Find the fixed point of the differential equation Check for

Find the fixed point of the differential equation. Check for the stability of each fixed points.

Solution

1. Fixed points is set of \'points where

x\'=x

So,

x=x-rx(1-x)

Giving, x=0,1

These are the two fixed points

2)

To study stability we need to ilnearize the fixed points

LInearizing about x=0

x\'=x-rx=x(1-r)

Hence for r>=1 we have an stable fixed point as x\'<0

For r<1,x\'>0 hence a unstabel fixed point

Now linearizing about :x=1

Let, x=1+X

(1+X)\'=1+X-r(1+X)(1-1-X)

X\'=1+X+rX(1+X)=1+X+rX>0 for all r>0

Hence unstable fixed point