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The simplest guess is using the Vehulst model: $r = r_0( 1 - N/K)$. Then the dynamic equation is $\frac{dN}{dt} = r_0N(1 - \frac{N}{K})$. From here, we can calculate the maximum point to be at $\frac {Kr_0}{4}$.

How have they got all this. Firstly, if our equation is $r$ in terms of $N$, how have they then differentiated and got it to be $\frac{dN}{dt}$.

Also, how do you calculate the maximum point from here? I would've assumed that you needed to differentiate a second time and equate this to $0$ and solve, but differentiating a second time would be mean we would need to differentiate with respect to $t$ and so the RHS would just be $0$, wouldn't it?

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Who are "they"? Where have they written this? –  Did Jan 31 '13 at 17:48
    
@Did It's in my lecture notes. I tend to write the stuff down quickly and then go through them again to understand them after as I don't pay attention to what I'm writing in the lecture so that's what I'm doing now. –  Kaish Jan 31 '13 at 17:53
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Obviously your notes refer to the $N$ point such that $dN/dt$ is maximal. And the dynamic equation is the refinement of the exponential growth $dN/dt=rN$ where $r$ depends on $N$ as the first equation indicates. –  Did Jan 31 '13 at 17:56

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Your notes refer to the $N$ point such that $dN/dt$ is maximal. This equation is a refinement of the exponential growth $dN/dt=rN$ where $r$ depends on $N$ as the first equation indicates.

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