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The information that I am given: p(12)=95,75 ; P(14)=98 ; and the carrying capacity K = 100. Question : What is r, as defined in:

deltaP/P = -r/K * P + r

The answer in the book is 0,325

The closest I have come is 0.388

Please let me know how to solve this problem. Thanks.

This is my working:

My attempt at answer [My answer2

caw
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1 Answers1

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Probably, the best thing to do would be to solve the differential equation and then plug in values of $t$ and $P$.

I believe you forgot a $dt$ in the denominator so I'll add that in.

$\displaystyle \frac{dP}{P\cdot dt} = -\displaystyle \frac{r}{K}(P)+r$

can be rewritten as $\displaystyle \frac{dP}{dt} = rP\left(1-\frac{P}{K}\right)$.

This is the more conventional forme that suggests that growth slows down as population (P) approaches carrying capacity (K).

Anyways, the next step is to solve the equation.

enter image description here

This is a clean solution, taken from https://sites.math.northwestern.edu/~mlerma/courses/math214-2-04f/notes/c2-logist.pdf, and in this example, $r$ is replaced with $k$. They represent the growth factor.

Now, all you have to do is plug in values of $t$ and $P$ and you can solve for $r$. Good luck!

Saketh Malyala
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  • This is exactly what I did. I used P(12) in the place of P(0) and P(14) in the place of P, and t=2 for 2 periods of time. It gave me k = 0.388 but the book says I should get 0.325. – caw Feb 17 '19 at 22:47
  • I am really pleased that I appear to be on the right track but can you suggest where I made my mistake please – caw Feb 17 '19 at 22:50
  • oh no!! that's where you went wrong – Saketh Malyala Feb 17 '19 at 23:02
  • you should be using P(12) and t=12, and P(14) and t=14 – Saketh Malyala Feb 17 '19 at 23:02
  • P(12) is not the beginning, rather it is 12 periods of time after the start, so t=12 necessarily – Saketh Malyala Feb 17 '19 at 23:03
  • I have put my revised attempt at an answer into the question post above and used p(12) and p(14) . I set up a simultaneous and this eliminates the need to know p(0). but am still getting 0.388 as an answer. Your comment would be muchly appreciated. – caw Feb 18 '19 at 06:12
  • How did they know "k = 0.031476" in the northwestern example? – Harry Iguana Nov 05 '23 at 12:21
  • This value is supplied within the problem. I don't think they already knew that. It is a parameter that is part of the differential equation @HarryIguana – Saketh Malyala Nov 06 '23 at 09:05