Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 10 (c)

MATH 180 December 2017

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Question 10 (c)

The differential equation $${\frac {dP}{dt}}=-P\left(1-{\frac {P}{T}}\right)\left(1-{\frac {P}{K}}\right),\quad P\geq 0,$$ where ${\textstyle 0<T<K}$, describes the rate of change of a population over time, subject to environmental constraints and the threat of extinction. (c) ${\textstyle T}$ is sometimes referred to as the population’s extinction threshold. Explain and justify in one or two sentences what happens to the population over time if the initial population ${\textstyle P_{0}}$ satisfies ${\textstyle P_{0}<T}$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Is the rate of change positive or negative?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. If ${\textstyle P_{0}<T}$, then ${\textstyle {\dfrac {dP}{dt}}<0}$. The population keeps decreasing and approaches ${\textstyle 0}$. Alternatively, we may write $$\lim \limits _{t\to \infty }P(t)=0.$$ Answer: ${\textstyle {\color {blue}{\mbox{the population tends to }}0}}$ over time.