Stuck on stability of critical points in logistic differential equations. It has
ID: 2855990 • Letter: S
Question
Stuck on stability of critical points in logistic differential equations. It has been awhile since i have taken calculus, so i am a bit over my head here. i have a problem which goes:
dx/dt = 3-x
and x critical = 3
i am asked to check the sign of dx/dt around this critical point in order to determine the stability of this critical point.
3-x<0 if x>3 and 3-x>0 if x<3
how do i tell if the sign of dx/dt around x=3 moves towards or away from the line x=3? this will apparently tell me whether the critical point is stable or unstable and i am supposed to be able to determine this right away (without a calculator) by what i have already stated, but i can't. what am i not seeing here?
Explanation / Answer
Here is the rule relating that...
c is the critical point...
(i) If f(x) < 0 on the left of c, and f(x) > 0 on the right of c, then the equilibrium solution x = c is unstable. (Visually, the arrows on the two sides are moving away from c.)
(ii) If f(x) > 0 on the left of c, and f(x) < 0 on the right of c, then the equilibrium solution x = c is asymptotically stable. (Visually, the arrows on the two sides are moving toward c.)
In this case, to the left of 3, i.e when x < 3, 3 - x was positive and to the right, it was negative
So, (ii) kicks in
And the point is ASYMPTOTICALLY STABLE ----> ANSWER