Hopf bifurcation in a diffusive population system with nonlocal delay effect. (English) Zbl 1477.35024
Summary: In this paper, we investigate the dynamics of a general nonlocal delayed reaction-diffusion equations with Dirichlet boundary condition. It is shown that a positive spatially nonhomogeneous equilibrium bifurcates from the trivial equilibrium. Then we obtain the stability of the positive spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcations with the change of the time delay by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. The stability and bifurcation direction of Hopf bifurcating periodic orbits are also derived by using the normal form theory and the center manifold reduction. Finally, we show some numerical simulations to illustrate our theoretical results.
MSC:
| 35B32 | Bifurcations in context of PDEs |
| 35B10 | Periodic solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
| 35R09 | Integro-partial differential equations |
| 37L10 | Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems |
| 92D25 | Population dynamics (general) |
Keywords:
stability and bifurcation directionReferences:
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