# How does exponential growth differ from logistic growth?

Sep 22, 2015

Logistic growth is also a suitably scaled and translated (in both the $x$ and $y$ axes) hyperbolic tangent function, which happens to have useful properties that might be exploited in certain mathematical models.

#### Explanation:

Note

$\sinh x = \frac{{e}^{x} - {e}^{- x}}{2}$

and

$\cosh x = \frac{{e}^{x} + {e}^{- x}}{2}$

so that

$\tanh x = \frac{\sinh x}{\cosh x} = \frac{{e}^{x} - {e}^{- x}}{{e}^{x} + {e}^{- x}}$

Dividing through by ${e}^{x}$ yields

$\frac{1 - {e}^{- 2 x}}{1 + {e}^{- 2 x}}$

Translating in the y-axis by 1 (in the positive direction) yields

(1 - e^(-2x))/(1 + e^(-2x)) + 1 = ((1 - e^(-2x))+ (1 + e^(-2x)))/(1 + e^(-2x)) = 2/((1 + e^(-2x))

Scaling this in the y-axis by $\frac{1}{2}$ yields

2/((1 + e^(-2x))) *1/2 = 1/((1 + e^(-2x))

Compare this with the answer given in the previous explanation shown below. This particular equation comprises a hyperbolic tangent function scaled and translated in the y-axis so that it lies between horizontal asymptotes $y = 0$ and $y = 1$. It provides a model of growth that satisfies particular requirements, including starting from close to zero and reaching a maximum upper limit (in this case, 1). As the upper limit is 1, it is not suitable to model the growth of discrete objects (eg the population of animals under certain circumstances) with these scaling factors. A suitable scaling value in the y-axis is chosen to model the upper limit of the population size (eg 100 has been chosen in the example given in the previous explanation shown below).

Scaled (as in this case) to lie between the asymptotes $y = 0$ and $y = 1$, it forms the basis of estimating values of $p$ for some binary outcome in the general linear model known as logistic regression (also known as binary regression), as used in inferential statistics. In the form given here (the outcome of scaling and translating in the y-axis), the function happens to be scaled in the x-axis by a factor of 2. Linear regression (on the log of the odds) is used to find suitable values of $\alpha$ and $\beta$ to scale and translate in the x-axis (to fit the observed outcome data) according to the expression (the standard form for logistic regression)

 1/((1 + e^(- (alpha + beta x)))

Note that $\alpha + \beta x$ is a first order constant coefficient polynomial in $x$ (that is, it is linear). The values of $\alpha$ and $\beta$ that are found by linear regression would be the parameters for some particular dataset.