電気通信大学  

  • Calculus 3, Applied Mathematics 6

    Waseda University  

  • 微積分3, 応用数学6

    早稲田大学  

  • Introductory Linear Algebra Ⅰ

    The University of Electro-Communications  

  • ベクトルと行列第一

    電気通信大学  

  • Introduction to Modern Mathematics A

    The University of Electro-Communications  

  • 現代数学入門A

    電気通信大学  

  • Analysis 1A

    Waseda University  

  • 解析学1A

    早稲田大学  

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Sub-affiliation

  • Faculty of Science and Engineering   Graduate School of Fundamental Science and Engineering

Research Institute

  • 2022
    -
    2024

    Waseda Research Institute for Science and Engineering   Concurrent Researcher

Internal Special Research Projects

  • 相互拡散(cross-diffusion)を伴う反応拡散方程式の解構造に対する研究

    2004  

     View Summary

    反応拡散方程式は、非線型解析学において研究が活発な分野のひとつです。特に「相互拡散(cross diffusion)」と呼ばれる非線型項を伴う反応拡散方程式系は、偏微分方程式論における従来の手法が直接的には通用しないケースが多く、更なる理論研究が待たれる状況にあります。私は、博士課程在籍時より「相互拡散を伴う反応拡散方程式」の研究に従事しています。今年度(2004年度)においては、数理生態学のモデルに現れる反応拡散方程式系 (P);Ut=△[(1+αV)U]+U(a-U-cV) in Ω×(0,T),Vt=μ△V+△[V/(1+βU)]+V(b+dU-V) in Ω×(0,T)の研究に力を注ぎました。方程式系(P)は、有界領域Ωの中に棲息する「食う食われるの関係」にある生物の個体数密度の時空的な変化を記述し、未知関数U=U(x,t)とV=V(x,t)はそれぞれ被食生物(prey)と捕食生物(predator)の場所x,時刻tにおける個体数密度を表します。第一式の相互拡散α△(UV)は、predator の個体数密度の高い地域で prey の空間的拡散が促進される状況を記述します。一方、第二式の非線型拡散△[V/(1+βU)] は prey の個体数密度が高い地域では predator の空間的拡散が鈍化する状況を記述します。これらの様な「非線型拡散」に対する解析理論は、国内外で模索中の段階にあり、(P)についても多くの未解決問題が残されています。中でも、(P)の時間的定常解(Ut=Vt=0を満たす解)を求めることは重要な問題です。正値定常解に対する解析は、数理生態学的な「共存定常状態」のみならず、非定常解のもつ時空的ダイナミクスの抽出にも役立ちます。 私は、ディレクレ境界条件の下で定常問題に取り組み「正値定常解が存在する十分条件」を係数パラメーターに与えました。この結果は、正値定常解がなす集合の大域的分岐構造を明らかにしています。例えば、preyの増殖率 a を分岐パラメーターとしたとき、ある分岐点Aを境にして a>A なら正値定常解が存在することが証明されました。また、非線型拡散(α,β)と正値定常解の関係も見出され、概して「αが大きいと正値定常解が存在しにくくなり、βが大きいと正値定常解が存在しやすくなる」ことが判明しました。上記の全ての結果は、門田智仁氏(昨年度修士修了)との共同研究に改良を重ねて得られたものであり、2004年9月に開催された日本数学会秋季総合分科会(於 北海道大学)の一般講演において口頭発表しました。さらに、論文としても完成済みであり、近く非線型解析の学術雑誌に投稿する予定です。 (P)の定常解集合を解析する上で、分数型相互拡散βと分岐構造の関係を解明することに興味がもたれます。そこで私は、前段の研究と併行して、「βが大きいケース」の(P)に対する解析を集中的に取り組みました。このケースでは「(P)でβを無限大に発散させた極限系」からの摂動が有効であることを発見し、正値定常解集合のなす大域分岐構造を詳細に抽出しました。ここで得られた分岐構造により、正値定常解の U 成分は、ある閾値 A’(>A)を境にして急激に増加することが分かりました。概して、大きい分数型相互拡散βの非線型効果により、正値定常解集合のなす分岐枝は、a=A’付近で「曲がる」ことが判明した訳です。ここまでの研究結果は、2004年10月に京都大学RIMSで開催された研究集会「反応拡散系に現れる時・空間パタ-ンのメカニズム」において発表しました。なお、この見地からの(P)の解析には、まだまだ進展の余地があり、現在も精力的に継続しております。