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Logistic Equation

The logistic equation plays a vital role in mathematics, especially for the study of growth model. Logistics equation is also known as Verhulst model which is defined as a population growth model, introduced by Pierre-Francois Verhulst in 1838.

The logistic equation is applied not only in mathematics, but also in a wide range of areas such as - biology, chemistry, neural networks, ecology, economics, bio-mathematics, demography, mathematical psychology, geoscience, probability theory and statistics, political science, sociology etc.

In this article below, we are going to focus on the concepts related to logistic equation. This includes the study of logistic function, logistic differential equation, logistic function, logistic regression equation etc. So, let us go ahead and get a conceptual knowledge about logistic equation in detail.

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Logistic Function

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logistic function is a sigmoid curve or an S-shaped curve. It is represented by the following equation:
$f(x)$ = $\frac{1}{1 + e^{-x}}$ 
Where, e is the base of natural logarithm or Euler's number. The variable x is within the real number range; i.e from $-\infty$ to $\infty$.

The graph of logistic function is demonstrated below:
Logistic Function Graph
It describes the growth function whose initial stage is exponential. As the saturation comes, the growth slows down. Also the growth stops at maturity.

Logistic Differential Equation

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The logistic differential equation is represented by the following relation:
$\frac{dP}{dt}$ = kP(1 - $\frac{P}{x}$ )
P be the population
k be the proportionality constant
x be the carrying capacity which is actually the point where the growth slows down and nearly reaches to a limit.

We shall solve this equation by separation of variables,

$\frac{dP}{dt}$ = kP(1 - $\frac{P}{x}$)

$\frac{dP}{dt}$ = kP ($\frac{x-P}{x}$)

$\frac{x dP}{P(x -  P)}$
= $k dt$

Integrating both sides,

$\int$ $\frac{x} {P(x -  P)}$ $dP$ = $\int k dt$

By using partial fractions,

$\int$ ($\frac{1}{P}$ - $\frac{1}{x -  P}$) $dP$ = $\int k dt$

$log\ P - log(x -  P) = k t + C$

$log$ ($\frac{P}{x - P}$) =$k t + C$

Logistic Growth Equation

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According to logistic growth equation, the population grows at some rate which is proportional to the population. Let us suppose that P be the population that is defined as the function of time (say t) and if r be the proportionality constant.

Then logistics growth model is given by:
$\frac{dP}{dt}$ = r P
The solution of this equation will be:
$P(t)$ =$P_{0} e^{rt}$
Where $P_{0}$ denotes the population when time t is zero. We can say that it represents approximately an exponential growth.

Logistic Regression Equation

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The logistic regression is the type of regression that measures the kind of relation between the categorical independent variable and dependent variable (or variables). These variables are generally not continuous variables.

The logistic regression is being utilized as the special case of generalized linear regression model. Therefore, it is analogous to the linear regression. This regression depends on some different assumptions based upon the relation of independent and dependent variables.

The logistic regression is actually defined as the statistical method used for the analysis of given data having one or more independent variables in order to find an outcome. Also, the outcomes are determined with the help of a binary variable i.e. a variable which has only two possibilities, such as: true and false, success and failure, pass or fail etc. The main purpose of logistic regression is finding a model that is best fit and reasonable in order to describe the given relation between independent and dependent variables.

In logistic regression equation, the logistic function is utilized in order to model the probability (say p) of some event which is affected by one or more independent variables.

An example of logistic regression equation would be:
p = P(a + bx)
Where, a and b are the parameters to be fitted in the model.
$f(p)$ = $\frac{1}{1 + e^{-p}}$
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