Applications of proportional calculus and a non-Newtonian logistic growth model




Proportional arithmetic, Proportional calculus and proportional derivative and integral , Geometric difference, Geometric integer, Proportional differential equations, Proportional wave equation, Proportional heat equation, Proportional logistic growth


On the set of positive real numbers, multiplication, represented by ⊕, is considered as an operation associated with the notion of sum, and the operation a ⨀ b = aln(b) represents the meaning of the traditional multiplication. The triple (R+, ⊕,⨀) forms an ordered and complete field in which derivative and integration operators are defined analogously to the Differential and Integral Calculus. In this article, we present the proportional arithmetic and we construct the theory of ordinary proportional differential equations. A proportional version of Gronwall inequality, Gompertz’s function, the q-Periodic functions, proportional heat, and wave equations as well as a proportional version of Fourier’s series are presented. Furthermore, a non-Newtonian logistic growth model is proposed.

Author Biographies

Manuel Pinto Jiménez, Universidad de Chile.

Dept. de Matemáticas, Fc. de Ciencias.

Ricardo Felipe Torres Naranjo, Universidad Austral de Chile.

Instituto de Ciencias Físicas y Matemáticas, Fc. de Ciencias.

William Campillay-Llanos, Universidad Católica del Maule.

Dept. de Matemática, Física y Estadística, Fc. de Cs. Básicas.

Felipe Guevara Morales, Universidad de Atacama.

Dept. de Matemática.


T. M. Apostol, Análisis matemático, 2nd ed. Barcelona: Reverté, 1996.

A. E. Bashirov, E. M. Kurpinar, and A. Özyapıcı, “Multiplicative calculus and its applications”, Journal of mathematical analysis and applications, vol. 337, no. 1, pp. 36-48, Jan. 2008, doi: 10.1016/j.jmaa.2007.03.081

A. E. Bashirov and M. Riza, “On complex multiplicative differentiation”, TWMS Journal of applied and engineering mathematics, vol. 1, no. 1, pp. 75-85, 2011. [On line]. Available:

C. Birch, “A new generalized logistic sigmoid growth equation compared with the Richards growth equation”, Annals of botany, vol. 83, no. 6, pp. 713-723, Jun. 1999, doi: 10.1006/anbo.1999.0877

K. Boruah and B. Hazarika, “Application of geometric calculus in numerical analysis and difference sequence spaces”, Journal of mathematical analysis and applications, vol. 449, no. 2, pp. 1265-1285, May 2017, doi: 10.1016/j.jmaa.2016.12.066

J. P. Ramis, A. Warusfel, and F. Moulin, Dirs., Cours de Mathématiques pures et appliquées: algèbre et géométrie, vol. 1. Bruxelles: De Boek, 2010.

M. S. Burgin, “Non-diophantine arithmetics or is it possible that 2 2 is not equal to 4?”, Home Page for Markin Burgin – UCLA Library. [Online]. Available:

M. Burgin, “Nonclassical models of the natural numbers”, Uspekhi matematicheskikh nauk, vol. 32, no. 6, pp. 209-210, 1977. [On line]. Available:

K. Boruah and B. Hazarika, “On some generalized geometric difference sequence spaces”, Proyecciones (Antofagasta, On line), vol. 36, no. 3, pp. 373-395, Sep. 2017, doi: 10.4067/S0716-09172017000300373

W. Campillay-Llanos, "Cálculo proporcional: un maravilloso recorrido por el mundo multiplicativo", Undergraduate Thesis, Universidad Metropolitana de Ciencias de la Educación, 2007. [On line]. Available:

W. Campillay-Llanos and F. Guevara-Morales, “Proportional algebraic curves, Geometry session,” in XXXI Jornada matemática de la zona sur, Universidad Austral de Chile, 25, 26 y 27 de abril de 2018, Valdivia, Chile. [On line]. Available:

W. Campillay-Llanos, F. Guevara-Morales, “Gelfond’s constant and proportional arithmetic”, in XXVIII Jornada matemática de la zona sur, Universidad del Bío-Bío, 22 al 24 de abril, Chillán, Chile, 2015. [On line]. Available:

W. Campillay-Llanos, F. Guevara-Morales, M. Pinto, and R. Torres, “Differential and integral proportional calculus: how to find a primitive for Gaussian function”, International journal of mathematical education in science and technology, pp. 1-14, 2020, doi:10.1080/0020739X.2020.1763489

E. J. Chapman and C. J. Byron, “The flexible application of carrying capacity in ecology”, Global ecology and conservation, vol. 13, Art ID. e00365, Jan. 201, doi: 10.1016/j.gecco.2017.e00365

F. Córdova-Lepe, “The multiplicative derivative as a measure of elasticity in economics”, TMAT Revista latinoamericana de ciencias e ingeniería, vol. 2, no. 3, pp. 1-8, Jan. 2006. [On line]. Available:

F. Córdova-Lepe and M. Pinto, “From quotient operation toward a proportional calculus”, International journal of mathematics, game theory and algebra, vol. 18, no. 6, pp. 527-536, 2009.

M. Czachor, “Relativity of arithmetic as a fundamental symmetry of physics”, Quantum studies: mathematics and foundations, vol. 3, no. 2, pp. 123-133, Jun. 2016, doi: 10.1007/s40509-015-0056-4

M. Czachor, “If gravity is geometry, is dark energy just arithmetic?”, International journal of theoretical physics, vol. 56, no 4, pp. 1364-1381, Apr. 2017, doi: 10.1007/s10773-017-3278-x

J. P. DeLong and D. A. Vasseur, “A dynamic explanation of size-density scaling in carnivores”, Ecology, vol. 93, no. 3, pp. 470-476, Mar. 2012, doi: 10.1890/11-1138.1

O. Dovgoshey, O. Martio, V. Ryazanov, and M. Vuorinen, “The cantor function”, Expositiones mathematicae, vol. 24, no. 1, pp. 1-37, Feb. 2006, doi: 10.1016/j.exmath.2005.05.002

R. Katz and M. Grossman, Non-Newtonian calculus. Pigeon Cove, MA: Lee Press, 1972.

M. Grossman, Bigeometric calculus: a system with a scale-free derivative, Rockport, MA: Archimedes Foundation, 1983.

M. Grossman, The first nonlinear system of differential and integral calculus, Rockport, MA: Mathco, 1979.

G. M. N’Guerekata, Ed., New Research on Evolution Equations, New York. NY: Nova Science Publishers, 2010.

J. Hofbauer and K. Sigmund, The theory of evolution and dynamical systems: mathematical aspects of selection, Cambridge: Cambridge University Press, 1988.

J. Kirkwood, Mathematical physics with partial differential equations, 2nd ed. London: Academic Press, 2018.

B. Kooi, M. Boer, and S. Kooijman, “On the use of the logistic equation in models of food chains”, Bulletin of mathematical biology, vol. 60, no. 2, pp. 231-246, 1998, doi: 10.1006/bulm.1997.0016

D. Laugwitz, “On the historical development of infinitesimal mathematics”, The American mathematical monthly, vol. 104, no. 7, pp. 654-663, 1997, doi: 10.2307/2975060

M. Mora, F. Córdova-Lepe, and R. Del-Valle, “A non-newtonian gradient for contour detection in images with multiplicative noise”, Pattern recognition letters, vol. 33, no. 10, pp. 1245-1256, Jul. 2012, doi: 10.1016/j.patrec.2012.02.012

S. Pawar, “The role of body size variation in community assembly”, Advances in ecological research, vol. 52, pp. 201-248, 2015, doi: 10.1016/bs.aecr.2015.02.003

M. Peleg, M. G. Corradini, and M. D. Normand, “The logistic (Verhulst) model for sigmoid microbial growth curves revisited”, Food research international, vol. 40, no. 7, pp. 808-818, Aug. 2007, doi: 10.1016/j.foodres.2007.01.012

O. Perron, “Die stabilitätsfrage bei differentialgleichungen”, Mathematische zeitschrift, vol. 32, pp. 703-728, Dec. 1930, doi: 10.1007/BF01194662

M. Pinto and W. Campillay-Llanos, “Proportional differential equations”, in VIII Congreso de Análisis Funcional y Ecuaciones de Evolución, Universidad de Santiago de Chile, 20 al 23 de Noviembre de 2013, Santiago, Chile. [On line]. Available:

M. Pinto and W. Campillay-Llanos, “Proportional logistic growth”, in 2do Workshop Modelamiento Matemático de Sistemas Biológicos, Universidad Tecnológica Metropolitana, 10 al 12 de Enero de 2018, Santiago Chile. [On line], Available:

V. D. Put and M. F Singer, Galois theory of linear differential equations, Berlin: Springer, 2003, doi: 10.1007/978-3-642-55750-7

A. Slavík, Product integration, its history and applications. Prague: Matfyzpress, 2007. [On line]. Available:

V. Volterra and B. Hostinsky, Opérations infinitésimales linéaire,. Paris: Gauthier-Villars, 1938.

J. S. Weitz and S. A. Levin, “Size and scaling of predator-preydynamics”, Ecology letters, vol. 9, no. 5, pp. 548-557, May 2006, doi: 10.1111/j.1461-0248.2006.00900.x

C. P. Winsor, “The Gompertz curve as a growth curve”, Proceedings of the national academy of sciences, vol. 18, no. 1, pp. 1-8, Jan. 1932, doi: 10.1073/pnas.18.1.1

T. Witelski and M. Bowen, Methods of mathematical modelling, Cham: Springer 2015.



How to Cite

M. Pinto Jiménez, R. F. Torres Naranjo, W. Campillay-Llanos, and F. Guevara Morales, “Applications of proportional calculus and a non-Newtonian logistic growth model”, Proyecciones (Antofagasta, On line), vol. 39, no. 6, pp. 1471-1513, Nov. 2020.