The Logistic Growth Equation: A Generalized Approach
Keywords:
Time kernels, Saturation time, Generalized derivatives, Generalized logistic growth, Adaptive numerical methods, Parametric sensitivity analysis.Abstract
This work presents an exhaustive study of the generalized logistic growth equation using local fractional derivatives defined by time kernels F(t,a). This paper analyzes how different time structures and variations in the fractional order a є (0,1] modify the saturation dynamics of the model. Four kernel families (algebraic and exponential) are considered, combined with three types of base growth rates: constant, dependent on the fractional order, and linearly increasing with time. The results, obtained through numerical integration using the adaptive RK45 method and semi-analytical solutions, reveal that the kernel choice critically determines the initial behavior of the system, potentially inducing accelerated growth or prolonged suppression (lags). Sensitivity analysis shows that the impact of the a parameter is predominant in the intermediate growth stages, although in certain exponential kernels, the differences persist even in the saturation regime. Additionally, the computational effort and saturation time are characterized, demonstrating that kernels with singularities at the origin or those that significantly delay growth impose a greater numerical demand.
AMS Subject Classification (2020): Primary 26A33, Secondary 34A08, 65L05, 92B05.
References
[1] Abreu Blaya R, Fleitas A, Nápoles Valdés JE, Reyes R, Rodríguez JM, Sigarreta JM. On the conformable fractional logistic models. Math Methods Appl Sci. 2020;43:4156–67.
[2] Castillo RE, Nápoles Valdes JE, Chaparro H. Omega derivative. Gulf J Math. 2024;16(1):55–67. doi: 10.56947/gjom.v16i1.1430
[3] Chatzarakis GE, Nápoles Valdes JE. Continuability of Liénard’s type system with generalized local derivative. Discontinuity Nonlinearity Complex. 2023;12(01):1–11.
[4] Fleitas A, Méndez Bermúdez JA, Nápoles Valdes JE, Sigarreta Almira JM. On fractional Liénard type systems. Rev Mex Fís. 2019;65(6):618–25.
[5] Fleitas A, Nápoles Valdés JE, Rodríguez JM, Sigarreta JM. Note on the generalized conformable derivative. Rev la UMA. 2021;62(2):443–57.
[6] Guzmán PM, Langton G, Lugo LM, Medina J, Nápoles Valdé JE. A new definition of a fractional derivative of local type. J Math Anal. 2018;9(2):88–98.
[7] Guzmán PM, Nápoles Valdes JE, Vivas-Cortez M. A new generalized derivative and related properties. Appl Math Inf Sci. 2024;18(5):923–32. doi: 10.18576/amis/180501
[8] Khalil R, Al Horani M, Yousef A, Sababheh M. A new definition of fractional derivative. J Comput Appl Math. 2014;264:65–70.
[9] Martínez F, Nápoles Valdes JE. Towards a non-conformable fractional calculus of n variables. J Math Appl. 2020;43(1):87–98.
[10] Nápoles Valdes JE, Guzmán PM, Lugo Motta Bittencourt LM. Some new results on nonconformable fractional calculus. Adv Dyn Syst Appl. 2018;13(2):167–75.
[11] Nápoles Valdés JE, Guzmán PM, Lugo LM, Kashuri A. The local generalized derivative and Mittag-Leffler function. Sigma J Eng Nat Sci. 2020;38(2):1007–17.
[12] Nápoles Valdes JE, Brisa AR, Lautaro S, Moragues SMAF. Numerical simulation of local generalized derivative for a harmonic oscillator. J Phys Appl Mech. 2022;1(1):1003.
[13] Nápoles Valdes JE, Roa PM. Numerical simulations in a generalized Liénard’s type system. Ann Math Phys. 2023;6(2):187–95. doi: 10.17352/amp.000101
[14] Nápoles Valdés JE, Torres MO. Numerical simulation of the local generalized derivative for the fractional logistic model. J Math Artif Intell. 2025;1(1):11–22.
[15] Podlubny I. Fractional differential equations. San Diego: Academic Press; 1998.
[16] Tsoularis A, Wallace J. Analysis of logistic growth models. Math Biosci. 2002;179(1):21–55.
[17] Verhulst PF. Notice sur la loi que la population suit dans son accroissement. Corresp Math Phys. 1838;10:113–21.
[18] Verhulst PF. Recherches mathématiques sur la loi d’accroissement de la population. Nouv Mém Acad R Sci Belles-Lett Brux. 1845;18:1–42.
[19] Vivas-Cortez M, Guzmán PM, Lugo LM, Nápoles Valdés JE. Fractional calculus: historical notes. Rev MATUA. 2021;8(2):1–13.
[20] Vivas Cortez M, Lugo LM, Nápoles Valdes JE, Samei ME. A multi index generalized derivative; some introductory notes. Appl Math Inf Sci. 2022;16(6):883–90.
[21] Zhao D, Luo M. General conformable fractional derivative and its physical interpretation. Calcolo. 2017;54:903–17. doi: 10.1007/s10092-017-0213-8
