Analysis of the dynamics of phytoplankton nutrient and whooping cough models with nonsingular kernel arising in the biological system.

• Dynamics of the phytoplankton nutrient and whooping cough models have been investigated. • The concept of fixed point theory has been used to derive the existence and uniqueness results. • Homotopy Perturbation Elzaki Transform Method (HPETM) has been used for obtaining the solutions. • A non-singular fractional operator has been used to characterize the fractional dynamics. In this study, the dynamics of the phytoplankton nutrient and whooping cough models have been examined. Mechanisms of transmission of whooping cough and phytoplankton nutrient models are defined in the Atangana-Baleanu-Caputo (ABC) fractional derivative sense. The first biological system is concerned with the dynamics of phytoplankton–nutrient interaction in the recycling of nutrients, and the second is the modeling of whooping cough in the human population. The essential characteristics of the titled models have been presented, and further, the transmissions of the models defined in the ABC sense are addressed. The concept of fixed point theory is used to derive the existence and uniqueness results of the titled models. Solutions are obtained using the homotopy perturbation Elzaki transform method (HPETM), and numerical results are computed. Graphical analysis of the effect of arbitrary order derivatives has been investigated in detail. [ABSTRACT FROM AUTHOR]