Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 10 (b)
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Question 10 (b)
The differential equation
where , describes the rate of change of a population over time, subject to environmental constraints and the threat of extinction.
(b) Explain and justify in one or two sentences what happens to the population over time if the initial population satisfies .
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Make a partition of interval and analyze the sign of on each interval.
For example, if , then increases on the interval.
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As mentioned in the hint, we distinguish three cases:
Case 1: . In this case, . The population increases to approach .
Case 2: . In this case, . The population remains as the constant .
Case 3: . In this case, . The population decreases to approach .
In conclusion, the population approaches over time, that is,
Answer: over time.