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

A phytoplankton-zooplankton-fish model with chaos control: In the presence of fear effect and an additional food

The interplay of phytoplankton, zooplankton, and fish is one of the most important aspects of the aquatic environment. In this paper, we propose to explore the dynamics of a phytoplankton-zooplankton-fish system, with fear-induced birth rate reduction in the middle predator by the top predator and an additional food source for the top predator fish. Phytoplankton-zooplankton and zooplankton-fish interactions are handled using Holling type IV and II responses, respectively. First, we prove the well-posedness of the system, followed by results related to the existence of possible equilibrium points. Conditions under which a different number of interior equilibria exist are also derived here. We also show this existence numerically by varying the intrinsic growth rate of phytoplankton species, which demonstrates the model's vibrant nature from a mathematical point of view. Furthermore, we performed the local and global stability analysis around the above equilibrium points, and the transversality conditions for the occurrence of Hopf bifurcations and transcritical bifurcations are established. We observe numerically that for low levels of fear, the system behaves chaotically, and as we increase the fear parameter, the solution approaches a stable equilibrium by the route of period-halving. The chaotic behavior of the system at low levels of fear can also be controlled by increasing the quality of additional food. To corroborate our findings, we constructed several phase portraits, time-series graphs, and one- and two-parametric bifurcation diagrams. The computation of the largest Lyapunov exponent and a sketch of Poincaré maps verify the chaotic character of the proposed system. On varying the parametric values, the system exhibits phenomena like multistability and the enrichment paradox, which are the basic qualities of non-linear models. Thus, the current study can also help ecologists to estimate the parameters to study and obtain such important findings related to non-linear PZF systems. Therefore, from a biological and mathematical perspective, the analysis and the corresponding results of this article appear to be rich and interesting.

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.