Logistic Growth Model
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, ▶MIT Gilbert Strang's, ▶The Organic Chemistry Tutor, ▶Krista King, ▶Bozeman Science
Intuition & Origin of Logistic Growth Model
Intuition & Origin of Logistic Growth Model
Refer to Khan academy: ▶Logistic models & differential equations (Part 1)
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
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:
"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".
Get the Original Population Function P(t)
It's gonna use the method
Separable Equations
, which introduced theinitial condition
asP₀
in this case.
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:
Solving Logistic Model Problems
Solving Logistic Model Problems
Example
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.
Example
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
Logistic Regression
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