A non-autonomous approach to study the impact of environmental toxins on nutrient-plankton system

Abstract

The interaction between phytoplankton and zooplankton has a significant impact on the marine ecology. The interplay between these two species is the building blocks for most of the food webs operating in an aquatic ecosphere. The environmental toxins released by different external sources also affect the phytoplankton-zooplankton dynamics. In the present study, we propose a model to explore the kinetics of a nutrient-phytoplankton-zooplankton-environmental toxins (NPZT) system. The defence mechanism of phytoplankton against zooplankton is reflected through modified Holling type IV response, whereas the consumption of nutrients by phytoplankton is outlined by Holling type II response. The external toxins are assumed to have the capability of reducing the birth rate of phytoplankton species after coming into contact with their cells. To make our model more pragmatic, seasonal variation in the parameters is also taken into account. Firstly, we do the analysis related to the autonomous model (non-seasonal) like; its boundedness, existence of equilibrium points, their stability analysis, and occurrence of Hopf-bifurcation. Further, for the non-autonomous model (seasonal), we analyze the existence of positive periodic solution and its global stability. Through numerical simulations, we observe that for the non-seasonal model, increasing the rate of suppressing phytoplankton’s growth by environmental toxin, and rate at which environmental toxin is added to system make it unstable through Hopf-bifurcation. These oscillations can be removed by raising phytoplankton’s inhibitory effect against zooplankton, and this increment also leads to the extinction of the zooplankton population, making zooplankton free equilibrium a stable one. Both models, non-seasonal as well as seasonal manifest different types of multistability, and this is an exciting character associated with non-linear models. We also note that the inclusion of seasonality in our system promotes the coexistence of all populations. Further, through numerical simulations, we show that making some of the parameters seasonal can cause the emergence of chaos in the system. To verify chaos, we sketch the Poincaré map and evaluate the maximum Lyapunov exponent. The seasonal model also shows the switching of stability through different periodic and chaotic windows on varying the maximum intrinsic growth rate for phytoplankton, and contact rate between environmental toxin and phytoplankton. To substantiate our results, we picture several time-series graphs, basins of attraction, one and two-parametric bifurcation diagrams. Thus we expect that the present work can assist biologists and mathematicians in studying nutrient-plankton systems in a more detailed and realistic manner. This study can also help researchers in the estimation of non-seasonal as well as seasonal parameters while studying these types of complex non-linear models. Therefore, the present work seems to be enriched from a mathematical and biological point of view.