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Differential equations and related rates — Question 218

Original QCE Vault practice · 10 marks

Q218 · Practice questionTechnology-freeComplex unfamiliar10 marks

QUESTION 218 (10 marks)

A managed algae population satisfies N′=N(1−N/100)−16N^{\prime}=N(1-N/100)-16 for N≥0N\ge0. The model stops at extinction N=0. The graph shows growth rate against population.
Harvested population growth parabola with zeros 20 and 80, maximum at N=50.
a)
Determine the equilibrium populations.
[2 marks]
b)
Classify both equilibria using the growth-rate signs.
[3 marks]
c)
Without solving, describe outcomes from initial populations 10 and 50.
[2 marks]
d)
The harvest is changed from 16 to a constant H, so N′=N(1−N/100)−HN^{\prime}=N(1-N/100)-H. Determine the greatest H allowing a positive equilibrium. At that harvest, assess the claim that every positive initial population approaches 50.
[3 marks]
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