> For the complete documentation index, see [llms.txt](https://solomons-mathbook.gitbook.io/calculus-basics/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://solomons-mathbook.gitbook.io/calculus-basics/comment-396484900/comment-396537354.md).

# Logistic Growth Model

```
#LogisticGrowth #LogisticGrowthModel #LogisticEquation #LogisticModel #LogisticRegression
```

This is a very famous example of Differential Equation, and has been applied to numerous of real life problems as a model. It's originally a `Population Model` created by `Verhulst`, as studying the `population's growth`.

Refer to lectures: [▶Khan academy](https://www.youtube.com/watch?v=oiDvNs15tkE), [▶MIT Gilbert Strang's](https://www.youtube.com/watch?v=TCkLSYxx21c), [▶The Organic Chemistry Tutor](https://www.youtube.com/watch?v=JgMvB22XQs0), [▶Krista King](https://www.youtube.com/watch?v=uemhtqZHnak), [▶Bozeman Science](https://www.youtube.com/watch?v=rXlyYFXyfIM)

## `Intuition & Origin of Logistic Growth Model`

Refer to Khan academy: [▶Logistic models & differential equations (Part 1)](https://www.khanacademy.org/math/ap-calculus-bc/bc-diff-equations/modal/v/modeling-population-with-differential-equations)

Let's let `P(t)` as the population's size in term of time `t`, and `dP/dt` represents the Population's growth.

### Malthus' Exponential growth theory of population

Mr. Malthus first introduced the exponential growth theory for the population by using a fairly simple equation: ![image](https://user-images.githubusercontent.com/14041622/41335333-06282114-6f1b-11e8-8535-a4c28551be02.png) Where `P` is the "Population Size", `t` is the "Time", `r` is the "Growth Rate".

### Verhulst's Logistic growth theory of population

Mr. Verhulst enhanced the `exponential growth theory of population`, as saying that the population's growth is **NOT ALWAYS** growing, but there is always a certain **LIMIT** or a `Carrying Capacity` to the **exponential growth**. And combining the `exponential growth` with a `limit`, it's then called the `Logistic Growth`.

And the logistic growth got its equation:

![image](https://user-images.githubusercontent.com/14041622/41333641-1f0e7c6a-6f15-11e8-839b-ee22a799810b.png) Where `P` is the "Population Size" (N is often used instead), `t` is "Time", `r` is the "Growth Rate", `K` is the **"Carrying Capacity"**. And the `(1 - P/K)` determines how close is the Population Size to the Limit `K`, which means as the population gets closer and closer to the limit, the growth gets slower and slower.

> "It explains how density dependent limiting factors eventually decrease the growth rate until a population reaches a Carrying Capacity ( K )."

### Carrying Capacity

Carrying Capacity means the "celling", the "limit", the "asymptote".

![image](https://user-images.githubusercontent.com/14041622/41334348-e75e1b88-6f17-11e8-99b7-6c46a3a60d01.png)

### Get the Original Population Function P(t)

> It's gonna use the method `Separable Equations`, which introduced the `initial condition` as `P₀` in this case.

We could directly solve the Logistic Equation as solving differential equation to get the `antiderivative`: ![image](https://user-images.githubusercontent.com/14041622/41335635-0711f95a-6f1c-11e8-8d4d-46ebb50874de.png)

But we still have a constant `C` in the `antiderivative`, which required us to introduce an `Initial Condition` to get rid of `C` and get the specific function:

![image](https://user-images.githubusercontent.com/14041622/41333555-d064b0ac-6f14-11e8-9bb4-51df158b18ac.png)

## `Solving Logistic Model Problems`

### Example

![image](https://user-images.githubusercontent.com/14041622/41399634-62ad394c-6fed-11e8-83d3-fbd99e345644.png) Solve:

* We know the Logistic Equation is `dP/dt = r·P(1-P/K)`.
* So twist the given derivative to the logistic form: `dy/dt = 10·y(1-y/600)`.
* Then we could see the `K = 600`, which is the limit, the Carrying capacity.

  ![image](https://user-images.githubusercontent.com/14041622/41399720-a8dc6ba4-6fed-11e8-9329-89113ec3d7c4.png)

### Example

![image](https://user-images.githubusercontent.com/14041622/41399831-eddc319e-6fed-11e8-948a-f9c68c3efb03.png) Solve:

* It's asking "growing fastest", means the Max value of Sale's function `S(c)`.
* For the max value of function, we let `S'(c) = dS/dc = 0`
* And we get `S = 0 or 20,000,000`
* So at two points they are getting fastest growing. 
* Yet we have to take the **AVERAGE** of the two points, which is `(0+20,000,000)/2 = 10,000,000`.

## `Logistic Regression`