9.4 Differential Equations
Population Growth Model
Question #1 (Easy): Population Carrying Capacity
( ) is known as the logistic differential equation, where is the carrying capacity, and is the
relative growth rate constant factor.
If the given equation is not in this form, needs to be factored in order to determine and .
The population follows after the equation, where is in years.
, ( )
1) What is the carrying capacity and the relative growth rate?
2) What is ( ) ?
3) When will the population reach 50% of the carrying capacity?
1) Given the equation, factor out the factor: ( ) ( ).
Then comparing to the logistic differential equation form, ( ) ( ), the
carrying capacity , and relative growth rate term
2) In order to find ( ) , first ( ) needs to be established. This involves solving the given differential
equation: ; ; into partial fractions: , then ( )
( ) ( )
; , therefore