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Math Help - {x'=x(1-x-y), y'=y(1.5-x-y)} System of Differential equations

  1. #1
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    {x'=x(1-x-y), y'=y(1.5-x-y)} System of Differential equations

    {x'=x(1-x-y), y'=y(1.5-x-y)} System of Differential equations-math2420_hw10_pr4.jpg

    I'm trying to evaluate this system, by finding and categorizing 4 equilibria points as stable, asymptotically stable, or unstable (nodes, saddles, foci, or centers). Then I have to determine if this describes a species coexistence or competitive exclusion scenario.

    I have found three of the 4 equil. points, and evaluated them in a Jacobi matrix. The fourth (positive equil. point) seems to not exist....as I solve the system to find that -0.5=0, which is obviously false.

    I'm afraid I may have made a mistake somewhere. Can I get a push in the right direction?
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  2. #2
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    Re: {x'=x(1-x-y), y'=y(1.5-x-y)} System of Differential equations

    I only get 3 roots for your system. What makes you think there is a fourth?
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  3. #3
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    Re: {x'=x(1-x-y), y'=y(1.5-x-y)} System of Differential equations

    Hi !
    The general solution of the EDOs system is shown in attachment.
    Attached Thumbnails Attached Thumbnails {x'=x(1-x-y), y'=y(1.5-x-y)} System of Differential equations-edossystem.jpg  
    Thanks from shane18
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  4. #4
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    Rational dynamical system with nonnegative paramaters

    let $A$ be a rational system of the form :$\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ z_{n+1}=\frac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3 }y_{n}+\lambda_{3}z_{n}}{A_{3}+B_{3}x_{n}+c_{3}y_{ n}+D_{3}z_{n}}{}\end{cases}$
    $n=0,1,....$

    with nonnegative paramaters and with nonnegative initial conditions such that the denominators are always positive.

    how i could show that :for every positive solution of the difference equation:
    $z_{n+1}= \frac{\alpha}{1+\prod_{i=0}^{k}z_{n-i}},n=0,1,.... $ has a finit limit ?



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