A note on fractional-type models of population dynamics
Abstract
The fractional exponential function is considered. General expansions in fractional powers are used to solve fractional population dynamics problems. Laguerretype exponentials are also considered, and an application to Laguerre-type fractional logistic equation is shown.
Keyword : fractional exponential function, Mittag-Leffler functions and generalizations, fractional population dynamic models, Laguerre-type exponentials, fractional Laguerre-type models
How to Cite
Caratelli, D., & Ricci, P. E. (2024). A note on fractional-type models of population dynamics. Mathematical Modelling and Analysis, 29(3), 480–492. https://doi.org/10.3846/mma.2024.19588
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
R. Almeida, Bastos Nuno R.O. and M.T.T. Monteiro. A fractional Malthusian growth model with variable order using an optimization approach. Statistics, Optimization & Information Computing, 6(1), 2018. https://doi.org/10.19139/soic.v6i1.465
L. Beghin and M. Caputo. Commutative and associative properties of the Caputo fractional derivative and its generalizing convolution operator. Communications in Nonlinear Science and Numerical Simulation, 89:105338, 2020. https://doi.org/10.1016/j.cnsns.2020.105338
G. Bretti and P.E. Ricci. Laguerre-type special functions and population dynamics. Applied Mathematics and Computation, 187(1):89–100, 2007. https://doi.org/10.1016/j.amc.2006.08.106
D. Caratelli, P. Natalini and P.E. Ricci. Fractional differential equations and expansions in fractional powers. Symmetry, 15:1842, 2023. https://doi.org/110.3390/sym15101842
G. Consolini and M. Materassi. A stretched logistic equation for pandemic spreading. Chaos, Solitons & Fractals, 140:110113, 2020. https://doi.org/10.1016/j.chaos.2020.110113
L. Cristofaro, R. Garra, E. Scalas and I. Spassiani. A fractional approach to study the pure-temporal epidemic type aftershock sequence (ETAS) process for earthquakes modeling. Fractional Calculus and Applied Analysis, 26(2):461–479, 2023. https://doi.org/10.1007/s13540-023-00144-5
G. Dattoli. Hermite–Bessel and Laguerre–Bessel functions: A by-product of the monomiality principle. In D. Cocolicchio, G. Dattoli and H.M. Srivastava(Eds.), Advanced Special Functions and Applications, Proc. Melfi School on Advanced Topics in Mathematics and Physics, 1999, pp. 147–164, Rome, Italy, 2000. Aracne Editrice.
P. Delerue. Sur le calcul symbolique à n variables et fonctions hyper-besseliennes (ii). Ann. Soc. Sci. Bruxelles, Ser. 1, n. 3:229–274, 1952.
I. Dimovski. Operational calculus for a class of differential operators. C. R. Acad. Bulgare Sci., 19:1111–1114, 1966.
V.A. Ditkin and A.P. Prudnikov. Integral Transforms and Operational Calculus. By V.A. Ditkin and A.P. Prudnikov. International series of monographs in pure and applied mathematics. Pergamon Press, 1965. https://doi.org/10.2307/3613667
M.M. Dzrbashjan. On the integral transformations generated by the generalized Mittag-Leffler function. Izv. AN Arm. SSR, 13(3):21–63, 1960.
M. D’Ovidio and P. Loreti. Solutions of fractional logistic equations by Euler’s numbers. Physica A: Statistical Mechanics and its Applications, 506:1081–1092, 2018. https://doi.org/10.1016/j.physa.2018.05.030
M. D’Ovidio, P. Loreti and S. Sarv Ahrabi. Modified fractional logistic equation. Physica A: Statistical Mechanics and its Applications, 505:818–824, 2018. https://doi.org/10.1016/j.physa.2018.04.011
A.M.A. El-Sayed, A.E.M. El-Mesiry and H.A.A. El-Saka. On the fractionalorder logistic equation. Applied Mathematics Letters, 20(7):817–823, 2007. https://doi.org/10.1016/j.aml.2006.08.013
R. Garra and F. Polito. On some operators involving Hadamard derivatives. Integral Transforms and Special Functions, 24(10):773–782, 2013. https://doi.org/10.1080/10652469.2012.756875
S. Gerhold. Asymptotics for a variant of the Mittag–Leffler function. Integral Transforms and Special Functions, 23(6):397–403, 2012. https://doi.org/10.1080/10652469.2011.596151
R. Gorenflo and F. Mainardi. Fractional Calculus, pp. 223–276. Springer Vienna, 1997. https://doi.org/10.1007/978-3-7091-2664-6_5
G. Groza and M. Jianu. Functions represented into fractional Taylor series. ITM Web of Conferences, 29:01017, 2019. ISSN 2271-2097. https://doi.org/10.1051/itmconf/20192901017
V. Kiryakova. Multiindex Mittag-Leffler functions, related Gelfond-Leontiev operators and Laplace type integral transforms. Fract. Calc. Appl. Anal, 2(4):445– 462, 1999.
V. Kiryakova. The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus. Comput Math. Appl., 59(5):1885–1895, 2010. https://doi.org/10.1016/j.camwa.2009.08.025
V. Kiryakova. A guide to special functions in fractional calculus. Mathematics, 9(1):106, 2021. https://doi.org/10.3390/math9010106
V. Kiryakova. Multiple Dzrbashjan-Gelfond-Leontiev fractional differintegrals. In Recent Advances in Applied Mathematics, Proc. Intern. Workshop RAAM ’96, Kuwait, pp. 281–294, May 4–7, 1996.
V.S. Kiryakova. Multiple (multiindex) Mittag–Leffler functions and relations to generalized fractional calculus. Journal of Computational and Applied Mathematics, 118(1–2):241–259, 2000. https://doi.org/10.1016/s0377-0427(00)00292-2
F. Mainardi, Y. Luchko and G. Pagnini. The fundamental solution of the spacetime fractional diffusion equation. Fract. Calc. Appl. Anal, 4:153–192, 2001.
A.C. McBride. V. Kiryakova generalized fractional calculus and applications (Pitman research notes in mathematics vol. 301, Longman 1994), 388 pp., 0 582 21977 9, £39. Proceedings of the Edinburgh Mathematical Society, 38(1):189–190, 1995. https://doi.org/10.1017/s0013091500006325
P. Natalini and P.E. Ricci. Laguerre-type linear dynamical systems. Ilirias J. Math., 4(1):24–40, 2015. Available on Internet: http://www.ilirias.com
J. Paneva-Konovska, V. Kiryakova, S. Rogosin and M. Dubatovskaya. Laplace transform (part 1) of the multi-index Mittag-Leffler-Prabhakar functions of Le Roy type. International Journal of Apllied Mathematics, 36(4), 2023. https://doi.org/10.12732/ijam.v36i4.2
P.E. Ricci and I. Tavkhelidze. An introduction to operational techniques and special polynomials. Journal of Mathematical Sciences, 157(1):161–189, 2009. https://doi.org/10.1007/s10958-009-9305-6
S.G. Samko, A.A. Kilbas and Marichev O. I. Fractional integrals and derivatives: Theory and applications. 1993. Available on Internet: https://api.semanticscholar.org/CorpusID:118631078
J.C.A. Soares, S. Jarosz and F.S. Costa. Fractional growth models: Malthus and Verhulst. C.Q.D. - Revista Eletrônica Paulista de Matemática, 22(2):162–177, 2022. https://doi.org/10.21167/cqdv22n22022162177
S.B. Yakubovich and Y.F. Luchko. The Hypergeometric Approach to Integral Transforms and Convolutions. Springer Netherlands, 1994. https://doi.org/10.1007/978-94-011-1196-6
L. Beghin and M. Caputo. Commutative and associative properties of the Caputo fractional derivative and its generalizing convolution operator. Communications in Nonlinear Science and Numerical Simulation, 89:105338, 2020. https://doi.org/10.1016/j.cnsns.2020.105338
G. Bretti and P.E. Ricci. Laguerre-type special functions and population dynamics. Applied Mathematics and Computation, 187(1):89–100, 2007. https://doi.org/10.1016/j.amc.2006.08.106
D. Caratelli, P. Natalini and P.E. Ricci. Fractional differential equations and expansions in fractional powers. Symmetry, 15:1842, 2023. https://doi.org/110.3390/sym15101842
G. Consolini and M. Materassi. A stretched logistic equation for pandemic spreading. Chaos, Solitons & Fractals, 140:110113, 2020. https://doi.org/10.1016/j.chaos.2020.110113
L. Cristofaro, R. Garra, E. Scalas and I. Spassiani. A fractional approach to study the pure-temporal epidemic type aftershock sequence (ETAS) process for earthquakes modeling. Fractional Calculus and Applied Analysis, 26(2):461–479, 2023. https://doi.org/10.1007/s13540-023-00144-5
G. Dattoli. Hermite–Bessel and Laguerre–Bessel functions: A by-product of the monomiality principle. In D. Cocolicchio, G. Dattoli and H.M. Srivastava(Eds.), Advanced Special Functions and Applications, Proc. Melfi School on Advanced Topics in Mathematics and Physics, 1999, pp. 147–164, Rome, Italy, 2000. Aracne Editrice.
P. Delerue. Sur le calcul symbolique à n variables et fonctions hyper-besseliennes (ii). Ann. Soc. Sci. Bruxelles, Ser. 1, n. 3:229–274, 1952.
I. Dimovski. Operational calculus for a class of differential operators. C. R. Acad. Bulgare Sci., 19:1111–1114, 1966.
V.A. Ditkin and A.P. Prudnikov. Integral Transforms and Operational Calculus. By V.A. Ditkin and A.P. Prudnikov. International series of monographs in pure and applied mathematics. Pergamon Press, 1965. https://doi.org/10.2307/3613667
M.M. Dzrbashjan. On the integral transformations generated by the generalized Mittag-Leffler function. Izv. AN Arm. SSR, 13(3):21–63, 1960.
M. D’Ovidio and P. Loreti. Solutions of fractional logistic equations by Euler’s numbers. Physica A: Statistical Mechanics and its Applications, 506:1081–1092, 2018. https://doi.org/10.1016/j.physa.2018.05.030
M. D’Ovidio, P. Loreti and S. Sarv Ahrabi. Modified fractional logistic equation. Physica A: Statistical Mechanics and its Applications, 505:818–824, 2018. https://doi.org/10.1016/j.physa.2018.04.011
A.M.A. El-Sayed, A.E.M. El-Mesiry and H.A.A. El-Saka. On the fractionalorder logistic equation. Applied Mathematics Letters, 20(7):817–823, 2007. https://doi.org/10.1016/j.aml.2006.08.013
R. Garra and F. Polito. On some operators involving Hadamard derivatives. Integral Transforms and Special Functions, 24(10):773–782, 2013. https://doi.org/10.1080/10652469.2012.756875
S. Gerhold. Asymptotics for a variant of the Mittag–Leffler function. Integral Transforms and Special Functions, 23(6):397–403, 2012. https://doi.org/10.1080/10652469.2011.596151
R. Gorenflo and F. Mainardi. Fractional Calculus, pp. 223–276. Springer Vienna, 1997. https://doi.org/10.1007/978-3-7091-2664-6_5
G. Groza and M. Jianu. Functions represented into fractional Taylor series. ITM Web of Conferences, 29:01017, 2019. ISSN 2271-2097. https://doi.org/10.1051/itmconf/20192901017
V. Kiryakova. Multiindex Mittag-Leffler functions, related Gelfond-Leontiev operators and Laplace type integral transforms. Fract. Calc. Appl. Anal, 2(4):445– 462, 1999.
V. Kiryakova. The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus. Comput Math. Appl., 59(5):1885–1895, 2010. https://doi.org/10.1016/j.camwa.2009.08.025
V. Kiryakova. A guide to special functions in fractional calculus. Mathematics, 9(1):106, 2021. https://doi.org/10.3390/math9010106
V. Kiryakova. Multiple Dzrbashjan-Gelfond-Leontiev fractional differintegrals. In Recent Advances in Applied Mathematics, Proc. Intern. Workshop RAAM ’96, Kuwait, pp. 281–294, May 4–7, 1996.
V.S. Kiryakova. Multiple (multiindex) Mittag–Leffler functions and relations to generalized fractional calculus. Journal of Computational and Applied Mathematics, 118(1–2):241–259, 2000. https://doi.org/10.1016/s0377-0427(00)00292-2
F. Mainardi, Y. Luchko and G. Pagnini. The fundamental solution of the spacetime fractional diffusion equation. Fract. Calc. Appl. Anal, 4:153–192, 2001.
A.C. McBride. V. Kiryakova generalized fractional calculus and applications (Pitman research notes in mathematics vol. 301, Longman 1994), 388 pp., 0 582 21977 9, £39. Proceedings of the Edinburgh Mathematical Society, 38(1):189–190, 1995. https://doi.org/10.1017/s0013091500006325
P. Natalini and P.E. Ricci. Laguerre-type linear dynamical systems. Ilirias J. Math., 4(1):24–40, 2015. Available on Internet: http://www.ilirias.com
J. Paneva-Konovska, V. Kiryakova, S. Rogosin and M. Dubatovskaya. Laplace transform (part 1) of the multi-index Mittag-Leffler-Prabhakar functions of Le Roy type. International Journal of Apllied Mathematics, 36(4), 2023. https://doi.org/10.12732/ijam.v36i4.2
P.E. Ricci and I. Tavkhelidze. An introduction to operational techniques and special polynomials. Journal of Mathematical Sciences, 157(1):161–189, 2009. https://doi.org/10.1007/s10958-009-9305-6
S.G. Samko, A.A. Kilbas and Marichev O. I. Fractional integrals and derivatives: Theory and applications. 1993. Available on Internet: https://api.semanticscholar.org/CorpusID:118631078
J.C.A. Soares, S. Jarosz and F.S. Costa. Fractional growth models: Malthus and Verhulst. C.Q.D. - Revista Eletrônica Paulista de Matemática, 22(2):162–177, 2022. https://doi.org/10.21167/cqdv22n22022162177
S.B. Yakubovich and Y.F. Luchko. The Hypergeometric Approach to Integral Transforms and Convolutions. Springer Netherlands, 1994. https://doi.org/10.1007/978-94-011-1196-6