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Stability and Hopf bifurcation in a class of nonlocal delay differential equation with the zero‐flux boundary condition
Author(s) -
Hu Wenjie
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5638
Subject(s) - mathematics , hopf bifurcation , center manifold , mathematical analysis , stability (learning theory) , saddle node bifurcation , delay differential equation , bifurcation , transcritical bifurcation , oscillation (cell signaling) , differential equation , physics , nonlinear system , quantum mechanics , machine learning , biology , computer science , genetics
The aim of this paper is to study the stability and Hopf bifurcation in a general class of differential equation with nonlocal delayed feedback that models the population dynamics of a two age structured spices. The existence of Hopf bifurcation is firstly established after delicately analyzing the eigenvalue problem of the linearized nonlocal equation. The direction of the Hopf bifurcation and stability of the bifurcated periodic solutions are then investigated by means of center manifold reduction. Subsequently, we apply our main results to explore the spatial‐temporal patterns of the nonlocal Mackey‐Glass equation. We obtain both spatially homogeneous and inhomogeneous periodic solutions and numerically show that the former is stable while the latter is unstable. We also show that the inhomogeneous periodic solutions will eventually tend to homogeneous periodic solutions after transient oscillations and increasing of the immature mobility constant will shorten the transient oscillation time.