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Delay driven Hopf bifurcation and chaos in a diffusive toxin producing phytoplankton‐zooplankton model
Author(s) -
Zhao Jianglin,
Yan Yong,
Huang Lizhuang,
Yang Run
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.5615
Subject(s) - mathematics , hopf bifurcation , center manifold , period doubling bifurcation , phytoplankton , chaotic , bifurcation , transcritical bifurcation , zooplankton , parameter space , delay differential equation , mathematical analysis , control theory (sociology) , differential equation , physics , nonlinear system , chemistry , geometry , ecology , computer science , organic chemistry , control (management) , quantum mechanics , artificial intelligence , nutrient , biology
This paper deals with a diffusive toxin producing phytoplankton‐zooplankton model with maturation delay. By analyzing eigenvalues of the characteristic equation associated with delay parameter, the stability of the positive equilibrium and the existence of Hopf bifurcation are studied. Explicit results are derived for the properties of bifurcating periodic solutions by means of the normal form theory and the center manifold reduction for partial functional differential equations. Numerical simulations not only agree with the theoretical analysis but also exhibit the complex behaviors such as the period‐3, 5, 6, 7, 8, 11, and 12 solutions, cascade of period‐doubling bifurcation in period‐2, 4, quasi‐periodic solutions, and chaos. The key observation is that time delay may control harmful algae blooms (HABs). Moreover, numerical simulations show that the chaotic states induced by the period‐doubling bifurcation are purely temporal, which is stationary in space and oscillatory in time. The investigations may provide some new insights on harmful phytoplankton blooms.