Chaotic dynamics of a plankton-fish system with fear and its carry over effects in the presence of a discrete delay

Abstract

Plankton-fish interactions are the central topic of interest related to marine ecology. Apart from the direct predation (lethal effect) of zooplankton by fish, there are some non-lethal implications in the zooplankton-fish relationship. Due to induced fear of predation, there can be a reduced reproduction rate of zooplankton species, and the effect of this non-lethal interconnection can be carried over to subsequent seasons or generations. In the current study, we tend to analyse the role of fish-induced fear in zooplankton with its carry-over effects (COEs) and a corresponding discrete delay (COE delay) in a phytoplankton-zooplankton-fish population model. We use Holling type IV and II functional responses to model the phytoplankton-zooplankton and zooplankton-fish interplay, respectively. In the well-posedness of the present biological system, firstly, we evaluate an invariant set in which the solutions of the model remain bounded. Then we prove its persistence under some ecologically well-behaved conditions. Next, we establish the conditions under which the different feasible equilibrium points exist; the existence of various interior equilibria is also set up here. To study the system's dynamical behavior, local and global stability analyses for the equilibria mentioned above are also discussed. Further, the theoretical conditions for Hopf and transcritical bifurcations in non-delayed and delayed models are determined. Impacts of non-lethal parameters, fear, and its carry-over effects, on the population densities are studied analytically and supported numerically. For intermediate values of COEs parameter, we notice that the system behaves chaotically, and decreasing (or increasing) it to low (or high) values, solution converges to interior equilibrium point through period-halving. Calculation of the largest Lyapunov exponent and drawing of Poincaré map validate the chaotic nature of the system. The chaos for medium values of COEs parameter can also be controlled by decreasing the fear parameter. Next, we numerically validate the theoretical result for transcritical bifurcation. We also note that our system shows the phenomenon of enrichment of paradox, and the attribute of multistability. In the delayed model, we observe that increasing delay can eliminate chaotic oscillations through amplitude death phenomenon. We draw various types of graphs and diagrams to assist our results. Thus we can say that the present study has various interesting characters related to non-linear models and can help biologists to study the plankton-fish models in a more detailed and pragmatic manner.