Question

How do I solve this Carrying Capacity problem for Calculus?

Answer 1

I don't remember how to do this problem.
http://tutorial.math.lamar.edu/Classes/DE/EquilibriumSolutions.aspx might help you though.

Answer 2

When you have a DE in this kind of a form:
\[\frac{ dy }{ dt }= ky(1-\frac{ y }{ L })\], you can pick out a couple bits of info and immediately skip to this as a solution:
\[y=\frac{ L }{ 1+be^{-kt} }\] L in these problems would be the carrying capacity, and is also the autonomous solution. All values above this will slope downwards towards L and all values below will slope upwards towards it. For your problem, you basically have L factored out. If you multiply the 1/1000 back in, you get a match of form, in which youcan then discern a lot of the info you need.

Answer 3

So I know that 1000 is the carrying capacity...For increasing and decreasing, do I just set it equal to zero and do interval testing?

Answer 4

And for concavity, do I just derive it again and do the same for the second derivative? Then what do I do for N=1500?

Answer 5

Well, youll just have to have two different graphs. In one case youll use N(0) = 200 to solve for b and then the same thing in the N(0) = 1500 case. Then youll do a couple of intervals to test to get a general idea of the slope of thegraph, but youll have two different things going on. As for the concavity yeah, 2nd derivative.

Answer 6

What is b?

Answer 7

So how does this original equation dN/dt change when N(0) = 1500 in comparison N(0) = 200?

Answer 8

This is also a good problem for ecology students.