### 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 |
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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 |

### 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