Abstract
The local dynamics with different topological classifications, bifurcation analysis, and chaos control for the phytoplankton–zooplankton model, which is a discrete analogue of the continuous-time model by a forward Euler scheme, are investigated. It is proved that the discrete-time phytoplankton–zooplankton model has trivial and semitrivial fixed points for all involved parameters, but it has an interior fixed point under the definite parametric condition. Then, by linear stability theory, local dynamics with different topological classifications are investigated around trivial, semitrivial, and interior fixed points. Further, for the discrete-time phytoplankton–zooplankton model, the existence of periodic points is also investigated. The existence of possible bifurcations around trivial, semitrivial, and interior fixed points is also investigated, and it is proved that there exists a transcritical bifurcation around a trivial fixed point. It is also proved that around trivial and semitrivial fixed points of the phytoplankton–zooplankton model there exists no flip bifurcation, but around an interior fixed point there exist both Neimark–Sacker and flip bifurcations. From the viewpoint of biology, the occurrence of Neimark–Sacker implies that there exist periodic or quasi-periodic oscillations between phytoplankton and zooplankton populations. Next, the feedback control method is utilized to stabilize chaos existing in the phytoplankton–zooplankton model. Finally, simulations are presented to validate not only obtained results but also the complex dynamics with orbits of period-8, 9, 10, 11, 14, 15 and chaotic behavior of the discrete-time phytoplankton–zooplankton model.
Highlights
The rapid population and rapid decline are features of phytoplankton
This infers the fact that phytoplankton–zooplankton model (8) may undergo a flip bifurcation by choosing γ as a bifurcation parameter if (γ, h, β, ω, ν) is located in the set
The work is about the topological classifications around fixed points, periodic points, bifurcations, and chaos control in the discrete-time phytoplankton–zooplankton model (8)
Summary
The rapid population and rapid decline are features of phytoplankton. Phytoplankton are temperature and nutrients sensitive. The growth rate of red tides depends upon different environmental factors such as temperature, nutrients, trace elements, and pollution. This mechanism describes the survival of that species in a changing environment. This model describes the mechanism of their rapid growth and recycling in the environment It is pointed out in [1] that a phytoplankton–zooplankton model explains the red tide environment as a constant system with population emerging with time, exemplified by ordinary differential equations. 8. In the present section, the existence of fixed points along with a linearized form of phytoplankton–zooplankton model (8) are given. The variation matrix V |FPZ(P,Z) around FPZ(P, Z) with respect to the map (10) is
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