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

MATH 180 December 2017

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Question 10 (a)

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. (a) Sketch a large graph of ${\frac {dP}{dt}}$ as a function of ${\textstyle P}$. You do not have to justify your sketch using derivatives.

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
We have a cubic polynomial. What are the intercepts?

Hint 2
For very large ${\textstyle P}$ (larger than ${\textstyle K}$), is ${\textstyle {\dfrac {dP}{dt}}}$ positive or negative?

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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. Let us work out the important features of the graph as indicated by the hints. From the factorized right hand side of the equation, we see that ${\textstyle {\dfrac {dP}{dt}}=0}$ exactly when ${\textstyle P=0}$, ${\textstyle 1-{\dfrac {P}{T}}=0}$ or ${\textstyle 1-{\dfrac {P}{K}}=0}$, that is, when ${\textstyle P=0}$, ${\textstyle T}$ or ${\textstyle K}$. Observe also that for ${\textstyle P>K}$, we have $P>0$, $1-{\dfrac {P}{T}}<0$, and $1-{\dfrac {P}{K}}<0$. This implies that ${\textstyle {\dfrac {dP}{dt}}<0}$ for all large ${\textstyle P}$. We gather the above information in the following graph. Q10a solution figure Answer: Please see the figure.