Results 1 to 2 of 2
Like Tree1Thanks
  • 1 Post By romsek

Math Help - Lotka-Volterra equations mistake

  1. #1
    Junior Member
    Joined
    Oct 2013
    From
    Moscow, Russia
    Posts
    41

    Lotka-Volterra equations mistake

    I have a problem with the Lotka-Volterra equations themselves. I believe that they might be wrong. Here is my reasoning - I would appreciate it if someone could find a flaw in it!


    The equations are generally of the form, as quoted from "A Modern Introduction to Differential Equations 2nd edition by Henry Ricardo":


    \frac{dx}{dt} = a_1x-a_2xy, \frac{dy}{dt}=-b_1y+b_2xy

    **My issue:** The $xy$ terms represent the number of possible interactions between two species. However, they only represent the number of possible *one-on-one* interactions between the two species. In order to account for *all* the possible interactions, such as $(x-1)$ predators acting on $2$ preys, shouldn't we arrive at \sum_{k=1}^x\sum_{j=1}^y {x \choose k}{y \choose j} = (2^x-1)(2^y-1) and thus \frac{dx}{dt} = a_1x-a_2(2^x-1)(2^y-1), \frac{dy}{dt}=-b_1y+b_2(2^x-1)(2^y-1)?


    Doesn't this make the number of interactions proportional not to the product of the number of predators and prey, but to their exponentiation?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,326
    Thanks
    893

    Re: Lotka-Volterra equations mistake

    Quote Originally Posted by abscissa View Post
    I have a problem with the Lotka-Volterra equations themselves. I believe that they might be wrong. Here is my reasoning - I would appreciate it if someone could find a flaw in it!


    The equations are generally of the form, as quoted from "A Modern Introduction to Differential Equations 2nd edition by Henry Ricardo":


    \frac{dx}{dt} = a_1x-a_2xy, \frac{dy}{dt}=-b_1y+b_2xy

    **My issue:** The $xy$ terms represent the number of possible interactions between two species. However, they only represent the number of possible *one-on-one* interactions between the two species. In order to account for *all* the possible interactions, such as $(x-1)$ predators acting on $2$ preys, shouldn't we arrive at \sum_{k=1}^x\sum_{j=1}^y {x \choose k}{y \choose j} = (2^x-1)(2^y-1) and thus \frac{dx}{dt} = a_1x-a_2(2^x-1)(2^y-1), \frac{dy}{dt}=-b_1y+b_2(2^x-1)(2^y-1)?


    Doesn't this make the number of interactions proportional not to the product of the number of predators and prey, but to their exponentiation?
    The LV system is a model, nothing more. In this case a model that is amenable to solution. The didn't have computers to simulate differential equations back in the 1920s so closed form solutions were important.

    Your system is also a model. Maybe it's more accurate maybe not. But it certainly seems to be much more difficult to solve if not impossible to find a closed form solution for.

    I suspect LV were after the simplest model they could use that captured the essence of the predator prey problem.
    Thanks from abscissa
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. lotka-volterra equations?? Euler
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: December 18th 2012, 08:49 PM
  2. Periodicity of Lotka-Volterra model
    Posted in the Advanced Applied Math Forum
    Replies: 4
    Last Post: March 7th 2011, 08:41 AM
  3. Lotka-Volterra Model
    Posted in the Advanced Applied Math Forum
    Replies: 2
    Last Post: November 27th 2008, 04:17 AM
  4. Lotka-Volterra Equations
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 25th 2008, 05:41 AM
  5. Lotka-Volterra equations
    Posted in the Calculus Forum
    Replies: 5
    Last Post: June 9th 2008, 04:03 AM

Search Tags


/mathhelpforum @mathhelpforum