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Question: Lotka-Volterra system

  1. Oct 4, 2006 #1
    Hi

    I need help intepreting the following.

    Given Lotka-Volterra model system

    [tex]\begin{array}{cc} x'_1 = (a-bx_2)x_1 \\ x'_2 = (cx_1 -d) x_2\end{array}[/tex]

    Look at the system on the open 1.Quadrant K; where a,b,c,d are all positive constants.

    Show that the system is integratable, which supposedly implies that there exist a [tex]C^1[/tex]-function [tex]F:U \rightarrow \mathbb{R}[/tex] where [tex]U \subseteq K[/tex] is open, and close in K.

    According to my professor "close" implies that for every point in K, there exist a sequence of socalled "limitpoints", who's elements belongs to K. Also as a consequence of "close" [tex]\nabla F \neq 0[/tex] for all [tex]x \in U[/tex], and F is constant on all trajectories of the system.

    What is my first step here? Do I prove that there exist a solution for the system only in K?

    Sincerley Yours
    Mathboy20
     
    Last edited: Oct 4, 2006
  2. jcsd
  3. Oct 19, 2006 #2
    Volterra

    To begin with you need to find a function of two variables (x_1, x_2) which is a constant of motion. i.e. its time derivative is zero. Then you worry where it is defined.
     
  4. Oct 19, 2006 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You might also want to look up the definition of "integrable" since that is what you are asked to show!
     
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