Population growth to infinity

Question

The population growth rate of the flea is modeled by $\frac{dP}{dt}= k(200-P)$. $t$ is measured in weeks. If the population is 20 at time 0, what is the population as time goes to infinity?

Here's what I have so far. \begin{align} \frac{dP}{200-P}&= k\, dt, \\ \ln(200-P)&= kt=+c \\ 200-P= e^{kt+c}&= we^{kt} \\ \text{when t=0} \quad 200-20&=w \implies w=180 \\ 200-P &=180e^{kt}\\ P&=200-(180e^{kt}). \end{align} When $t$ goes to infinity is where I am getting stuck. Thanks.

It is very difficult to follow your work. Please see this Mathjax post for suggestions on how to format your question. — John Douma, Apr 07 '19 at 16:24

score 1 · Answer 1 · answered Apr 07 '19 at 16:26

1

You did not do the integration correctly as when you integrate wrt P it is should be -ln(200-P). Remember that the negative P on the bottom changes the sign when you integrate. Because of that you have something that is increasing to infinity rather than having the exponential component decrease to zero. Try again and watch your signs. ;)

answered Apr 07 '19 at 16:26

Chris Moorhead

187

So since k goes to zero than 200- 180e^0= 20. Hope I'm correct and thanks! – user943583 Apr 07 '19 at 17:05

Population growth to infinity

1 Answers1