Fractional Poisson and Gamma Models in Rainfall Analysis: Impact on Climate Change and Marine Ecosystems

Abstract

The study of rainfall patterns and their impact on climate change and marine ecosystems is essential to understanding contemporary environmental challenges. This chapter introduces two innovative models, fractional Poisson and fractional Gamma, to analyze complex rainfall data, effectively handling zero-inflated occurrences and reducing data overdispersion. These models offer greater accuracy and flexibility in representing climate variations and impacts, contributing to more reliable climate prediction tools. The second part of the chapter explores the dynamic interaction between phytoplankton and oxygen production in oceans, illustrating how rising water temperatures disrupt this delicate balance. By integrating Brownian motion into the mathematical modeling, we provide a framework that allows for a deeper understanding of how climate change affects oceanic oxygen levels and the sustainability of marine ecosystems. In the final phase, we enhance the modeling by adding fractional Brownian motion, enabling a more precise representation of long-memory effects and interwoven impacts within the ecosystem. The findings of this study open new avenues for developing effective strategies to maintain ecosystem stability, providing scientific tools to support environmental policies in addressing climate change.