- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
Logistic growth and phase lines practice question 6
A fish stock of tonnes grows logistically and is harvested at a constant 32 tonnes per year, so that the equation below holds. Find the steady states. If the stock is 150 tonnes when harvesting begins, what does it approach in the long run?
Give the long-run stock in tonnes.
Two steps: find where the stock stops changing, then decide which of those a stock of 150 tonnes heads for. What tells you which way it moves?
Expand and set the right side to zero: . Multiplying by gives a quadratic with whole-number roots.
Factor the quadratic to get the two steady states, then find the sign of at to see which way the stock moves.
A whole number.
Theory UBC Math 100 §13.1Key results
Worked solution
Strategy
Find the steady states as the zeros of the right-hand side, then use its sign between them to see which way a stock of 150 tonnes moves and where it stops.
Working
Multiply through by −1250 to clear the decimals.
Harvesting moves the steady states of the unfished stock, 0 and 500, to 100 and 400.
Negative below 100, positive between 100 and 400, negative above 400: 400 is stable and 100 is unstable.
The stock starts between the steady states, so it rises and approaches 400.
Answer
The answer is steady states at 100 and 400 tonnes; the stock approaches 400 tonnes.
Check
At 400 tonnes growth 0.4 · 400 · 0.2 = 32 a year replaces the catch, and the stock gains 10 tonnes a year at 350 but loses 14 at 450.
Where this goes wrong
Choosing the nearer steady state, 100. It is unstable: just above 100 the stock grows, so from 150 tonnes it climbs away from 100 to the stable steady state at 400.
Leans on
Solving a quadratic equation, from Functions, graphs and inverses
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