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

Kousuke Kuto*

*この研究の対応する著者

研究成果: Article査読

33 被引用数 (Scopus)

抄録

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 RN (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 Γp 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 Γp 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.

本文言語English
ページ(範囲)943-965
ページ数23
ジャーナルNonlinear Analysis: Real World Applications
10
2
DOI
出版ステータスPublished - 2009 4
外部発表はい

ASJC Scopus subject areas

  • 分析
  • 工学(全般)
  • 経済学、計量経済学および金融学(全般)
  • 計算数学
  • 応用数学

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