Exploring the complexity and chaotic behavior in plankton-fish system with mutual interference and time delay

Anti-predator defense is an important mechanism that preys use to reduce the stress of constant struggle in a high concentration of predator and commonly established through evolution that supports prey organisms against predators. In the current study, we explore a three-tier plankton-fish interaction model using two kinds of function form, Monod-Haldane and Beddington-DeAngelis type. We introduce a discrete-time delay in the top predator population due to gestation. Our main objective persuades in this article is to address the role of inhibitory effect, mutual interference and gestation delay on the system dynamics in the presence of intermediate and top predators population. We perform theoretical analyses such as positivity and boundedness along with the local stability conditions of the delayed plankton-fish system. We also derive the condition of stability and direction of Hopf-bifurcation by using normal form theory and center manifold theorem. Our numerical computation demonstrates the dynamical outcome such as periodic and chaotic solutions of the model system without and with time delay validates our analytical findings. We also draw bifurcation diagrams that show the complexity of different parameters of model system. Interestingly, extinction is noticed in the top predator owing to the defense of phytoplankton. Model system exhibits irregular behavior when the inhibitory effect of phytoplankton is high or the value of gestation period of fish is high. We explore the significance of time delay with defense in our study which promotes chaotic phenomena in plankton system. Further, we notice the occurrence of double Hopf-bifurcation in a certain range of predator's interference with variation in the coefficient of time delay.