Chaos control in a multiple delayed phytoplankton–zooplankton model with group defense and predator’s interference

Abstract

Phytoplankton–zooplankton interaction is a topic of high interest among the interrelationships related to marine habitats. In the present manuscript, we attempt to study the dynamics of a three-dimensional system with three types of plankton: non-toxic phytoplankton, toxic producing phytoplankton, and zooplankton. We assume that both non-toxic and toxic phytoplankton are consumed by zooplankton via Beddington–DeAngelis and general Holling type-IV responses, respectively. We also incorporate gestation delay and toxic liberation delay in zooplankton’s interactions with non-toxic and toxic phytoplankton correspondingly. First, we have studied the well-posedness of the system. Then, we analyze all the possible equilibrium points and their local and global asymptotic behavior. Furthermore, we assessed the conditions for the occurrence of Hopf-bifurcation and transcritical bifurcation. Using the normal form method and center manifold theorem, the conditions for stability and direction of Hopf-bifurcation are also studied. Various time-series, phase portraits, and bifurcation diagrams are plotted to confirm our theoretical findings. From the numerical simulation, we observe that a limited increase in inhibitory effect of toxic phytoplankton against zooplankton can support zooplankton’s growth, and rising predator’s interference can also boost zooplankton expansion in contrast to the nature of Holling type IV and Beddington–DeAngelis responses. Next, we notice that on variation of toxic liberation delay, the delayed system switches its stability multiple times and becomes chaotic. Furthermore, we draw the Poincaré section and evaluate the maximum Lyapunov exponent in order to verify the delayed system’s chaotic nature. Results presented in this article might be helpful to interpret biological insights into phytoplankton–zooplankton interactions.