### 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 language | English |
---|---|

Pages (from-to) | 3993-4024 |

Number of pages | 32 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 47 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2015 Jan 1 |

Externally published | Yes |

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### 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

**Limiting structure of shrinking solutions to the stationary shigesada-kawasaki-teramoto model with large cross-diffusion.** / Kuto, Kousuke.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Limiting structure of shrinking solutions to the stationary shigesada-kawasaki-teramoto model with large cross-diffusion

AU - Kuto, Kousuke

PY - 2015/1/1

Y1 - 2015/1/1

N2 - 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.

AB - 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.

KW - A priori estimate

KW - Asymptotic behavior

KW - Bifurcation

KW - Blow up

KW - Cross-diffusion

KW - Nonlinear elliptic system

UR - http://www.scopus.com/inward/record.url?scp=84947460424&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84947460424&partnerID=8YFLogxK

U2 - 10.1137/140991455

DO - 10.1137/140991455

M3 - Article

VL - 47

SP - 3993

EP - 4024

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 5

ER -