Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect

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• 作者：Shangjiang Guo ; ^{shangjguo@hnu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Shuling Yan
• 关键词：34K18 ; 35B32 ; 35B35 ; 35K57 ; 35Q92 ; 92D40
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 January 2016
• 年：2016
• 卷：260
• 期：1
• 页码：781-817
• 全文大小：1115 K

文摘

The dynamics of a diffusive Lotka–Volterra type model for two species with nonlocal delay effect and Dirichlet boundary conditions is investigated in this paper. The existence and multiplicity of spatially nonhomogeneous steady-state solutions are obtained by means of Lyapunov–Schmidt reduction. The stability of spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcation with the changes of the time delay are obtained by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic orbits are derived. Finally, our theoretical results are illustrated by a model with homogeneous kernels and one-dimensional spatial domain.