Limiting structure of shrinking solutions to the stationary shigesada-kawasaki-teramoto model with large cross-diffusion

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6 Citations (Scopus)

Abstract

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 -δ subject to the homogeneous Neumann boundary condition.

Original languageEnglish
Pages (from-to)3993-4024
Number of pages32
JournalSIAM Journal on Mathematical Analysis
Volume47
Issue number5
DOIs
Publication statusPublished - 2015 Jan 1
Externally publishedYes

Fingerprint

Cross-diffusion
Shrinking
Limiting
Boundary conditions
Limiting Behavior
Neumann Boundary Conditions
Coexistence
Differential equations
Bifurcation
Competing Species
Lotka-Volterra Model
Competition Model
Connected Set
Segregation
Shrinkage
Model
Blow-up
Positive Solution
Infinity
Tend

Keywords

  • A priori estimate
  • Asymptotic behavior
  • Bifurcation
  • Blow up
  • Cross-diffusion
  • Nonlinear elliptic system

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

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title = "Limiting structure of shrinking solutions to the stationary shigesada-kawasaki-teramoto model with large cross-diffusion",
abstract = "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 -δ subject to the homogeneous Neumann boundary condition.",
keywords = "A priori estimate, Asymptotic behavior, Bifurcation, Blow up, Cross-diffusion, Nonlinear elliptic system",
author = "Kousuke Kuto",
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