### Abstract

We consider the following quasilinear elliptic system: (EQUATION PRESENT) where Ω is a bounded domain in ℝ. This system is a stationary problem of a prey-predator model with non-linear diffusion δ(^{v}/ _{1+βu} ), and u (respectively v) denotes the population density of the prey (respectively the predator). Kuto [15] has studied this system for large β under the restriction b > (1 + γ)λ_{1}, where λ_{1} is the least eigenvalue of -δ with homogeneous Dirichlet boundary condition. The present paper studies two shadow systems and gives the complete limiting characterization of positive solutions as β → ∞ without any restriction on b.

Original language | English |
---|---|

Pages (from-to) | 725-752 |

Number of pages | 28 |

Journal | Differential and Integral Equations |

Volume | 22 |

Issue number | 7-8 |

Publication status | Published - 2009 Jul |

Externally published | Yes |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Differential and Integral Equations*,

*22*(7-8), 725-752.

**Limiting characterization of stationary solutions for a prey-predator model with nonlinear diffusion of fractional type.** / Kuto, Kousuke; Yamada, Yoshio.

Research output: Contribution to journal › Article

*Differential and Integral Equations*, vol. 22, no. 7-8, pp. 725-752.

}

TY - JOUR

T1 - Limiting characterization of stationary solutions for a prey-predator model with nonlinear diffusion of fractional type

AU - Kuto, Kousuke

AU - Yamada, Yoshio

PY - 2009/7

Y1 - 2009/7

N2 - We consider the following quasilinear elliptic system: (EQUATION PRESENT) where Ω is a bounded domain in ℝ. This system is a stationary problem of a prey-predator model with non-linear diffusion δ(v/ 1+βu ), and u (respectively v) denotes the population density of the prey (respectively the predator). Kuto [15] has studied this system for large β under the restriction b > (1 + γ)λ1, where λ1 is the least eigenvalue of -δ with homogeneous Dirichlet boundary condition. The present paper studies two shadow systems and gives the complete limiting characterization of positive solutions as β → ∞ without any restriction on b.

AB - We consider the following quasilinear elliptic system: (EQUATION PRESENT) where Ω is a bounded domain in ℝ. This system is a stationary problem of a prey-predator model with non-linear diffusion δ(v/ 1+βu ), and u (respectively v) denotes the population density of the prey (respectively the predator). Kuto [15] has studied this system for large β under the restriction b > (1 + γ)λ1, where λ1 is the least eigenvalue of -δ with homogeneous Dirichlet boundary condition. The present paper studies two shadow systems and gives the complete limiting characterization of positive solutions as β → ∞ without any restriction on b.

UR - http://www.scopus.com/inward/record.url?scp=84886243026&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84886243026&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84886243026

VL - 22

SP - 725

EP - 752

JO - Differential and Integral Equations

JF - Differential and Integral Equations

SN - 0893-4983

IS - 7-8

ER -