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The factor $g(N) =\displaystyle r \left(1-\frac{N}{K}\right)$ in the logistic equation $\displaystyle \frac{dN}{dt} = \displaystyle r \left(1-\frac{N}{K}\right)N$ is a per capita growth rate. Smith observed that in cultures of the unicellular alga Daphnia magna g decreases at a nonlinear rate as $N$ increases. To account for this fact, smith suggested that the growth rate depends on the rate at which food is utilized: $g(N) = r \left(\displaystyle\frac{T-F}{T}\right)$ where $F$ is the rate of utilization when the population size is $N$, and $T$ is the maximal rate, when the population has reached a saturated level. He further assumed that $F=c_{1}N + c_{2}\displaystyle \frac{dN}{dt}$ where $c_{1}$ and $c_{2}$ are both positive , and as long as $\displaystyle \frac{dN}{dt}>0$.
(a) Explain this assumption for $F$.
(b) Show that the modified logistic equation is then $\displaystyle\frac{dN}{dt}=rN\left[\frac{K-N}{K+eN}\right]$ where $e =r \displaystyle\frac{ c_{2}}{c_{1}}$ and $K = \displaystyle\frac{T}{c_{1}}$.

(c) Sketch the expression in square brackets as a function of $N$.
(d) What would be the qualitative behavior of this population growth?

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Something is going on with some of the text at the bottom of the question. You might want to try and clean it up because it runs all the way off of the page. And also a note: \textit does not work in here, I believe the equivalent command is to use underscores like this: _enter text here_ –  night owl Jun 23 '11 at 7:11
    
Homework? Not a problem, but 1. please add the "homework" tag, and 2. please rephrase the question so it doesn't sound like you are ordering us to do something. –  Gerry Myerson Jun 23 '11 at 7:19
    
I see you are fairly new to this community and welcome, glad to have you here. As Gerry mentioned, you might want to not be so abrasive when posing questions here on the site. I'm sure everyone here is more than happy to assist you, but some questions (and not just yours) can read off as asking in "forceful way" and is not really favored upon in this community. Just as a heads up, try to ask your questions in a more passive style of tone rather than the following: (Do this, do that) style and you should be just fine. There are community wiki's on meta that goes over this in full :) –  night owl Jun 23 '11 at 8:48
    
while you're editing, do something about the part of (c) that refers to "square brackets." In the current incarnation of your question, there are no square brackets to be found anywhere. –  Gerry Myerson Jun 23 '11 at 12:38
    
In your first three sentences, we can just replace $K=T$ and $F=N$. Any assumption on $F$ needs to be justified for (a), but this is not a mathematical problem-it is input to the math. Please show your work going to (b)-it is a detailed question. –  Ross Millikan Jun 24 '11 at 4:58

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