Dynamics and stability of a fractional-order renewable resource model with generalized Mittag-Leffler kernel and fractal dimension

Kottakkaran Sooppy Nisar; Anum Zehra; Evren Hincal; Mustafa Bayram; Muhammad Farman

doi:10.46481/jnsps.2027.3812

Authors

Kottakkaran Sooppy Nisar Department of Mathematics, College of Science and Humanities in Al Kharj, Prince Sattam bin Abdulaziz University, Al Kharj, 16277, Saudi Arabia; Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman, Jordan
Anum Zehra Department of Mathematics, The Women University Multan, Multan, Pakistan; Department of Mathematics, Mathematics Research Center, Near East University, Mersin-10, Turkey
Evren Hincal Department of Mathematics, Mathematics Research Center, Near East University, Mersin-10, Turkey; Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan
Mustafa Bayram Department of Computer Engineering, Biruni University, 34010 Topkapi, Istanbul, Turkey
Muhammad Farman [email protected] Department of Mathematics, Mathematics Research Center, Near East University, Mersin-10, Turkey; Faculty of Medicine, Department of Biostatistics and Medical Informatics, Karadeniz Technical University, Trabzon, Turkey; International Center for Interdisciplinary Research in Sciences, The University of Lahore, Lahore, Pakistan

Keywords:

Model of biomass, Stability, Modeling, Fixed-point theory, Fractional derivative

Abstract

Fractional-order derivatives have a wide range of applications in real-world problems. In this work, we focus on a renewable resource in biomass management with population effects using a generalized Mittag-Leffler kernel. We modify the system into a time-fractional-order biomass model with population impacts on resource biomass while accounting for the suffocation effect, and analyze its dynamics under the Atangana--Baleanu Caputo (ABC) fractional operator. We establish positivity, boundedness, existence, and uniqueness of the fractional-order model using fixed-point theory. We confirm global stability by means of the fractional derivative of a Lyapunov function and compute Ulam--Hyers stability using the spectral radius of the associated matrix. The proposed model has a unique solution according to the Banach contraction principle. The numerical scheme for the fractal-fractional Atangana--Baleanu Riemann--Liouville (ABR) extension of the ABC model is constructed using a two-step Lagrange polynomial in the generalized form of the Mittag-Leffler kernel. Simulations are carried out at various parameter values, fractional orders, and fractal dimensions. The numerical results reveal that the dynamics vary significantly with fractional order, leading to either convergence to a steady state or oscillatory behavior. The study offers a framework for capturing population-resource interactions and demonstrates how fractional order can inform control of biomass consumption under population pressure. These insights are useful for planning and decision-making regarding the social impact of biomass.

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