Discrete-time phytoplankton–zooplankton model with bifurcations and chaos

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.

Bifurcations of a discrete-time Phytoplankton-Zooplankton model

“…In the following, we prove the possibility of an occurrence of a Neimark-Sacker, (N-S), bifurcation around the fixed point around * * , * = * there may exist the N-S bifurcation by choosing as a bifurcation parameter if β, ω, γ, ν ∈ Ω 4 [11] The following theorem guarantees that, around * Theorem 3.2.2.…”

“…Furthermore, a supercritical Neimark-Sacker bifurcation will occur if < 0 and a subcritical Neimark-Sacker bifurcation if > 0. [11]. The occurrence of an N-S bifurcation has been proven.…”

“…Over the past few decades, academics have developed an interest in phytoplankton due to its relevance in the maritime ecosystem, as the basis of its food chain and the effects of hazardous algal blooms, such as red tides, [1][2][3][4][5][6][7][8][9][10][11][12].…”

Bifurcation Analysis and 0-1 Chaos Test of a Discrete T System

“…A limited number of contributions have recently been made to the study of the dynamics of 3-D discrete systems (Khan and Javaid 2021;Abdelaziz et al 2020;Din and Ishaque 2019;Feng et al 2021;Hu et al 2014;Ishaque et al 2019;Qin et al 2016;Xin et al 2010). For example, a discrete-time SIR epidemic models discussed in (Abdelaziz et al 2020;Hu et al 2014), in (Xin et al 2010) the authors investigated discrete financial system and in (Qin et al 2016), the authors studied discrete chaotic system.…”

“…The explicit Flip-NS bifurcation criterion, center manifold theory, and bifurcation theory were all used by the researchers in these works to focus their efforts on figuring out the direction and stability of Flip and NS bifurcation. The studies in (Khan and Javaid 2021;Din and Ishaque 2019;Ishaque et al 2019) investigated discrete population models. In (Feng et al 2021), the authors explored NS bifurcation for discrete food chain model.…”