Innovative Approaches To Population Dynamics: The Role Of Fractional Operator And Soliton Solutions

Biological population models are crucial in numerous fields, significantly enhancing our understanding of public health, environmental dynamics, and ecology. These models play a vital role in ecology by aiding in the prediction and management of species population growth, interactions and the impact of changing environmental conditions on ecosystems. These models aid in making informed decisions including resource management, environmental protection and biodiversity conservation for investigating dynamics of population. Recently, the fractional calculus has gained significant attention across different scientific fields, including ecology and biology. Fractional derivatives, which generalize classical derivatives, model the processes that exhibit long-range interactions or memory effects, characteristics often exist in real-world ecological and biological systems. These advancements have led to the development of more sophisticated and precise population models. In this study, we employ the conformable derivative operator, a type of fractional derivative that retains many of the useful properties of classical calculus, to model the complex dynamics of biological populations. This approach offers a new perspective on understanding population behavior over time and space, while accounting for the inherent nonlinearity and irregularities in ecological systems. Exploring analytical solutions for fractional partial differential equations remains a significant challenge due to the complex nature of these equations. To address this, numerous mathematical methods have been developed. In this study, we utilize two powerful techniques — the Modified Sardar Sub-Equation Method and the new Kudryashov method — to derive a wide range of soliton solutions for the biological population model. These include dark, bright, singular, periodic-singular, combined dark–bright, and combined dark-singular soliton solutions. Each type of soliton solution gives unique insights into different population dynamics, such as rapid population growth, declines, or more complex relations between population groups. We generated three-dimensional (3D) plots, density plots, contour plots, and two-dimensional (2D) plots to provide a detailed visualization of the population dynamics under various conditions. We also performed a comparative analysis of parameters by using 2D plots for clearer understanding of how different factors influence population growth and interactions. This study not only builds upon existing research by providing new analytical solutions for population models but also aims to offer deeper insights into the complex factors that govern population dynamics.