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Math Help - Competition system

  1. #1
    MHF Contributor alexmahone's Avatar
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    Competition system

    \frac{dx}{dt}=a_1x-b_1x^2-c_1xy=x(a_1-b_1x-c_1y)

    \frac{dy}{dt}=a_2y-b_2y^2-c_2xy=y(a_2-b_2y-c_2x)

    a_1, a_2, b_1, b_2, c_1, c_2 are positive constants.

    Consider the nonzero critical point (x_E, y_E), which is the solution of the system a_1-b_1x-c_1y=0 and a_2-b_2x-c_2y=0.

    The stability of (x_E, y_E) depends on whether c_1c_2<b_1b_2 or c_1c_2>b_1b_2. Could someone please direct me to a proof of this or guide me through it if it is not too computationally tedious?
    Last edited by alexmahone; December 25th 2011 at 08:30 AM.
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  2. #2
    Grand Panjandrum
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    Re: Competition system

    Quote Originally Posted by alexmahone View Post
    \frac{dx}{dt}=a_1x-b_1x^2-c_1xy=x(a_1-b_1x-c_1y)

    \frac{dy}{dt}=a_2y-b_2y^2-c_2xy=y(a_2-b_2y-c_2x)

    a_1, a_2, b_1, b_2, c_1, c_2 are positive constants.

    Consider the nonzero critical point (x_E, y_E), which is the solution of the system a_1-b_1x-c_1y=0 and a_2-b_2x-c_2y=0.

    The stability of (x_E, y_E) depends on whether c_1c_2<b_1b_2 or c_1c_2>b_1b_2. Could someone please direct me to a proof of this or guide me through it if it is not too computationally tedious?
    Consider a series expansion about the critical point.

    CB
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  3. #3
    MHF Contributor alexmahone's Avatar
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    Re: Competition system

    Quote Originally Posted by CaptainBlack View Post
    Consider a series expansion about the critical point.

    CB
    But there is no series in my question!
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  4. #4
    Grand Panjandrum
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    Re: Competition system

    Quote Originally Posted by alexmahone View Post
    But there is no series in my question!
    Let (x_0,y_0) be the critical point. Put (x,y)=(x_0+\varepsilon, y_0+\delta). Now consider the solution to the system as a power series in \varepsilon and \delta (and truncate after the first non-zero (lowest order) non-constant terms)

    Then if: \partial_t\varepsilon  \ne 0 and \partial_t\delta  \ne 0 you have a stable critical point iff:

    \partial_t\varepsilon < 0

    and

    \partial_t\delta  < 0

    CB
    Last edited by CaptainBlack; December 26th 2011 at 12:51 AM.
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