Dynamical Study of an Eco-Epidemiological Delay Model for Plankton System with Toxicity.

In this paper, we analyze the complexity of an eco-epidemiological model for phytoplankton-zooplankton system in presence of toxicity and time delay. Holling type II function response is incorporated to address the predation rate as well as toxic substance distribution in zooplankton. It is also presumed that infected phytoplankton does recover from the viral infection. In the absence of time delay, stability and Hopf-bifurcation conditions are investigated to explore the system dynamics around all the possible equilibrium points. Further, in the presence of time delay, conditions for local stability are derived around the interior equilibria and the properties of the periodic solution are obtained by applying normal form theory and central manifold arguments. Computational simulation is performed to illustrate our theoretical findings. It is explored that system dynamics is very sensitive corresponding to carrying capacity and toxin liberation rate and able to generate chaos. Further, it is observed that time delay in the viral infection process can destabilize the phytoplankton density whereas zooplankton density remains in its old state. Incorporation of time delay also gives the scenario of double Hopf-bifurcation. Some control parameters are discussed to stabilize system dynamics. The effect of time delay on (i) growth rate of susceptible phytoplankton shows the extinction and double Hopf-bifurcation in the zooplankton population, (ii) a sufficiently large value of carrying capacity stabilizes the chaotic dynamics or makes the whole system chaotic with further increment.

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.