Stability and hopf bifurcation of coexistence steady-states to an skt model in spatially heterogeneous environment

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12 Citations (Scopus)

Abstract

This paper is concerned with the following Lotka-Volterra cross-diffusion system ? in a bounded domain ω ⊂ RN with Neumann boundary conditions ∂vu = ∂vv = 0 on ∂ω. In the previous paper [18], the author has proved that the set of positive stationary solutions forms a fishhook shaped branch ⌈ under a segregation of p(x) and d(x). In the present paper, we give some criteria on the stability of solutions on ⌈. We prove that the stability of solutions changes only at every turning point of ⌈ if ? is large enough. In a different case that c(x) > 0 is large enough, we find a parameter range such that multiple Hopf bifurcation points appear on ⌈.

Original languageEnglish
Pages (from-to)489-509
Number of pages21
JournalDiscrete and Continuous Dynamical Systems
Volume24
Issue number2
DOIs
Publication statusPublished - 2009 Jun 1
Externally publishedYes

Fingerprint

Heterogeneous Environment
Hopf bifurcation
Stability of Solutions
Coexistence
Hopf Bifurcation
Cross-diffusion System
Lotka-Volterra System
Turning Point
Bifurcation Point
Segregation
Neumann Boundary Conditions
Stationary Solutions
Bounded Domain
Branch
Boundary conditions
Model
Range of data
Form

Keywords

  • Coexistence states
  • Heterogeneous environment
  • Hopf bifurcation
  • Limiting system
  • Lyapunov-schmidt reduction
  • Skt model
  • Stability

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

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abstract = "This paper is concerned with the following Lotka-Volterra cross-diffusion system ? in a bounded domain ω ⊂ RN with Neumann boundary conditions ∂vu = ∂vv = 0 on ∂ω. In the previous paper [18], the author has proved that the set of positive stationary solutions forms a fishhook shaped branch ⌈ under a segregation of p(x) and d(x). In the present paper, we give some criteria on the stability of solutions on ⌈. We prove that the stability of solutions changes only at every turning point of ⌈ if ? is large enough. In a different case that c(x) > 0 is large enough, we find a parameter range such that multiple Hopf bifurcation points appear on ⌈.",
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AB - This paper is concerned with the following Lotka-Volterra cross-diffusion system ? in a bounded domain ω ⊂ RN with Neumann boundary conditions ∂vu = ∂vv = 0 on ∂ω. In the previous paper [18], the author has proved that the set of positive stationary solutions forms a fishhook shaped branch ⌈ under a segregation of p(x) and d(x). In the present paper, we give some criteria on the stability of solutions on ⌈. We prove that the stability of solutions changes only at every turning point of ⌈ if ? is large enough. In a different case that c(x) > 0 is large enough, we find a parameter range such that multiple Hopf bifurcation points appear on ⌈.

KW - Coexistence states

KW - Heterogeneous environment

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KW - Lyapunov-schmidt reduction

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