Solutions to Real Quiz 9 of Dr. Z.’s Dynamical Models in Biology class

Name: Dr. Z.

1. Find all the equilibrium points and stable equilibrium points of the following one-dimensional dynmaical system

\frac{dx}{dt} = −(x + 2)(x + 1)(x − 1)(x − 2)(x − 5) .

Sol. to 1: The equilibrium points are the solutions of

−(x + 2)(x + 1)(x − 1)(x − 2)(x − 5) = 0 ,

where are {−2, −1, 1, 2, 5}. The derivative of the function on the right is (by the product rule of calculus) −((x+2)′(x+1)(x−1)(x−2)(x−5)+(x+2)(x+1)′(x−1)(x−2)(x−5)+(x+2)(x+1)(x−1)′(x−2)(x−5)+(x+2)(x+1)(x−1)(x−2)′(x−5)+(x+2)(x+1)(x−1)(x−2)(x−5)′) f′(x) = −((x+1)(x−1)(x−2)(x−5)+(x+2)(x−1)(x−2)(x−5)+(x+2)(x+1)(x−2)(x−5)+(x+2)(x+1)(x−1)(x−5)+(x+2)(x+1)(x−1)(x−2)) When x = −2 we have

f′(−2) = −(−2 + 1)(−2 − 1)(−2 − 2)(−2 − 5) = −(−1)(−3)(−4)(−7) > 0 ,

since this is positive x = −2 is stable. When x = −1 we have

f′(−1) = −((−1 + 2)(−1 − 1)(−1 − 2)(−1 − 5)) = (−1)(1)(−2)(−6) < 0

since this is postive x = −1 is ustable. Similarly, When x = 1 it is stable, when x = 2 it is unstable when x = 5 it is stable. Ans: The stable equilibrium points are {−2, 1, 5}.