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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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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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