A generalization of the exponential function to model growth

Abstract

We generalize the exponential function to model instantaneous relative growth. The modified function is defined by a linear relationship between a continuous quantity (rather than time) and logarithmic relative growth. The corresponding formula is ln[q'(t)/q(t)] = a+b·q(t) where q'(t)/q(t) is instantaneous relative growth of a quantity q, t refers to time, a denotes initial logarithmic relative growth, and b is a shape parameter in terms of its sign, as well as a scaling parameter in terms of its magnitude. For calculating q(t), the exponential integral Ei[-b·q] = ∫(exp[-b·q]/q)dq is needed. The problem of taking the inverse of Ei[x] = zEi is addressed. In order to distinguish two possible solutions for given zEi, we define the two inverse functions Ei(-1)x > 0[zEi] and Ei(-1)x < 0[zEi]. An indirect method for their numerical evaluation is developed. With the generalized exponential function, one can model sigmoid growth (b < 0), exponential growth (b = 0), and explosive growth (b > 0), where the term "explosive growth" refers to a relative growth rate that increases with time. The resulting formula of generalized exponential growth is where qC is a calibrating quantity at time tC. For b = 0, the two functions equal the (standard) exponential function q(t) = qC· exp[(t-tc)·exp[a]]. In the case of sigmoid growth, the inflection point quantity is -1/b, which depends only on one parameter (b). Negative growth can be modeled by substituting t-tC with tC - t. Any two points of logarithmic relative growth can be connected unambiguously with the generalized exponential function, to derive the corresponding function of q(t). Furthermore, we derive formulas for the conversion of a segmented curve of logarithmic relative growth as a function of time, into an equivalent growth curve of q(t).Finally, the generalized exponential function is compared with a 2nd-degree polynomial and the nonlinear Schnute function. In conclusion, the generalized exponential function is useful for modeling a path of changing relative growth continuously, and to translate it into a growth curve of quantity as a function of time.

Keywords

Explosive growth; Exponential function; Exponential growth; Exponential integral Ei[x]; Growth curves; Inverse of the exponential integral function; Schnute function; Sigmoid growth

Published in

International Journal of Applied Mathematics
2018, volume: 48, number: 2, pages: 152-167
Publisher: International Association of Engineers

SLU Authors

• von Rosen, Dietrich

  • Department of Energy and Technology, Swedish University of Agricultural Sciences
    
  • Linköping University

UKÄ Subject classification

Other Mathematics
Probability Theory and Statistics

Permanent link to this page (URI)

https://res.slu.se/id/publ/130246