# 007A Sample Midterm 2, Problem 4 Detailed Solution

Assume  ${\displaystyle N(t)}$  denotes the size of a population at time  ${\displaystyle t}$  and that  ${\displaystyle N(t)}$  satisfies the equation:

${\displaystyle {\frac {dN}{dt}}=3N{\bigg (}1-{\frac {N}{20}}{\bigg )}.}$

Let  ${\displaystyle f(N)=3N{\bigg (}1-{\frac {N}{20}}{\bigg )},~N\geq 0.}$  Graph  ${\displaystyle f(N)}$  as a function of  ${\displaystyle N}$  and identify all equilibria. That is, all points where  ${\displaystyle {\frac {dN}{dt}}=0.}$

Background Information:
If  ${\displaystyle ab=0,}$  then  ${\displaystyle a=0}$  or  ${\displaystyle b=0.}$

Solution:

Step 1:
First, we start by graphing  ${\displaystyle f(N).}$
Step 2:
Now, we want to solve
${\displaystyle 0=3N{\bigg (}1-{\frac {N}{20}}{\bigg )}.}$
Then, we have  ${\displaystyle 3N=0}$  or  ${\displaystyle 1-{\frac {N}{20}}=0.}$
This means either  ${\displaystyle N=0}$  or  ${\displaystyle N=20.}$
Therefore, the equilibria occur at
${\displaystyle N=0}$  and  ${\displaystyle N=20.}$

${\displaystyle N=0}$  and  ${\displaystyle N=20}$