I have the equation:

\(P(t)=\frac{K}{9e^{-rt}+1}\)

this equation was derived from solving the logistic growth equation:

\(\frac{dP}{dt}=rP\big(1-\frac{P}{K}\big)\)

where,

r is the rate of growth = 1.13

K is the maximum population = 10,000

I've been asked to change the growth rate to:

\(r= 0.83 + \frac{0.3}{1+0.3t}\)

My question is,

Do I have to solve the entire logistic growth equation again, that is,

or can I just substituted r into the derived equation and rearrage, that is,

I wanted to get some clarification as I've tried doing both process with a simpler equation, however the answer I get quite is complex and I'm not sure If I've done it correctly.

In my opinion I don't think it matters if I substitute the new r straight into P(t) ... but i'm not overly confident about that

Thank you.

vest4R
Mar 30, 2018

#1**+2 **

I don't see how you can avoid doing the integration again, since r is now a function of t. The result I get when so doing is as follows:

Alan
Mar 30, 2018

#2**+1 **

thanks for your reply Alan. I was thinking that might be the case.

I see your image has been blocked. Would you be able to unblock it please?

vest4R
Mar 30, 2018