FOR PART C, can you explain how they got the quadratic equation
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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FOR PART C, can you explain how they got the quadratic equation
Transcribed Image Text:Problem 4: The growth of a certain population (in millions) over time, P(t), is modelled by the
differential equation
P' = 6P - P² - h.
The parameter h in the equation is the harvesting rate (in millions per year).
(a) Determine if the population will grow, be steady, or decline in the following cases
i) h=5, P(0) = 1,
ii) h=5, P(0) = 5,
iii) h=10, P(0) = 5.
Solution: I need to compute the derivative in each case. i) P' = 6-1-5 = 0 so the
population stays constant. ii) P' = 30-25-5 = 0 so also here the population stays constant,
iii) P' = 30 - 25 - 10 = -5 so the population will decline.
(b) What should the harvest rate be if we want P = 4 to be an equilibrium population?
I need that 0 = 24 16 h this means that h = 8.
(c) For what values of the harvest rate, h, does a stable equilibrium population exist?
Equilibrium population is a solution of a quadratic equation 0 = 6P – P² - h. This equation
has a solution if the discriminant 36 4h is non-negative. This gives a condition h≤ 9. If
h = 9 then there is only one equilibrium population and the function 6P - P² - h does not
change sign around this point. So this is an unstable equilibrium. For h < 9 there are two
equilibria solutions, as we saw in the lecture the large one is stable in this case. So h < 9.
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