Results 1 to 8 of 8

Math Help - Differential Equation of a Population Model

  1. #1
    Newbie
    Joined
    Oct 2012
    From
    Cambridge Ontario
    Posts
    12

    Differential Equation of a Population Model

    "An example of a differential equation that serves as a slightly more realistic model
    of population growth than the exponential growth model is the Lotka-Volterra DE,
    dy/dt= ky − ay^2, (1)
    Once again, y(t) denotes the population of a species at time t. The term −ay^2, which decreases the rate of growth, represents
    competition between population members for resources. In the special case a = 0, there is no competition, resulting in exponential growth.

    (a) Show that the constant population y(t) = k/a is a solution to Eq. (1).
    (b) Show that the following function, y(t) = y0/[(1 − by0)e^(−kt) + by0], t >= 0, (2)
    where b = a/k, is a solution to DE in (1). Also show that it satisfies the
    (positive) initial condition y(0) = y0 > 0"

    I've been working on this question for what seems like an eternity and I just can't fully understand how to prove. I need a hint, or something, please someone give me a push in the right direction.
    Last edited by Jtmartz; November 2nd 2012 at 03:13 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Jan 2008
    From
    UK
    Posts
    484
    Thanks
    66

    Re: Differential Equation of a Population Model

    To show that a function satisfies a differential equation you may simply substitute in the function and any necessary derivatives.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2012
    From
    Cambridge Ontario
    Posts
    12

    Re: Differential Equation of a Population Model

    Yes, but it has never specified whether a is a constant or not - so i don't know what the derivative of y(t)=k/a is!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Jan 2008
    From
    UK
    Posts
    484
    Thanks
    66

    Re: Differential Equation of a Population Model

    a is a constant.

    If it doesn't say so in your notes or book then check another source.

    By the way, writing e-kt instead of e^{-kt} is not a good idea. You could at least write e^(-kt) but better still, use LaTeX. LaTeX Help
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Oct 2012
    From
    Cambridge Ontario
    Posts
    12

    Re: Differential Equation of a Population Model

    i didnt even notice that, thank u - but at this point im really looking for steps. Ive exhausted my knowledge and my text book and sources online I am in need of a walkthorugh of this question. Would be greatly appreciated
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Jan 2008
    From
    UK
    Posts
    484
    Thanks
    66

    Re: Differential Equation of a Population Model

    Post your working.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Oct 2012
    From
    Cambridge Ontario
    Posts
    12

    Re: Differential Equation of a Population Model

    i believe i got a) down, but b) is troublemaker.

    i found dy/dt of y(t) to be ((e^(-kt))(1-by0))/[((e^(-kt))-by0(e^(-kt))+(by0))]^2

    and honestly everything I've tried to work it out has been a failure and it just doesnt work out.

    The closest method i have had was setting the derivative i found above equal to the derivative given, with the y values substituted with y(t)... which makes it massive and bulky. But, essentially, with a bit of algebra, ive found ((e^(-kt))(1-by0)) = ((e^(-kt))(ky0-a(y0^2))) but i feel like that is wrong
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member
    Joined
    Jan 2008
    From
    UK
    Posts
    484
    Thanks
    66

    Re: Differential Equation of a Population Model

    Your dy/dt is wrong. It's also almost unreadable. Use the quotient rule.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Model with First Order Differential Equation
    Posted in the Differential Equations Forum
    Replies: 7
    Last Post: May 31st 2011, 07:18 PM
  2. differential equation to find population
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: April 24th 2011, 07:59 AM
  3. Differential equation - population of a city
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: November 13th 2010, 07:16 PM
  4. Solve the Mathematical Model (Population Differantial Equation)
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: February 14th 2009, 12:23 AM
  5. [SOLVED] population growth differential equation problem
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: February 10th 2009, 06:24 AM

Search Tags


/mathhelpforum @mathhelpforum