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In this study, we are concerned with the dynamics of a new established fractional-order delayed zooplankton–phytoplankton system. The existence and uniqueness of the solution are proved via Banach fixed point theorem. Non-negativeness of the solution is studied by mathematical inequality technique. The boundedness of the solution is analyzed by virtue of constructing an appropriate function. A novel delay-independent sufficient condition ensuring the stability and the onset of Hopf bifurcation for the established fractional-order delayed zooplankton–phytoplankton system is derived by means of Laplace transform, stability criterion and bifurcation knowledge of fractional-order differential equation. The global stability condition for the involved fractional-order delayed zooplankton–phytoplankton system is built by using a suitable positive definite function. Taking advantage of hybrid control tactics, we effectively control the time of occurrence of Hopf bifurcation for the established fractional-order delayed zooplankton–phytoplankton system. The study manifests that delay plays a vital role in controlling the stability and the time of occurrence of Hopf bifurcation for the involved fractional-order delayed zooplankton–phytoplankton system and the fractional-order controlled zooplankton–phytoplankton system involving delays. To verify the correctness of established chief results, computer simulation figures are distinctly displayed. The derived conclusions of this research are entirely new and possess potential theoretical value in preserving the balance of biological population. Up to now, there are few publications on detailed and comprehensive dynamic analysis on fractional-order delayed zooplankton–phytoplankton system via various exploration ways. [ABSTRACT FROM AUTHOR]

• A novel necessary and sufficient condition for the stability is expressed in an easily verified algebraical criterion. • The Laplace transform method and a method of imbedding the characteristic equation are used. • Our results yield a delay independent stability condition for discrete delays, which coincides with the well-known real-valued differential equations. • For fractional-order complex-valued integro-differential equation, the stability criterion is delay-dependent. In this paper, we investigate the global asymptotic stability of fractional-order complex-valued differential equations with distributed delays. Based on the Laplace transform method, a novel necessary and sufficient condition for the stability is established by imbedding the characteristic equation into two-dimensional complex system. The algebraical criterion is expressed by the fractional exponent, coefficients and the delay. Finally, two numerical examples are given to show the feasibility and effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]