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Does this limit exist?

  1. Aug 12, 2009 #1
    1. The problem statement, all variables and given/known data

    A population develops according to logistic equation:
    [tex]\frac{dy}{dt} = y(0.5-0.01y)[/tex]

    Determine the following:
    [tex]\lim_{t\rightarrow\infty} y(t)[/tex]


    3. The attempt at a solution

    By finding an equilibrium solution to the differential equation, we see that dy/dt = 0 when y = 0 or y = 50.

    But does the limit exist? What if y = 0 initially?
     
  2. jcsd
  3. Aug 12, 2009 #2

    Cyosis

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    Homework Helper

    You should try to solve the differential equation first.
     
  4. Aug 12, 2009 #3
    But do you have to do the differential to determine the limit? Because we can see that if y=0 and t goes to infinity, y will still equal 0. and if y does not equal 0 initially, then y will go to 50.
     
  5. Aug 12, 2009 #4

    Cyosis

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    You have picked two fixed values for y which indeed satisfy the differential equation, however these values aren't the only solutions. The more interesting solutions are the ones that still depend on t.
     
  6. Aug 12, 2009 #5
    I'm sorry, I don't understand.

    I solved the differential equation:

    y(t) = 50 / (1+Ae^(-0.5t)

    where A = (50-y(0)) / y(0)

    so isn't it still dependent on the value of y(0)?
     
    Last edited: Aug 12, 2009
  7. Aug 12, 2009 #6
    OH. so now if we take the limit, as t approaches infinity, then A will go to 0, no matter what. is that correct?
     
  8. Aug 12, 2009 #7
    Yea, more correctly, Ae^(-0.5t) goes to zero
     
  9. Aug 12, 2009 #8
    ah okay, thanks.

    i still don't get the concept though. how can the limit of the equation be 50 if the initial population is 0?
     
  10. Aug 12, 2009 #9
    Since the equation is modeling the development of a population, that would imply that y(0) cannot be zero; otherwise you have no population to begin with.
     
  11. Aug 12, 2009 #10
    haha, i suppose. thanks very much cyosis and fightfish.
     
  12. Aug 12, 2009 #11

    Mark44

    Staff: Mentor

    No, it's not. As fightfish said, Ae-.5t goes to 0 as t approaches infinity. This, however, does not imply that A is 0 or is approaching zero. A is a constant in the problem.
     
  13. Aug 12, 2009 #12
    sorry, i'll be more specific next time...
     
  14. Aug 12, 2009 #13
    >> how can the limit of the equation be 50 if the initial population is 0?

    You need to object to nietsche, who said the solution is:
    y(t) = 50 / (1+Ae^(-0.5t)
    that is NOT the solution if y(0)=0...
     
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