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Math Help - rabbit population

  1. #1
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    rabbit population

    dN/dt = cN^1.01

    (i) show that, if there are N(0) = 2 rabbits initially, and 16 rabbits after 3 months, then the population N(t) become unbounded at a finite time t
    (ii) does the population become unbounded far any initial condition N(0) > 0, and any c>0, expain your answer

    i solved the DE and got

    N(t) = [(0.68t - 99.31) / -100]^(-100)
    how do i show that the population becomes unbounded at a finite time t

    can someone give me a detailed explanation of both parts??
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  2. #2
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    Quote Originally Posted by razorfever View Post
    dN/dt = cN^1.01

    (i) show that, if there are N(0) = 2 rabbits initially, and 16 rabbits after 3 months, then the population N(t) become unbounded at a finite time t
    (ii) does the population become unbounded far any initial condition N(0) > 0, and any c>0, expain your answer

    i solved the DE and got

    N(t) = [(0.68t - 99.31) / -100]^(-100)
    how do i show that the population becomes unbounded at a finite time t

    can someone give me a detailed explanation of both parts??
    Another way to write your solution is

    N(t) = \left(\frac{100}{D - ct}\right)^{100}

    where you have determined D \approx 99.31 and c \approx 0.68.

    Now you can see from this that when the denominator is zero then N become unbounded (division by zero) so this will happen at the critical time

    t_{critical} = \frac{D}{c}

    For part one the critical time is

    t_{critical} \approx \frac{99.31}{0.68} \approx 146

    For part 2, if N(0) > 0 then

    D = \frac{100}{N(0)^{0.01}}

    which is positive and if c is positive then also t_{critical} will be positive.


    Hope this helps
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