Math 260 Lab #6 Biochemical switch

Save your work regularly--you never know when Mathematica may crash.  

Email your completed file to tleise@amherst.edu by Monday 4pm.

Consider the differential equation

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

the dimensionless form of a differential equation describing a biochemical switch controlled by expression of a gene.

Assume r>0 and s≥0.

In class we derived the bifurcation diagrams for r (with s fixed for different cases).  
Your job is to derive the bifurcation diagrams for s (with r fixed for different cases), which we sketched but didn’t justify.

Exercise. Plot the bifurcation diagram (with stability marked) for the biochemical switch with respect to the parameter s, for the following cases of fixed r:

(a) 0 < r < (1/2) (choose a representative value of r)
(b) r = (1/2)
(c) (1/2) < r < ((3(Sqrt[3]))/8) (choose a representative value of r)
(d) r = ((3(Sqrt[3]))/8)
(e) r > ((3(Sqrt[3]))/8) (choose a representative value of r)