# Please explain about logistic equation

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• Oct 5th 2012, 11:50 AM
angelme
Please explain about logistic equation
Please explain about logistic equation

dN/dt = (a-bN)N where a and b are positive number

if W(0)=N0

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
• Oct 5th 2012, 01:13 PM
TheEmptySet
Re: Please explain about logistic equation
If you set the derviative equal to zero you will get the constant solutions to the equation. They are

$N=\frac{a}{b}$ and $N=0$

This divides the plane into three parts.

If you pick an N value less than zero lets call it $m > 0$ and test it in the ODE you get

$\frac{dN}{dt}\bigg_{t =-m}(a+bm)(-m)=-m(a+bm)$

This is negative so the derivative is always negative in this regrion

If you pick an N value $0 \le m \le \frac{a}{b}$

$\frac{dN}{dt}\bigg_{t =m}\underbrace{(a-bm)}_{\text{This is positive}}$

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 $N=0$ is unstable all solutions move away from it.

The horizontal line $N =\frac{a}{b}$ is stable all solutions approch it as time increases.

Try to sketch the above it will help.
• Oct 5th 2012, 10:32 PM
angelme
Re: Please explain about logistic equation
I still don't get it....why is it divided into 3 parts and how you get (dN/dt)t
Could you please explain more?

Thank you
• Oct 6th 2012, 05:24 AM
TheEmptySet
Re: Please explain about logistic equation
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.

Attachment 25077
The plot would look like this.