### Abstract

This paper is concerned with the following Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment (SP) {(Δ [(1 + k ρ (x) v) u] + u (a - u - c (x) v) = 0, in Ω,; Δ v + v (b + d (x) u - v) = 0, in Ω,; ∂
_{ν}
u = ∂
_{ν}
v = 0, on ∂ Ω .) Here Ω 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 ρ (x) > 0 is a smooth function with ∂
_{ν}
ρ = 0 on ∂ Ω. 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 Ω. Hence, the set Γ
_{p}
of positive solutions (with bifurcation parameter b) forms a bounded line in a spatially homogeneous case that ρ, c and d are constant. This paper proves that if a and | b | are small and k is large, a spatial segregation of ρ (x) and d (x) causes Γ
_{p}
to form a ⊂-shaped curve with respect to b. A crucial aspect of the proof involves the solving of a suitable limiting system as a, | b | → 0 and k → ∞ by using the bifurcation theory and the Lyapunov-Schmidt reduction.

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

Pages (from-to) | 943-965 |

Number of pages | 23 |

Journal | Nonlinear Analysis: Real World Applications |

Volume | 10 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2009 Apr 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Bifurcation
- Cross-diffusion
- Heterogeneous environment
- Limiting system
- Lyapunov-Schmidt reduction
- Multiple coexistence states
- Perturbation

### ASJC Scopus subject areas

- Analysis
- Engineering(all)
- Economics, Econometrics and Finance(all)
- Computational Mathematics
- Applied Mathematics

### Cite this

**Bifurcation branch of stationary solutions for a Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment.** / Kuto, Kousuke.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Bifurcation branch of stationary solutions for a Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment

AU - Kuto, Kousuke

PY - 2009/4/1

Y1 - 2009/4/1

N2 - This paper is concerned with the following Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment (SP) {(Δ [(1 + k ρ (x) v) u] + u (a - u - c (x) v) = 0, in Ω,; Δ v + v (b + d (x) u - v) = 0, in Ω,; ∂ ν u = ∂ ν v = 0, on ∂ Ω .) Here Ω 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 ρ (x) > 0 is a smooth function with ∂ ν ρ = 0 on ∂ Ω. 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 Ω. Hence, the set Γ p of positive solutions (with bifurcation parameter b) forms a bounded line in a spatially homogeneous case that ρ, c and d are constant. This paper proves that if a and | b | are small and k is large, a spatial segregation of ρ (x) and d (x) causes Γ p to form a ⊂-shaped curve with respect to b. A crucial aspect of the proof involves the solving of a suitable limiting system as a, | b | → 0 and k → ∞ by using the bifurcation theory and the Lyapunov-Schmidt reduction.

AB - This paper is concerned with the following Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment (SP) {(Δ [(1 + k ρ (x) v) u] + u (a - u - c (x) v) = 0, in Ω,; Δ v + v (b + d (x) u - v) = 0, in Ω,; ∂ ν u = ∂ ν v = 0, on ∂ Ω .) Here Ω 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 ρ (x) > 0 is a smooth function with ∂ ν ρ = 0 on ∂ Ω. 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 Ω. Hence, the set Γ p of positive solutions (with bifurcation parameter b) forms a bounded line in a spatially homogeneous case that ρ, c and d are constant. This paper proves that if a and | b | are small and k is large, a spatial segregation of ρ (x) and d (x) causes Γ p to form a ⊂-shaped curve with respect to b. A crucial aspect of the proof involves the solving of a suitable limiting system as a, | b | → 0 and k → ∞ by using the bifurcation theory and the Lyapunov-Schmidt reduction.

KW - Bifurcation

KW - Cross-diffusion

KW - Heterogeneous environment

KW - Limiting system

KW - Lyapunov-Schmidt reduction

KW - Multiple coexistence states

KW - Perturbation

UR - http://www.scopus.com/inward/record.url?scp=56549113683&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=56549113683&partnerID=8YFLogxK

U2 - 10.1016/j.nonrwa.2007.11.015

DO - 10.1016/j.nonrwa.2007.11.015

M3 - Article

VL - 10

SP - 943

EP - 965

JO - Nonlinear Analysis: Real World Applications

JF - Nonlinear Analysis: Real World Applications

SN - 1468-1218

IS - 2

ER -