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Math Help - logistic model for population

  1. #1
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    logistic model for population

    Hello. This is a sample question for a test I have coming up on Monday.
    It being the weekend, my prof doesn't have office hours at this point.

    Consider the logistic model of population dynamics xn+1 = xn + Rxn(1 - xn/K),
    where R is the growth rate, K is the carrying capacity, and xn is the population size at time n. Given fixed K, find the values of the growth rate R for which the population becomes extinct.

    I would normally try to offer what I think I should do, but I am really clueless, so even a simple suggestion could put me in the right direction. Thanks for any help.
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  2. #2
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    would R be between -1 and 0?

    if its less than -1, youre getting into negative values
    and if you are are 0, it doesnt change
    if you are above 0, then it is increasing

    by having R between [-1,0]
    then you are approaching zero population

    is this the right thinking?
    if so, is there a better way to prove it rather than using trial examples?
    Last edited by chrisc; October 24th 2009 at 01:32 PM.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by chrisc View Post
    Hello. This is a sample question for a test I have coming up on Monday.
    It being the weekend, my prof doesn't have office hours at this point.

    Consider the logistic model of population dynamics xn+1 = xn + Rxn(1 - xn/K),
    where R is the growth rate, K is the carrying capacity, and xn is the population size at time n. Given fixed K, find the values of the growth rate R for which the population becomes extinct.

    I would normally try to offer what I think I should do, but I am really clueless, so even a simple suggestion could put me in the right direction. Thanks for any help.
    When x_n is small the term x_n/K is negligable and the difference equation becomes:

    x_{n+1}=(1+R)x_n

    Now if -1<R<0 the population is shrinking, but as we have a continuous model of population this never becomes zero, so we have no extinction (in reality the population will drop below 1 (or 2 ) and that would be extinction).

    If R\le -1 the population will go extinct in one step.

    CB
    Last edited by CaptainBlack; October 25th 2009 at 08:26 AM.
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post
    If R\le 0 the population will go extinct in one step.
    Do you mean if

    R\le -1

    because like you said, if its in the interval of [-1, 0] is will approach zero but never get there (and if it is 0, it will never change)

    Yes

    CB
    Last edited by CaptainBlack; October 25th 2009 at 08:25 AM.
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