### Abstract

We study the positive solution set of the following quasilinear elliptic system: where ω is a bounded domain in R^{N}, a, b, c, d, and μ are positive constants, and β is a nonnegative constant. This system is the stationary problem associated with a prey-predator model with the strongly coupled diffusion δ( v/1+βu ), and u (respectively v) denotes the population density of the prey (respectively the predator). In the previous paper by Kadota and Kuto [10], we obtained the bifurcation branch of the positive solutions, which extends globally with respect to the bifurcation parameter a. In the present paper, we aim to derive the nonlinear effect of large β on the positive solution continuum. We obtain two shadow systems in the limiting case as β → ∞. From the analysis for the shadow systems, we prove that in the large β case, the positive solutions satisfy ||u||_{∞} = O(1/β) if a is less than a threshold number, while the positive solutions can be approximated by a positive solution of the associated system without the strongly coupled diffusion if a is large enough.

Original language | English |
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Pages (from-to) | 145-172 |

Number of pages | 28 |

Journal | Advances in Differential Equations |

Volume | 12 |

Issue number | 2 |

Publication status | Published - 2007 Dec 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**A strongly coupled diffusion effect on the stationary solution set of a prey-predator model.** / Kuto, Kousuke.

Research output: Contribution to journal › Article

*Advances in Differential Equations*, vol. 12, no. 2, pp. 145-172.

}

TY - JOUR

T1 - A strongly coupled diffusion effect on the stationary solution set of a prey-predator model

AU - Kuto, Kousuke

PY - 2007/12/1

Y1 - 2007/12/1

N2 - We study the positive solution set of the following quasilinear elliptic system: where ω is a bounded domain in RN, a, b, c, d, and μ are positive constants, and β is a nonnegative constant. This system is the stationary problem associated with a prey-predator model with the strongly coupled diffusion δ( v/1+βu ), and u (respectively v) denotes the population density of the prey (respectively the predator). In the previous paper by Kadota and Kuto [10], we obtained the bifurcation branch of the positive solutions, which extends globally with respect to the bifurcation parameter a. In the present paper, we aim to derive the nonlinear effect of large β on the positive solution continuum. We obtain two shadow systems in the limiting case as β → ∞. From the analysis for the shadow systems, we prove that in the large β case, the positive solutions satisfy ||u||∞ = O(1/β) if a is less than a threshold number, while the positive solutions can be approximated by a positive solution of the associated system without the strongly coupled diffusion if a is large enough.

AB - We study the positive solution set of the following quasilinear elliptic system: where ω is a bounded domain in RN, a, b, c, d, and μ are positive constants, and β is a nonnegative constant. This system is the stationary problem associated with a prey-predator model with the strongly coupled diffusion δ( v/1+βu ), and u (respectively v) denotes the population density of the prey (respectively the predator). In the previous paper by Kadota and Kuto [10], we obtained the bifurcation branch of the positive solutions, which extends globally with respect to the bifurcation parameter a. In the present paper, we aim to derive the nonlinear effect of large β on the positive solution continuum. We obtain two shadow systems in the limiting case as β → ∞. From the analysis for the shadow systems, we prove that in the large β case, the positive solutions satisfy ||u||∞ = O(1/β) if a is less than a threshold number, while the positive solutions can be approximated by a positive solution of the associated system without the strongly coupled diffusion if a is large enough.

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UR - http://www.scopus.com/inward/citedby.url?scp=58749086962&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:58749086962

VL - 12

SP - 145

EP - 172

JO - Advances in Differential Equations

JF - Advances in Differential Equations

SN - 1079-9389

IS - 2

ER -