Red blood cells are formed from stem cells in the bone marro

Red blood cells are formed from stem cells in the bone marrow. The red blood cell density x satisfies the differential equation

dx/dt = (2x / (1 + x²)) - x

Find all equilibria, and determine their stability

Equilibria are determined by setting

dx/dt=0

So we get

2x/(1+x^2)-x=0

x(2/(1+x^2)-1)=0

x=0,2-1-x^2=0

x=0,x^2=1

x=0,-1,1

x=0

Linearizing about x=0 gives

dx/dt=2x/1-x=x

Hence unstable about x=0

x=1

dx/dt=2x/(1+x^2)-x=x(1-x^2)/(1+x^2)=x(1-x)(1+x)(1+x^2)

So linearizing about x=1 . let x=1+y

dy/dt=1*y*1*1=y

Hence unstable

3. ABout x=-1

x=-1+y

dy/dt=-1*2*y(1+1)=-4y

Hence stable