- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Logistic growth and phase lines practice question 2
The differential equation below has steady states and . Which statement describes their stability?
A steady state is stable when solutions that start near it move towards it. What does the sign of on either side of each steady state tell you?
Take one test value in each interval, , and , and find the sign of there. Positive means increases: the arrow on the state-space diagram points right.
Theory UBC Math 100 §13.1Key results
Worked solution
Strategy
The sign of the right-hand side between the steady states says which way y moves there. A steady state is stable when the arrows on both sides point at it.
Working
The sign can change only at a steady state, so one test value per interval is enough.
Arrows point right where y increases and left where it decreases: both point at 1, and both point away from 4.
Answer
The answer is y = 1 is stable and y = 4 is unstable.
Check
The derivative of (y − 1)(y − 4) with respect to y is 2y − 5: −3 at y = 1, stable, and 3 at y = 4, unstable.
Where this goes wrong
Assuming the larger steady state is stable, as it is for the logistic equation. Here dy/dt > 0 above 4, so a solution starting just above 4 grows away from it: read the arrows from the sign of the right-hand side every time.
Leans on
Signs of a factored function, from Curve sketching
Saved in this browser only.