TY - JOUR

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

AU - Kuto, Kousuke

N1 - Funding Information:
The author was partially supported by a Grant-in-Aid for Scientific Research (No. 18740093), The Ministry of Education, Culture, Sports, Science and Technology, Japan.

PY - 2009/4

Y1 - 2009/4

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 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.

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 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.

KW - Bifurcation

KW - Cross-diffusion

KW - Heterogeneous environment

KW - Limiting system

KW - Lyapunov-Schmidt reduction

KW - Multiple coexistence states

KW - Perturbation

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U2 - 10.1016/j.nonrwa.2007.11.015

DO - 10.1016/j.nonrwa.2007.11.015

M3 - Article

AN - SCOPUS:56549113683

VL - 10

SP - 943

EP - 965

JO - Nonlinear Analysis: Real World Applications

JF - Nonlinear Analysis: Real World Applications

SN - 1468-1218

IS - 2

ER -