Please explain about logistic equation
dN/dt = (a-bN)N where a and b are positive number
if W(0)=N_{0 N=0 and N=a/b is stable or unstable? I try to simplify the eq. to: N(a-bN) = Na(1-(b/a)N) but it does not explain anything, please help }
Please explain about logistic equation
dN/dt = (a-bN)N where a and b are positive number
if W(0)=N_{0 N=0 and N=a/b is stable or unstable? I try to simplify the eq. to: N(a-bN) = Na(1-(b/a)N) but it does not explain anything, please help }
If you set the derviative equal to zero you will get the constant solutions to the equation. They are
$\displaystyle N=\frac{a}{b}$ and $\displaystyle N=0$
This divides the plane into three parts.
If you pick an N value less than zero lets call it $\displaystyle m > 0$ and test it in the ODE you get
$\displaystyle \frac{dN}{dt}\bigg_{t =-m}(a+bm)(-m)=-m(a+bm)$
This is negative so the derivative is always negative in this regrion
If you pick an N value $\displaystyle 0 \le m \le \frac{a}{b} $
$\displaystyle \frac{dN}{dt}\bigg_{t =m}\underbrace{(a-bm)}_{\text{This is positive}} $
So the product is positive this gives that the derivative is always positive here
If you check the last region you will find the derivative is always negative
The horizontal line $\displaystyle N=0$ is unstable all solutions move away from it.
The horizontal line $\displaystyle N =\frac{a}{b}$ is stable all solutions approch it as time increases.
Try to sketch the above it will help.
The critical values of the derivative are found by setting it equal to zero.
If you solve that equation you get. N=0 or N =a/b
If you draw a plot they will be horizontal lines intersecting the vertical axis.
This divides the plance into three parts. The derivative can only change from positive to negative when it crosses these horizontal lines.
What this means is, that is where the function changes from increasing to decreasing and from decreasing to increasing.
See attached.
The plot would look like this.