This research work is dedicated to studying the dynamics of a coupled plankton-oxygen model in the framework of three non-linear differential equations. As we know that the ocean dynamics have a firm impact on the global climate change and on the creation of the environment. Also, it is recorded that about 70% of the environmental oxygen is manufactured in the oceans due to the photosynthetic bustling of phytoplankton. All the same, the rate of oxygen manufacture based on the temperature of the water and hence can be dominance by global warming. To study the proposed Oxygen-Phytoplankton-Zooplankton system, we use a very recent fractional numerical algorithm. We discuss the existence of the solution for the given model problem because in the case of fractional-order models, the proof of solution existence always becomes an important task. After that, we perform many novel 2-D and 3-D graphs by using Mathematica and Python software to fulfill the requirements of the numerical simulations. We use a generalised form of the well known Liouville-Caputo fractional derivative. Given numerical algorithm is very recent, short, easy, and reliable to use or to apply to the non-linear dynamical models. The main motive of this research work is to study the Plankton-Oxygen dynamics under the climate change by using the proposed fractional-order model with capturing the memory effects and discuss some novel results for the literature on such ecological topics.

In this study, our aim is to explore the dynamics of COVID-19 or 2019-nCOV in Argentina considering the parameter values based on the real data of this virus from March 03, 2020 to March 29, 2021 which is a data range of more than one complete year. We propose a Atangana–Baleanu type fractional-order model and simulate it by using predictor–corrector (P-C) method. First we introduce the biological nature of this virus in theoretical way and then formulate a mathematical model to define its dynamics. We use a well-known effective optimization scheme based on the renowned trust-region-reflective (TRR) method to perform the model calibration. We have plotted the real cases of COVID-19 and compared our integer-order model with the simulated data along with the calculation of basic reproductive number. Concerning fractional-order simulations, first we prove the existence and uniqueness of solution and then write the solution along with the stability of the given P-C method. A number of graphs at various fractional-order values are simulated to predict the future dynamics of the virus in Argentina which is the main contribution of this paper.