The graphs to the right show two functions,
P ( c ) = P m c 2 k 2 + c 2 , D ( c ) = r c {\displaystyle P(c)=P_{m}{\frac {c^{2}}{k^{2}+c^{2}}},\quad D(c)=rc} where P m = 5 {\displaystyle P_{m}=5} , k = 10 {\displaystyle k=10} and r = 1 / 5 {\displaystyle r=1/5} . Suppose that c ( t ) {\displaystyle c(t)} is the concentration of a substance involved in a chemical reaction and satisfies the equation d c d t = P ( c ) − D ( c ) . {\displaystyle {\frac {dc}{dt}}=P(c)-D(c).}
Suppose that P m {\displaystyle P_{m}} and k {\displaystyle k} are as given above. For what value of r {\displaystyle r} does the differential equation for c ( t ) {\displaystyle c(t)} have only two steady states?
