On the qualitative study of a discrete-time phytoplankton-zooplankton model under the effects of external toxicity in phytoplankton population

Abstract

The current manuscript studies a discrete-time phytoplankton-zooplankton model with Holling type-II response. The original model is modified by considering the condition that the phytoplankton population is getting infected with an external toxic substance. To obtain the discrete counterpart from a continuous-time system, Euler's forward method is applied. Moreover, a consistent discrete-time phytoplankton-zooplankton model is obtained by using a nonstandard difference scheme. The boundedness character for every positive solution is discussed, and the local stability of obtained system about each of its fixed points is discussed. The existence of period-doubling bifurcation at a positive equilibrium point is discussed for the discrete system obtained by Euler's forward method. In addition, the comparison of the consistent discrete-time version with its inconsistent counterpart is provided. It is proved that the discrete-time system obtained by using a nonstandard scheme is dynamically consistent as there is no chance for the existence of period-doubling bifurcation in that system. In order to control the period-doubling bifurcation and Neimark-Sacker bifurcation, an improved hybrid control strategy is applied. Finally, we have provided some interesting numerical examples to explain our theoretical results.