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April 20, 2014

April 20, 2014

Posted by **Mel** on Friday, May 25, 2012 at 9:03am.

Question 3. In ecology, the logistic equation is often written in the form

dN divided by dt = rN(1-N/K)

where N = N(t) stands for the size of the population at time t, the constants r

and K stand for the intrinsic growth rate and the carrying capacity of the species,

respectively.

A pond on a sh farm has carrying capacity of 1000 sh, intrinsic growth rate

0.3 (when time is measured in months) and is originally stocked with 120 sh.

(a) Set up a logistic equation for the sh population N(t) in the pond, with t

measured in months.

(b) Find the size of the population when t = 10.

(c) Can the population reach 1000 at any future time?

even just recommendations on how to begin would be appreciated!

thanks :)

- Math -
**MathMate**, Friday, May 25, 2012 at 9:33amdN/dt = rN(1-N/K)

is a separable equation by rewriting it as:

dN/[rN(1-N/K)] = dt

=>

(K/r)[1/(N(K-N))]dN = dt

split into partial fractions

(1/r)[1/N + 1/(K-N)]dN = dt

Now integrate

(1/r)(ln(N)+ln(K-N))=t + C

Rewriting,

ln(N*(K-N))=r(t+C)

At this point, substitute r and K to find constant C at time t=0.

Hint: C is between 38 and 39.

After that, use the equation to solve for required unknowns.

- Math -
**Mel**, Friday, May 25, 2012 at 9:42amthank you so much!

- Math :) -
**MathMate**, Friday, May 25, 2012 at 9:52amYou're welcome!

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