Math

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Someone please help! Have spent ages stressing over this and still can't figure out how to do it! need asap.

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 -

    dN/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 -

    thank you so much!

  • Math :) -

    You're welcome!

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