Fractional Logistic Growth with Memory Effects: A Tool for Industry-Oriented Modeling

TL;DR

The paper develops a generalized logistic growth model that embeds memory effects through the Atangana-Baleanu Caputo (ABC) fractional derivative of order $\mu$ and incorporates a proportional time delay $\lambda$, addressing nonlocal dynamics in constrained growth. It introduces a Hybrid Sumudu Variational (HSV) method to obtain semi-analytical solutions, deriving an ST-based iterative scheme that preserves the fractional memory structure for both linear and nonlinear delayed dynamics. The analysis yields explicit solutions in the case $\lambda=0$ and a HSV-based iterative framework for $0<\lambda\leq1$, illustrating how the memory parameter $\mu$ and delay parameter $\lambda$ modulate growth, memory fading, and stability. The work demonstrates that the ABC operator offers smooth fading-memory dynamics and robust Hyers-Ulam stability, enabling effective modeling in engineering, medicine, and social systems, with potential extensions to data-fitting and stochastic or network-based settings.

Abstract

The logistic growth model is a classical framework for describing constrained growth phenomena, widely applied in areas such as population dynamics, epidemiology, and resource management. This study presents a generalized extension using Atangana-Baleanu in Caputo sense (ABC)-type fractional derivatives. Proportional time delay is also included, allowing the model to capture memory-dependent and nonlocal dynamics not addressed in classical formulations. Free parameters provide flexibility for modeling complex growth in industrial, medical, and social systems. The Hybrid Sumudu Variational (HSV) method is employed to efficiently obtain semi-analytical solutions. Results highlight the combined effects of fractional order and delay on system behavior. This approach demonstrates the novelty of integrating ABC-type derivatives, proportional delay, and HSV-based solutions for real-world applications.

Figures (7)

• Figure 1: Convergence of HSV method.
• Figure 2: Classical logistic model.
• Figure 3: Exact solution of the ABC-type logistic model for $\lambda = 0$, showing $z(t,\mu)$ as a function of time $t$ and fractional order $\mu$. Smaller $\mu$ values enhance memory effects, slowing growth, while larger $\mu$ reduce memory influence, leading to faster dynamics approaching the classical logistic behavior. The logarithmic scale highlights gradual acceleration over time.
• Figure 4: Comparison of logistic models with three different fractional operators.
• Figure 5: 3D surface of the ABC-type logistic model with proportional delay, showing $z(t,\mu)$ over time $t$ and fractional order $\mu$. Smaller $\mu$ values enhance memory effects and slow growth, while larger $\mu$ reduce memory, approaching classical logistic dynamics.

Theorems & Definitions (6)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• Definition 2.6