INNOVATIVE APPROACHES TO POPULATION DYNAMICS: THE ROLE OF FRACTIONAL OPERATOR AND SOLITON SOLUTIONS

Muhammad Bilal Riaz, Abdelbaki Choucha (), Rajaa Taher Matoog (), Saad Althobaiti (), Tomas Kozubek () and Ifrah Iqbal ()
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Muhammad Bilal Riaz: IT4Innovations, VSB — Technical University of Ostrava, Ostrava, Czech Republic2Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon
Abdelbaki Choucha: Department of Material Sciences, Faculty of Sciences, Amar Teleji Laghouat University, Algeria Laboratory of Mathematics and Applied Sciences, Ghardaia University, Algeria
Rajaa Taher Matoog: Mathematics Department, Faculty of Sciences, Umm Al-Qura University, Makkah, Saudi Arabia
Saad Althobaiti: Department of Science and Technology, University College Ranyah, Taif University, Ranyah, 21975, Saudi Arabia
Tomas Kozubek: IT4Innovations, VSB — Technical University of Ostrava, Ostrava, Czech Republic
Ifrah Iqbal: Department of Mathematics, University of Okara, Okara, Pakistan

FRACTALS (fractals), 2025, vol. 33, issue 04, 1-19

Abstract: 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.

Keywords: Conformable Derivative; Modified Sardar Subequation Method (MSSEM); New Kudryashov’s Method (NKM); Fractional Biological Population Model (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X25400894

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